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A Comparison of Three Rounding Algorithms for IEEE Floating-Point Multiplication.

AbstractA new IEEE compliant floating-point rounding algorithm for computing the rounded product from a carry-save representation of the product is presented. The new rounding algorithm is compared with the rounding algorithms of Yu and Zyner [26] and of Quach et al. [17]. For each rounding algorithm, a logical description and a block diagram is given, the correctness is proven, and the latency is analyzed. We conclude that the new rounding algorithm is the fastest rounding algorithm, provided that an injection (which depends only on the rounding mode and the sign) can be added in during the reduction of the partial products into a carry-save encoded digit string. In double precision format, the latency of the new rounding algorithm is $12$ logic levels compared to $14$ logic levels in the algorithm of Quach et al. and $16$ logic levels in the algorithm of Yu and Zyner.

Introduction Every modern microprocessor includes a oating-point multiplier that complies with the IEEE 754 Standard [13]. The latency of the FP multiplier is critical to the oating-point performance since a large portion of the FP instructions consists of FP multiplications. For example, Oberman reports that FP multiplications account for 37% percent of the FP instructions in benchmark applications [17]. A lot of research has been devoted to optimizing the latency of adding the partial products to produce the product, e.g. [1, 2, 6, 9, 15, 16, 18, 19, 20, 21, 26, 28, 29, 30]. More recently, work on rounding the product according to the IEEE 754 Standard has been published [4, 7, 10, 22, 23, 24, 25, 31, 33, 34]. Assuming that the multiplier outputs a carry-save encoded digit string representing the exact product, the following natural question arises: What is the fastest method to compute the rounded product given the exact product represented by carry-save encoded digit string? We consider and compare three rounding algorithms: (a) the algorithm of Quach et al. [23], which we denote by the QTF algorithm; (b) the algorithm of Yu and Zyner [31], which we denote by the YZ algorithm; and (c) a new algorithm that is based on injection based rounding [10], which we denote by the ES algorithm. We provide block diagrams of these rounding algorithms, optimized for speed. We measure the latency of the algorithms in logic levels to enable technology independent comparisons. The main building blocks of these algorithms are similar, and consist of a compound adder and the computation of a sticky and carry bits. Thus, the costs of the three algorithms are similar, and the interesting question is nding the fastest algorithm. We focus on double precision multiplication in which each signicand is represented by 53 bits. The algorithms assume that the signicands are normalized, namely, in the range [1; 2), and therefore, their product is in the range [1; 4). We do not consider the cases that deal with denormal or special values since supporting denormal values can be obtained by using an extended exponent range [14, 25, 32] and the computation on special values can be done in parallel [12]. The three algorithms share the following techniques: 1. The product represented by a carry-save encoded digit string of 106 digits in the case of double precision is partitioned into a lower part and an upper part. The upper part is added by a compound adder that computes the binary representations of sum and sum+ ulp, where ulp denotes a unit in the last position, and sum denotes the sum of the upper part. A carry-bit, a round-bit, and a sticky-bit is computed from the lower part. 2. The rounding decision is computed in two paths: the non-over ow path works under the assumption that the exact product is in the range [1; 2), and over ow path works under the assumption that the product is in the range [2; 4). Although the sum of the upper part, denoted by sum, does not equal the exact product, the most signicant bit of sum controls the selection between these two paths. The main dierences between the three rounding algorithms are outlined, as follows. 1. The rounding decision. The QTF and ES algorithms simplify the rounding decision by an early addition of a value (this value is called the prediction in the QTF algorithm and the injection in the ES algorithm). In the QTF algorithm, the prediction depends on the rounding mode and on the carry-save digit positioned 53 digits to the right of the radix point. In the ES algorithm, the injection depends only on the rounding mode and we assume that it is added in with the partial products, and thus the product already includes the injection. The rounding decision in the YZ algorithm is based on customary rounding tables. 2. The position in which the carry-save encoded product is partitioned into a lower part and an upper part diers in the three algorithms. In the YZ algorithm the lower and upper parts are separated by a \buer" of three carry-save digits in positions [51 : 53], where the position of a digit denotes how many digits it is to the right of the radix point. In the other two algorithms the upper part consists of positions and the lower part consists of position [53 : 104]. The latencies of the proposed designs that implement these algorithms in terms of logic levels are the following: The latency of the ES algorithm is 12 logic levels, the latency of the QTF algorithm is 14 logic levels, and the latency of the YZ algorithm is 16. Note that we modied and adapted the QTF and YZ algorithms for minimum latency. Supporting all four rounding modes of the IEEE 754 Standard is an error prone task. We therefore provide correctness proofs of all three algorithms which formalize and clarify the tricky aspects. From this point of view, the YZ algorithm is easiest to prove and the QTF algorithm is the most intricate (especially the rounding decision logic). The paper is organized as follows. In Section 2, preliminary issues are described, such as: notation, conventions we use regarding IEEE rounding, and the general setting. In Section 3, a straightforward rounding algorithm is reviewed. This algorithm is described to provide an outline the task of rounding after the exact product is computed. It does not attempt to parallelize the task of rounding, and therefore, has a long latency. In Sections 4-6, each rounding algorithm is described, proven, and analyzed. In Section 7, we discuss how the latencies of the algorithms increase as the precision is increased. In Section 8, a summary and conclusion is given. Due to space limitations some of the sections are omitted and can be found in the full version [11]. Preliminaries Notation Let x i x binary string. By we denote the binary string x z1 x z 1 +1 x z 2 . We also sometimes refer to x i as x[i]. Since we deal with fractions, we index binary encoded bit strings by x so that x i is associated with the weight 2 i . The value encoded by x[z denoted by jx[z IEEE rounding The IEEE-754-1985 Standard denes four rounding modes: round toward 0, round toward +1, round toward 1, and round to nearest (even). Based on the sign of the number the rounding modes round toward +1 and round toward 1 can be reduced to the rounding modes RZ (round to zero) and RI (round to innity) [23]. Thus leaving only three rounding modes: RI, RZ, and RNE (round to nearest - even). Furthermore, Quach et al. [23] suggested to implement RNE by round to nearest (up), denoted by RNU. The rounding mode RNU is dened as follows. If x is between two successive representable r RNU The reason that RNE can be implemented by RNU is that r RNU (x) 6= r RNE (x) and the least signicant bit (LSB) of the binary encoding of y 2 is 1. Therefore, obtaining r RNE (x) from r RNU (x) can be accomplished by \pulling down" the LSB, when For the sake of clarity, we dene rounding to zero (RZ) of signicands in the range [1; 4) in double precision. Note, that this denition excludes the post-normalization shift that takes place when the number is in the binade [2; 4). Denition 1 Let x 2 [1; 4), then r RZ (x) is dened by r RZ where x div 2 ' is the integer q that satises: General setting In this paper we consider a double precision multiplier. We assume that the signicands are prenormalized, namely, that the values of the two signicands are in the range [1; 2) and that each signicand is represented by a binary string with bits in positions [0 : 52]. The exact product of the two signicands is in the range [1; 4) and is encoded by a binary string with bits in positions (Note that the weight of the bit in position [ 1] is 2). For the sake of simplicity, we ignore the exponent and sign-bit paths. Floating point multipliers perform the computation in two phases. In the rst phase, an addition tree reduces the partial products to a carry-save encoded digit string that represents the exact product. In the second phase, a binary string representing the rounded product is computed from the carry-save encoded string. This paper discusses implementations of the second phase. 3 Naive IEEE rounding In this section we review a simple but slow IEEE compliant algorithm for rounding after multiplication [5]. 3.1 Description The input consists of two binary strings sum and carry each having 106 bits which are indexed from 1 to 104. The sum of the binary numbers represented by sum and carry equals the exact product exact 2 [1; 4). Rounding is computed as follows (the computation of the exponent string is omitted): 1. Reduce the rounding mode to one of three rounding modes based one the sign of the product. 2. 2:1-compression. The sum and carry strings are added to obtain a single binary string namely Note that since the exact product is in the range [1; 4), the the most signicant bit of X is in position [ 1]. 3. Normalization. If jXj 2, then jX jXj. This is implemented by a conditional shift by at most one position to the right. Note that X 0 is indexed from 0 to 4. Compute sticky. The sticky-bit equals 5. Compute rounding decision. The rounding decision rd 2 f0; 1g is based on the rounding mode, the bits: and the sticky-bit. Note that the rounding mode at this stage already incorporates the sign. 6. Increment. Compute be the binary string that represents the sum. 7. Post-normalize. If jY The signicand string of the rounded product is given by Y 0 . 3.2 Delay analysis The latency of steps 1, 3, and 5 of the naive rounding procedure is at least logarithmic in the length of the binary strings sum and carry. The other steps require only constant delay. If every pipeline stage can accommodate at most one logarithmic depth circuit, then an implementation of the naive rounding procedure requires at least 3 pipeline stages. 4 The ES rounding algorithm In this section we review injection based rounding [10], and present an implementation for double precision that requires (under assumptions specied in Sec. 4.5) only 12 logic levels. 4.1 Injection based rounding Rounding by injection reduces the rounding modes RI and RNU to RZ [10]. The reduction is based on adding an injection that depends only on the rounding mode, as follows: The eect of adding injection is summarized in the following equation: where mode 2 fRZ;RNU;RIg. Figure 1 depicts the reduction of RNU and RI to RZ by injection assuming that the number to be rounded is in the range [1; 2). If the exact product, denoted by exact, is in the range [2; 4), then the injection must be xed in order to make the reduction to RZ correct. The correction amount, denoted by inj correct, is dened by Therefore, if X is in the range [2; 4) the eect of adding the injection and the correction amount is summarized in the following equation: where mode 2 fRZ;RNU;RIg. Our assumption is that injection is added in the multiplier adder array, and therefore This completes the description of injection based rounding for numbers in the range [1; 4). 4.2 The rounding algorithm In this section we present the new ES algorithm for rounding in oating-point multiplication that is based on injection based rounding. Figure 2 depicts a block diagram of the ES rounding algorithm. The rounding algorithm works under the assumption that the sum and carry-strings already include the injection (but not the injection correction) and proceeds as follows: 1. The sum and carry-strings are divided into a high part and a low part. The high part consists of positions and the low part consists of positions [53 : 104]. 2. The low part is input to the box that computes the carry, round and sticky-bits, dened as follows: where 52 104 is the binary string that satises: 3. The higher part is input to a line of Half Adders and produces the output (X sum and Note that the bit LX is in position 52, and that no carry is generated to position 2, because the exact product is less than 4 (even after adding the injection). 4. are input to the Compound Adder, that outputs the sum and the incremented sum jY 5. The Increment Decision box receives the round-bit (R), the carry-bit (C[52]), the LSB (LX ), the MSB (Y 0[ 1]) and the rounding modes (RN , RI). The output signal inc indicates whether Y 0 or Y 1 is to be selected. 6. The most signicant bits Y 0[ 1] and Y 1[ 1] indicate whether Y 0 and Y 1 are in the range [2; 4). Depending on these bits, Y 0 and Y 1 are normalized as follows: shif t right(Y shif t right(Y 7. The rounded result (except for the least signicant bit) is selected between Z0 and Z1 according to the increment decision inc, as follows: 8. In case the rounding mode is RNE, the least signicant bit needs to be corrected since RNE and RNU do not always result with the same least signicant bit. The correction of the least signicant bit is computed by two parallel paths; one path working under the assumption that the rounded result over ows (i.e. greater than or equal to 2), and the other path working under the assumption that the rounded result does not over ow. The path that computes the correction of the LSB under the \no-over ow" assumption is implemented by the box called \x L (novf)". The inputs of the \x L (novf)" box are the round bit R, the sticky bit S, and a signal RNE indicating whether the rounding mode is round to nearest even. When the output, denoted by not(pd), equals zero the LSB should be pulled down. The path that computes the correction of the LSB under the \over ow" assumption is implemented by the box called \x L (ovf)". The inputs of the \x L (ovf)" box are the LX bit, the carry-bit C[52], the round bit R, the sticky bit S, and the signal RNE. When the output, denoted by not(pd'), equals zero the LSB should be pulled down. Note that the pull down signals are inactive if the rounding mode is not RNE. 9. The least signicant bit of the rounded result before xing the LSB (in case of a discrepancy between RNE and RNU) equals one of three values: (a) if the rounded result does not over ow, then the LSB equals LX C[52]; (b) if the rounded result over ows and the increment decision is not to increment, then the (c) if the rounded result over ows and the increment decision is to increment, then the LSB The xing of the LSB is implemented by combining (using AND-gates) the pull-down signals with the corresponding candidates for the LSB signals. The outputs of the 3 AND-gates are denoted by: L 0 (inc), L 0 (ninc), and L(inc). For the sake of clarity, we introduce the signal L(ninc), which equals L(inc). 10. The LSB of the rounded result equals L(ninc) if no over ow occurred and no increment took place. The LSB of the rounded result equals L(inc) if no over ow occurred and an increment took place. The LSB of the rounded result equals L 0 (ninc) if an over ow occurred and no increment took place. The LSB of the rounded result equals L 0 (inc) if an over ow occurred and an increment took place. According to the 4 cases, the LSB of the rounded result is selected depending on the over ow signals and the increment decision 4.3 Details In this section we describe the functionality of three boxes in gure 2, that have not been fully described yet. Fix L (novf). This box belongs to the path that assumes that the product is in the range [1; 2). Recall that there might be a discrepancy between RNE and RNU when a tie occurs, namely, when the exact product equals the midpoint between two successive representable numbers. Let exact denote the value of the exact product, the \Fix L (novf)" generates a signal not(pd) that satises: exact 2 [1; occurs and RNE) When a tie occurs there are two possibilities: (a) If RNU and RNE agree, then both yield a rounded result with a LSB equal to zero. Pulling down the LSB in this case is not required, but causes no damage. (b) If RNU and RNE disagree, then the LSB of the RNU result must be pulled down. Without the addition of the injection, a tie occurs when an injection of 2 53 is already included, a tie occurs when Therefore the not(pd) signal is dened by: Fix L' (ovf). This box belongs to the path that assumes that the product is in the range [2; 4). The \Fix L' (ovf)" generates a signal not(pd') that satises: exact 2 [2; occurs and RNE) The dierence between not(pd') and not(pd) is that not(pd') is used under the assumption that the product is greater than or equal to 2. Without the addition of the injection, a tie occurs (in case of over ow) when LX an injection of 2 53 is already included, a tie occurs when LX Therefore the not(pd') signal is dened by: Increment Decision. The increment decision box has two paths, depending on whether an over ow occurs. The path working under the assumption that no over ow occurs (i.e. Y 0[ produces an increment decision, if LX 2. The path working under the assumption that an over ow occurs (i.e. Y 0[ needs to take into account the correction of the injection, denoted by inj correct. It produces an increment decision, if LX 2. Therefore the inc signal is dened by: 4.4 Correctness proof The tricky part in our algorithm is the correctness of the inc signal. As long as the bit Y 0[ 1] indicates correctly whether the exact product is greater than or equal to 2, Equations (1) and (2) imply that the inc signal is correct. But one should also consider the cases that Y 0[ 1] fails to indicate correctly the binade of the exact product. Namely: (a) Y 0[ and the exact product is greater than or equal to 2; and (b) Y 0[ and the exact product (without the injection) is less than 2. The source of such errors is due to the fact that jY does not always equal the the 53 most-signicant bits of the exact product. Recall that The lower part of the product (corresponding to positions [53 : 104] in the registers sum and carry) as well as LX do not eect the value of Y 0[ 1]. However, the injection might have an eect on Y 0[ 1] since it is added-in in the multiplier array, depending on how the multiplier array is implemented (Wallace tree, etc. The following claim shows that when such mismatches occur, the rounded product equals 2. Moreover, in these cases both paths: the one working under the assumption that no over ow occurs, and the one working under the assumption that over ow occurs, yield the result 2. Therefore, correct rounding is obtained even when Y 0[ 1] fails to indicate correctly the binade of the exact product. exact denote the exact product, and let sum and carry satisfy jsumj exact injection. Then correct rounding of exact can be computed as follows: r mode r RZ (exact r RZ (exact Proof: We consider two main cases: (a) Y 0[ (a) Suppose Y 0[ exact < 2 then the claim follows from Eq. (1). If exact 2, then exact . The reason for this is the possible contribution of LX 2 52 2 f0; 2 52 g and have Therefore, exact The correction of the injection satises 0 inj correct 2 52 , therefore: exact According to Eq. (2), in this case r mode However, in this case r RZ (exact because rounding to zero maps both intervals: 2. (b) Suppose Y 0[ exact 2 then the claim follows from Eq. (2). If exact < 2, then since injection 2 [0; 2 52 ), it follows that exact The proof now follows the proof in case (a). 2 proves that Y 0[ 1] can be used for controlling the selection of the right rounded result. The following claim proves that our implementation of the computation of r mode (exact) is correct. Note, that the claim does not deal with xing the LSB to obtain RNE from RNU. r RZ (exact 2. If Y 0[ r RZ (exact exact where tail 2 [0; 2 52 ). This implies that r RZ (exact The inc signal in this case equals 1 i the addition of LX and C[52] generates a carry to position 51. If inc = 0, then simple addition takes place: r RZ (exact If inc = 1, there are two cases: in the rst case, the increment does not cause an over ow, and again simple addition takes place. If an over ow is caused, then since only 53 bits are output, the bit L x C[52] is discarded. This completes the proof of the rst part of the lemma. Suppose Y 0[ Therefore, exact This implies that r RZ (exact and the lemma follows. 2 4.5 Delay analysis In the section we present a delay analysis of the rounding algorithm depicted in Fig. 3. Our analysis is based on the following assumptions: 1. Consider a carry look-ahead adder, and let dCLA denote the delay of the 53-bit adder measured in logic levels. We assume, that the MSB of the sum has a delay of at most dCLA 1 logic levels. This assumption is easy to satisfy if the carry look-ahead adder of Brent and Kung is used [3]. Otherwise, satisfying this assumption may require arranging the parallel-prex network so that the MSB is ready one logic level earlier. 2. The compound adder is implemented so that the delay of the sum is dCLA and the delay of the incremented sum is dCLA + 1. This can be obtained by oring the carry-generate and carry-propagate signals [27, Lemma 1]. 3. Consider the box in which the carry, round and sticky bits are computed. According to the rst assumption, since the widths of this box and the compound adder are similar, the delay of the carry bit is dCLA 1 logic levels and the delay of the round bit is dCLA logic levels. The delay of the sticky bit is estimated to be dCLA 2 logic levels, based on the fast sticky bit computation presented in [31]. 4. We assume that the delay associated with buering a fan-out of 53 is one logic level. Figure 3 depicts the block diagram of the injection based rounding algorithm annotated with timing estimates. We assigned dCLA the value of 8 logic levels. This implies that the sticky bit is valid after 6 logic levels, the carry-bit C[52] is valid after 7 logic levels, and the round-bit is valid after 8 logic levels. Similarly, the sum Y 0 is valid after 9 logic levels, the MSB Y 0[ 1] is valid after 8 logic levels, the incremented sum Y 1 is valid after 10 logic levels, and the MSB Y 1[ 1] is valid after 9 logic levels. Figure 4 depicts implementations of the Fix L (novf), Fix L' (ovf), and Increment Decision boxes annotated with timing estimates. These timing estimates are used in Fig. 3 to obtain the estimated delay of 12 logic levels for the rounded product. 5 The YZ rounding algorithm In this section we review and analyze the rounding algorithm of Yu and Zyner, which was reported to have been implemented in the ULTRASparc RISC microprocessor [31]. We refer to this algorithm as the YZ rounding algorithm. 5.1 Description Figure 5 depicts a block diagram of the YZ rounding algorithm. This description diers from the description in [31] in two ways: 1. In [31] the sum output by the 3-bit adder has only three bits. We believe that this is a mistake, and that the sum should have four bits (we denote this sum by Z[50 : 53]). 2. The sum and the incremented sum in [31] is fed to a 4 : 1-mux, which selects one of them either shifted to the right or not. We propose to normalize the sum and the incremented sum before the selection takes place. This early normalization helps reduce the delay of the rounding circuit at the cost of two shifters rather than one. The algorithm is described below: 1. The sum and carry-strings are divided into a high part and a low part. The high parts consist of positions and the low parts consist of positions [54 : 104]. 2. The low part is input to the box that computes the carry and sticky bits, dened as follows: where 53 104 is the binary string that satises: 3. The higher part is input to a line of Half Adders and produces the output X sum Note that no carry is generated to position [ 2], because the exact product is less than 4. 4. The high part X sum divided into two parts. Positions are fed into the Compound Adder, that outputs the sum Y and the incremented sum jY are added with the carry bit C[53] to produce the sum Z[50 : 53]. 5. The processing of Z[50 : 53] is split into two paths; one working under the assumption that the rounded product will not over ow (i.e. less than 2), and the other path working under the assumption that the rounded product will over ow. The no-over ow path computes a rounding decision, rd[52], in the round dec. (novf) box. The rounding decision rd[52] is added with Z[50 : 52] in the -novf box to produce the sum In Claim 3 we prove that this 3 bit addition does not produce a carry bit in position 49. The sum Z novf [50 : 52] has two roles: positions are the result bits in positions [51 : 52] if no over ow occurs, and position [50] is used to detect if a carry is generated in position [50] if no over ow occurs. The bit Z novf [50] decides whether the upper or the incremented sum Y 1[0 : 50] should be selected in the no-over ow case. The over ow path computes a rounding decision, rd 0 [51], in the round dec. (ovf) box. The rounding decision rd 0 [51] is added with Z[50 : 51] in the -ovf box to produce the sum In Claim 3 we prove that this 2 bit addition does not produce a carry bit in position 49. The sum Z ovf [50 : 51] has two roles: position [51] serves as the result bit in position [52] if over ow occurs, and position [50] is used to decide whether an increment should take place in the upper part 6. The decision which path should be chosen is made by the select decision box. First, an over ow signal ovf is computed as follows: The over ow signal ovf determines whether Z ovf [50] or Z novf [50] is chosen as the carry-bit that eects position [50], and therefore, determines the increment decision inc: Z ovf [50] if Z novf [50] if 7. The two least signicand bits of the rounded product are computed as follows: If no over ow occurs Therefore the lower mux selects these bits for result[51 : 52] when If an over ow occurs [51]. The bit result[51] depends on whether an increment takes place or not: Y 0[50] if inc = 0: Note that inc = Z ovf [50] if ovf = 1. Since the signal Z ovf [50] is ready earlier than inc, we use Z ovf [50] to control the selection: Y 1[50] if Z ovf Y 0[50] if Z ovf The selection between Y 1[50] and Y 0[50] is done by the sel multiplexer in Fig. 5. 8. The most signicant bits Y 0[ 1] and Y 0[ 1] indicate whether Y 0 and Y 1 are in the range [2; 4). Depending on these bits, Y 0 and Y 1 are normalized as follows: shif t right(Y shif t right(Y 9. The rounded result (except for the least signicant bit) is selected between Z0 and Z1 according to the increment decision inc signal, as follows: 5.2 Correctness In this section we provide a proof that adding the rounding decision does not generate a carry-bit in position 49. This claim applies both to the no-over ow path and to the over ow path. denote the sum that is output by the 3-bit adder as depicted in Fig. 5. Let rd[52] denote the rounding decision for the no-over ow path, and let rd 0 [51] denote the rounding decision for the over ow path. Then, Proof: The partial compression [8] caused by the half-adder line implies that (jX sum This follows by the fact that X sum [i] and X carry [i cannot be both equal to one. Adding C[53] increases the above range by 2 4 , yielding that The contribution of rd 0 [51] 2 2 is in the range [0; 4=16], and therefore, Eq. 13 follows. The contribution of rd[52] 2 3 is in the range [0; 2=16], and therefore, Eq. 14 follows. 2 5.3 Delay analysis Figure 6 depicts the YZ rounding algorithm annotated with timing estimates. We use the same assumptions on the delays of signals that are used in Sec. 4.5. We argue that at least 16 logic levels are required. The path in which the sum and incremented sum are computed does not lie on the critical path. The critical path consists of the carry-bit computation, the 3-bit adder, the round dec. (novf) box, the -novf box, the select decision box, a driver, and the upper mux. We considered the following optimizations to minimize delay for a lower bound on the required number of logic levels: 1. The 3-bit adder is implemented by conditional sum adder; the late carry-in bit C[53] selects between the sum and the incremented sum. This is a fast implementation because the bits of X carry and X sum are valid after one logic level and the carry-bit C[53] is valid after 7 logic levels. 2. The rounding decision boxes are implemented by cascading two levels of multiplexers that are controlled by Z[52 : 53] in the no-over ow path and by by Z[51 : 52] in the over ow path. In the over ow path, Z[53] is combined with sticky-bit, and hence the rounding decision required 3 logic levels. In the no-over ow path, only 2 logic levels are required. 3. The addition of the rounding decision bit required only one logic level using a conditional sum adder. 4. The inc signal is valid after 3 more logic levels, due to the need to compute the signal ovf in two logic levels (see Eq. 11), and one selection according to Eq. 12. 5. The inc signal passes through a driver due to the large fanout. This driver incurs a delay of one logic level, and controls the upper mux to output the result after 16 logic levels. 6 The QTF rounding algorithm Quach et al. [23] presented methods for IEEE compliant rounding. Their technique is a generalization of the rounding algorithm of Santoro et al. [24]. In this section we present a rounding algorithm that is based on the method of Quach et al. while aiming for minimum delay. Apart from reducing the rounding modes to RZ;RNU and RI, the key idea used in the methods of Quach et al. and Santoro et al. is to inject a prediction bit that is based on the rounding mode and the values of sum[53] and carry[53]. The injection of the prediction bit reduces the number of possibilities of the rounded result. In this section we deviate from Quach et al. [23] in the following points: 1. The presentation in the paper of Quach et al. is separated according to the rounding mode. Since we are investigating rounding algorithms that support all the rounding modes, we integrated the rounding modes into one algorithm. 2. Quach et al. suggest several options for the choice of the prediction logic in RNU. Only one possibility was suggested in modes RZ and RI. Since the prediction logic lies on the critical path, we chose to simplify the prediction logic as much as possible by dening: pred 3. Quach et al. separate the rounding decision and the compound adder. They use a 3-way compound adder that computes sum; sum 2. The correct sum is selected by the control logic. We are interested in a faster design, and therefore, we break the 3-way adder into a Half-Adder line, a 2-way compound adder, and a mux. The control logic uses an output of the 2-way compound adder, and the LSB (in case of no over ow) is generated by the control logic as well as the increment decision. 6.1 Description Figure 7 depicts a block diagram of a rounding algorithm that we suggest based on Quach et al. [23]. There are many similarities between the rounding algorithm based on injection rounding and the rounding algorithm based on Quach et al., so we point out the dierences and the new notations. Before being input to the compound adder, the high part of the sum and carry pass through two lines of Half-Adders. The rst line makes room for the prediction bit. The second pass enables separating the bit LX 0 in position [52] (this is, in fact, part of a 3-way compound adder). The increment decision has two paths: one for over ow and the other for no-over ow. The MSB Y 0[ 1] selects which path outputs the increment decision inc. In addition the increment decision computes the LSB (before xing for RNE) in case an over ow does not occur. 6.2 Details In this section we describe the details of the increment decision box and the LSB-x for RNE box. Increment decision box. The outputs of the increment decision box are the increment decision inc and the bit L that equals the LSB of the rounded product before xing in case no over ow occurs. The increment decision is partitioned into two paths. One for the case that an over ow occurs which computes the signal inc ovf , and the other path for the case that no over ow occurs which computes the signal incnovf . The following equations dene the signals inc ovf ; inc novf , and inc. inc (R pred pred inc (S _ R _ pred pred The output inc equals inc novf or inc ovf bit according to bit Y 0[ 1]. inc novf if Y 0[ inc ovf if Y 0[ The bit L, which equals the LSB of the rounded product (before xing) in case no over ow occurs, is dened by: R LX 0 C[52] if RNU pred if RI Note that the case of RI is complicated due to the possibility that pred 6= C[52]. If pred = C[52], then pred 6= C[52], the the eect of the wrong prediction is reversed by pred C[52]. LSB-x for RNE. The LSB-x for RNE box outputs two signals: not(pd) is used to pull-down the LSB if a \tie" occurs, but no over ow occurs, and not(pd 0 ) is used to pull-down the LSB if a \tie" and an over ow occur. These signals are dened as follows: In contrast to injection based rounding no injection or prediction is contained in the LX 0 , R and S-bit computation in RNE. If no over ow occurs, a \tie" occurs 0, in which case the LSB should be pulled down for RNE. Therefore, If over ow occurs, a \tie" occurs i which case the LSB should be pulled down for RNE. Therefore, 6.3 Correctness In this section we prove the correctness of the selection signal inc. The proof is divided into two parts. In the rst part, we assume that Y 0[ the exact product is in the range [2; 4). In the second part, we prove that even if the Y 0[ 1] signals over ow incorrectly, then the selection signal inc is still correct. correctly whether the exact product is in the range [2; 4). Then the inc signal signals correctly whether an increment is required for rounding. Proof: We consider separately the cases of over ow and no over ow. For each case we consider the three possible rounding modes. The question which we address is whether the rounding decision in conjunction with the compression of the lower part of the carry-save representation produces a carry into position 51. The inc signal should be 1 i a carry is generated into position 51. Suppose no over ow occurs, namely Y 0[ 1. In rounding mode RZ, only truncation takes place, and therefore, a carry into position 51 is generated 2. 2. In rounding mode RNU, the rounding decision is to increment (in position 52) if the round-bit equals 1. This increment generates a carry to position 52, and hence, a carry is generated into position 51 2. 3. In rounding mode RI, the rounding decision is to increment (in position 52) if One needs to take into account the prediction that was already added to the product. We consider two sub-cases: (a) If pred, then the contributions of pred and C[52] cancel out, and therefore, C[52] should be ignored. The rounding decision generates a carry into position 51 i (R (b) If C[52] 6= pred, then this implies that Therefore, the rounding decision without the prediction would have been to increment in position 52. pred = 1, this increment already took place, and an additional carry should not be generated into position 51. Suppose that over ow occurs, namely, Y 0[ 1. In rounding mode RZ, since only truncation takes place, this case is identical to the case of no over ow. 2. In rounding mode RNU, the rounding decision is determined by the bit in position 52 which Therefore, there are two cases: either a carry is generated into position 51 since LX 0 or a carry is generated into position 51 by the rounding decision. Combining these cases implies that a carry is generated into position 51 i 3. In rounding mode RI, we consider two cases: (i) If pred, then we may ignore C[52] and the prediction since their contributions cancel out. In this case, the rounding decision is to increment i or (L X pred, then We consider two sub-cases: (a) If LX then the eect of the prediction was restricted to changing LX 0 from 0 to 1. Therefore, the rounding decision is based on R _ S. Since the rounding decision is to increment. (b) If LX then the eect of the prediction was to generate a carry into position 51 in the second half-adder line and to change LX 0 from 1 to 0. This means that without the prediction, LX 0 would have been equal to 1, which implies that the rounding decision would have been to increment. Since an increment already took place, an additional increment is not required.The selection between inc ovf and inc novf is controlled by Y 0[ 1], although Y 0[ 1] might not signal correctly the case of over ow. The following claim shows that when Y 0[ 1] does not signal over ow correctly, both choices are equal, and hence, the inc signal is correct. does not signal an over ow correctly, namely, Y 0[ exact < 2, or Y 0[ exact 2. Then, inc inc novf . Proof: The proof is divided into two cases: 1. Y 0[ exact < 2. This case can only occur when pred Therefore, it is restricted to rounding mode RI. Since pred 2 52 it follows that LX This implies that in this case inc ovf = inc novf , as required. 2. Y 0[ exact 2. This discrepancy can only occur if Therefore, is smaller than 2 and a multiple of 2 52 , it follows that This implies that LX 1. Consider the three rounding modes: In RZ, inc inc novf . In RI, if excluding the possibility of this case. In RNU, since and and the claim follows.6.4 Delay analysis Figure 8 depicts the rounding algorithm based on Quach et al. [23] with delay annotation. The delay assumptions that are used here are similar to those used in the two previous rounding algorithms. The rounding algorithm depicted in Fig. 8 uses a prediction logic which lies on the critical path. The delay of the prediction logic is two logic levels. Following Quach et al., Fig. 8 depicts a non-optimized processing order in which the post- normalization shift takes place after the round selection. The increment decision box is assumed to be organized as follows: The bits S, C[52], R and Y 0[ 1] are valid after 6; 7; 8; 10 logic levels, respectively. To minimize delay we implement the rounding equations by 4 levels of multiplexers, so that the results can be selected conditionally the signals as they arrive. Thus, a total delay of 15 logic levels is obtained. By performing post-normalization before the round selection takes place, one logic level can be saved to obtain a total delay of 14 logic levels. 7 Higher Precisions How do these rounding algorithms scale when higher precisions are used? One can see that the parts in the presented rounding algorithms that depend on the length of the signicands are: the half-adders, the compound adder, the sticky, round, and carry-bit computation, the selection multiplexers, and the drivers for amplifying the signals that control the wide multiplexers. When precision is increased, the widths of upper and lower parts of the carry and save strings grow, but they still stay almost equal to each other. This implies that our assumptions on the relative delay of the carry-bit computation and the compound adder do not need to be changed. Moreover, it is expected that as precision grows, the gap between the delay of computing the carry-bit and the sticky-bit grows, so that the sticky-bit computation will not lie on the critical path. This implies that a rst order estimate (ignoring additional delay due to increased fanout and interconnection length) of the delays of the rounding algorithms for precision p can be stated as follows: 1. The delay of the injection based rounding algorithm is 4 logic levels plus the delay of the sum computation of the p-bit compound adder dCLA (p). 2. The delay of the YZ rounding algorithm is 8 3. The delay of the rounding algorithm based on Quach et al.[23] with the optimization (in which the post-normalization takes place before the selection) is 6 levels. 8 Summary and Conclusions A new IEEE compliant oating-point rounding algorithm for computing the rounded product from a carry-save representation of the product is presented. The new rounding algorithm is compared with two previous rounding algorithms. To make the comparison as relevant as possible, we considered optimizations of the previous algorithms which improve the delay. For each rounding algorithm, a logical description and a block diagram is given, the correctness is proven, and the latency is analyzed. Our conclusion is that the new ES rounding algorithm is the fastest rounding algorithm, provided that an injection is added in during the reduction of the partial products into a carry-save encoded digit string. With the ES algorithm the rounded product can be computed in 12 logic levels in double precision (i.e. when the signicands are 53 bits long). In \precision independent" terms, the critical path consists of a compound adder and 4 additional logic levels. If the injection is not added in during the reduction of the partial products into a carry-save encoded digit string, then an extra step of adding in the injection is required. This step amounts to a carry-save addition, and the latency associated with it is that of a full-adder, namely, 2 logic levels. Thus, if the injection is added in late, then the latency of the ES rounding algorithm is 14 logic levels. The addition of the injection during the reduction of the partial products can be accomplished without a slowdown or with a very small slowdown. The justication for this is: (a) The partial products are usually obtained by Booth recoding and by selecting (e.g. 5:1 multiplexer), and hence, are valid much later than the injection; and (b) The delay of adding the partial products does not increase strictly monotonically as a function of the number of partial products. The delay incurred by adding in the injection, if any, depends on the length of the signicands and on the organization of the adder tree. The other two rounding algorithms do not require an injection, and in double precision, the latency of the QTF rounding algorithm is 14 logic levels. The critical path consists of a compound adder and 6 additional logic levels. The YZ rounding algorithm ranks as the slowest rounding algorithm, with a latency of 16 logic levels, and the critical path consists of a compound adder and 8 additional logic levels. --R Area and Performance Optimized CMOS Multipliers. Fast Multiplication: Algorithms and Implementation. A Regular Layout for Parallel Adders. Method for rounding using redundant coded multiply result. Some schemes for parallel multipliers. Method and apparatus for rounding in high-speed multipliers Recoders for partial compression and rounding. Fast multiplier bit-product matrix reduction using bit- ordering and parity generation A Dual Mode IEEE multiplier. A comparison of three rounding algorithms for IEEE oating- point multiplication Parallel method and apparatus for detecting and completing oating point operations involving special operands. IEEE standard for binary oating-point arithmetic Multistep gradual rounding. Design strategies for optimal multiplier circuits. Design Issues in High Performance Floating Point Arithmetic Units. The SNAP project: Design of oating point arithmetic units. A method for speed optimized partial product reduction and generation of fast parallel multipliers using an algorithmic approach. Reducing the number of counters needed for integer multiplication. Generation of high speed CMOS multiplier- accumulators Floating Point Multiplier Performing IEEE Rounding and Addition in Parallel. On fast IEEE rounding. Rounding algorithms for IEEE multipliers. How to half the latency of IEEE compliant oating-point multiplication A reduced-Area Scheme for Carry-Select Adders A very fast multiplication algorithm for VLSI implementation. A suggestion for parallel multipliers. A new design technique for column compression multipliers. oating point multiplier Method and apparatus for partially suporting subnormal operands in oating point multiplication. Shared rounding hardware for multiplier and division/square root unit using conditional sum adder. Circuitry for rounding in a oating point multiplier. 9L 13L 13L 14L 15L 14L 13L 16L 16L 9L 8L 10L --TR --CTR Peter-Michael Seidel , Guy Even, Delay-Optimized Implementation of IEEE Floating-Point Addition, IEEE Transactions on Computers, v.53 n.2, p.97-113, February 2004 Ahmet Akkas , Michael J. Schulte, Dual-mode floating-point multiplier architectures with parallel operations, Journal of Systems Architecture: the EUROMICRO Journal, v.52 n.10, p.549-562, October 2006 Nhon T. Quach , Naofumi Takagi , Michael J. Flynn, Systematic IEEE rounding method for high-speed floating-point multipliers, IEEE Transactions on Very Large Scale Integration (VLSI) Systems, v.12 n.5, p.511-521, May 2004

floating-point multiplication;IEEE 754 Standard;floating-point arithmetic;IEEE rounding

350326

Integer Multiplication with Overflow Detection or Saturation.

AbstractHigh-speed multiplication is frequently used in general-purpose and application-specific computer systems. These systems often support integer multiplication, where two $n$-bit integers are multiplied to produce a $2n$-bit product. To prevent growth in word length, processors typically return the $n$ least significant bits of the product and a flag that indicates whether or not overflow has occurred. Alternatively, some processors saturate results that overflow to the most positive or most negative representable number. This paper presents efficient methods for performing unsigned or two's complement integer multiplication with overflow detection or saturation. These methods have significantly less area and delay than conventional methods for integer multiplication with overflow detection or saturation.

Introduction 1.1 Multiplication Multiplication is an essential arithmetic operation for general purpose computers and digital signal processors. High-performance systems support parallel multiplication in hardware. Various high-speed parallel multipliers have been proposed and realized. Most parallel multiplier designs can be divided into two classes; array multipliers and tree multipliers. Array multipliers consist of an array of similar cells that generate and accumulate the partial products [3]. Tree multipliers generate all partial products in parallel, use a tree of counters to reduce the partial products to sum and carry vectors and then sum these vectors, using a fast carry-propagate adder. Several methods have been developed for reducing the partial products [1], [2], The regular structure of array multipliers facilitates their implementation in VLSI technology. The delay of array multipliers, however, is proportional to the operand length. On the other hand, tree multipliers offer a delay proportional to the logarithm of the operand length. The main drawback of tree multipliers is their irregular interconnection structure, which makes them difficult to implement in VLSI. Thus, tree multipliers are preferred for high performance systems, while array multipliers are preferred for systems requiring less area. New implementations and optimizations of parallel multipliers are still active research areas [5]. More detail descriptions of array and tree multipliers are given in the next chapter. 1.2 Overflow To avoid grow in word length, most instruction set architectures and high-level languages require that arithmetic operation return results with the same length as their input operands. If the result of an integer arithmetic operation on n-bit numbers cannot be represented by n bits, overflow occurs and needs to be detected. For integer multiplication, the method for overflow detection also depends on whether the operands are signed or unsigned integers. For unsigned multiplication, overflow only occurs if the result is larger than the largest unsigned n-bit number. For signed integer multiplication overflow also occur if the result is smaller than the minimum representable n-bit number. For two's complement multiplication there is also a difference between fractional and integer overflow detection. Since overflow only occurs for two's complement fractional numbers when \Gamma1 is multiplied by \Gamma1, it is easy to detect overflow when multiplying two's complement fractions. It is an important design issue in computer architecture is to decide what to do when overflow occurs. Typically, overflow results in an overflow flag being set. This overflow flag can then be used to signal an arithmetic exception [9]. 1.3 Saturation In most general purpose processors, overflow is handled by setting an exception flag. More recent implementations for digital signal processing and multimedia applications saturate results that overflow to the most positive or most negative representable number [11], [12]. For two's complement integers this is \Gamma2 negative numbers and 2 positive numbers. For unsigned integers, results that are too large saturate to 2 1.4 Thesis Overview Previous studies have focussed on overflow detection in two's complement addition [13], multi-operand addition [14], fractional arithmetic operations [10] and generalized signed-digit arithmetic[15]. This thesis presents efficient techniques for integer multiplication with overflow detection or saturation. Most existing computers detect overflow in integer multiplication by a computing 2n-bit product and then testing the most significant bits to see if overflow has occurred. The methods proposed in this thesis only calculate n or n bits of the product. This leads to a significant reduction in area and delay. Chapter 2 presents array multipliers, tree multipliers, and conventional methods for overflow detection. Chapter 3 introduces new methods for overflow detection and saturation for unsigned integer multiplication. Chapter 4 focuses on overflow detection and saturation for two's complement integer multiplication. Chapter 5 presents the component counts and area and delay estimates for unsigned and two's complement parallel multipliers that use either the conventional or the proposed methods for overflow detection. Chapter 6 discusses future work and gives conclusions. Chapter 2 Previous Research 2.1 Unsigned Parallel Multipliers Multiplication of two n-bit unsigned numbers is shown in Figure 2.1, where Multiplication produces a 2n-bit product. If the n least significant bits are used for the result, then overflow occurs when the actual product uses more than n bits. In other words, overflow occurs when the product is greater or equal to 2 n . With conventional methods, overflow is detected after the 2n-bit result is pro- duced. This is done by simply logically ORing together the n most significant bits, a n-2 a n-1 n-2 a n-1 a n-2 n-2 a n-1 a n-2b b 1 a n-1 a n-2b b 0 Figure 2.1: Multiplication of A and B. such that where V is one if overflow occurs and + denotes logical OR. Although calculating 2n product bits and then detecting overflow leads to unnecessary area and delay, most computers that provide integer multiplication with overflow detection use this approach. If the system requires saturation, the saturated least significant product bits are computed as which sets the product to 2 n\Gamma1 when overflow occurs. 2.1.1 Unsigned Array Multipliers In general, array multipliers are slower than tree multipliers. In spite of this speed disadvantage, however, array multipliers are often used due their regular layout, low area, and simplified design. A block diagram of an unsigned 8 by 8 array multiplier is shown in Figure 2.2. Each diagonal in Figure 2.2 corresponds to a column in the multiplication matrix, in Figure 2.1. A modified half adder (MHA) is a half adder with an AND gate, and a modified full adder (MFA) is a full adder with an AND gate. The AND gates generate the partial products. Full adders and half adders add the generated partial products. Sum outputs are connected diagonally and carry outputs are connected vertically. The last row of adders, which are connected from left to right, generates the n-most significant product bits . The critical path through this multiplier is shown with dashed lines. Since almost half of the latency is due to the bottom row of adders, this row may be replaced by a fast carry-propagate adder. Although this decreases the overall delay, it has a negative impact on the design's regularity. An n by n unsigned array multiplier uses n 2 AND gates, (n HAs. The conventional method for overflow detection requires the n most significant product bits to be calculated. These product bits are then OR'ed together to produce the overflow flag, as shown in Figure 2.2. The conventional method for saturating multiplication is accomplished by ORing V with p 0 to p n\Gamma1 , as shown in Figure 2.3. AND AND AND AND AND AND AND AND AND AND AND AND AND MHA MHA MHA MHA MHA MHA MHA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA AND MFA MFA FA c s FA c s FA c s AND MFAa 7 a 6 a 5 a 4 a 3 a 2 a 1 a 0 c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s MFA MFA HA c s FA c s FA c s FA c s Figure 2.2: Unsigned Array Multiplier with Conventional Overflow Detection. AND AND AND AND AND MHA MHA MHA MHA MHA MHA MHA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA AND MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA FA c s FA AND AND AND AND MFA FA AND AND AND AND a 7 a 6 a 5 a 4 a 3 a 2 a 1 a 0 c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c c c c c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s MFA MFA HA c FA c s FA c s FA c s Figure 2.3: Unsigned Array Multiplier with Conventional Saturation. 2.1.2 Unsigned Tree Multipliers Tree multipliers have three main parts; partial product generation, partial product reduction, and final carry-propagate addition. Various reduction schemes have been developed over the years. Two of the most well-known methods for multiplier tree designs are those proposed by Wallace [2] and Dadda [1]. Wallace's strategy combines three rows of partial product bits using (3; 2) and (2; 2) counters to produce two rows. Dadda's strategy leads to a simpler counter tree but requires a larger final carry-propagate adder. There are hybrid approaches between these two methods that offers cost and speed trade-offs for VLSI implementations. These reduction schemes differs in the number and placement of counters in the tree and the size of the final carry propagate adder. The tree multipliers presented in this thesis use Dadda's method, since it allows component counts to easily be determined based on n. Since our overflow detection method does not depend on the reduction strategy, similar savings are expected for other tree multipliers. Figure 2.4 shows a dot diagram of an 8 by 8 unsigned array multiplier. Dot diagrams are often used to illustrate reduction strategies in tree multipliers [2]. With this technique, a dot represents a partial product bit, a plain diagonal line represents a full adder and a crossed diagonal line represents a half adder. The two bottom rows of the dot diagram corresponds to sum and carry vectors that are combined using the final carry-propagate adder to produce the product. Dadda multipliers require n 2 AND gates, (n and a Figure 2.4: Dadda Reduction Scheme. CPA. The number of stages, based on n; is shown in Table 2.1. For example a 24 by 24 Dadda multiplier requires seven reduction stages. The worst case delay path is equal to the delay of partial product generation, plus the delay for the reduction stages, plus the delay for the final carry-propagate addition. With the conventional method, overflow is detected with a tree of (n \Gamma 1) 2-input gates. The delay of the tree of OR gates is equivalent to dlog 2 ne 2-input OR gates, as shown in Figure 2.5. Range of n s 43 Table 2.1: Number of Stages s for n-Bit Dadda Tree Multipliers.V101214 pp Figure 2.5: Tree of 2-input OR Gates for 2.2 Two's Complement Multipliers Two's complement numbers A and B and their product P have the values where \Gammab a a overflow occurs when Multiplication of two's complement numbers generates signed partial products as shown in Figure 2.6. Since a negative weights, they should be subtracted, rather than added. This makes the design difficult to implement because it requires adder and subtracter cells. Consequently, several techniques have been proposed to handle partial products with negative and positive weight, such as the Baugh-Wooley Algorithm[6] and its variations [7], [8] and, Booth's Algorithm [16]. The Baugh-Wooley algorithm provides a method a 1 a 0 - a - a a 0 a 1 a 0 n-1 x a n-1 a a Figure 2.6: Two's Complement Multiplication Matrix. for modifying the partial product matrix, so that all the partial product bits have positive weights. This algorithm and its modified form are often used to perform two's complement multiplication.b b 1 a n-1 a a a 1 b n-2 a a a n-2 a n-1 n-2 bn-1a n-2 n-2 a Figure 2.7: Modified Two's Complement Multiplication Matrix. Two's complement multiplication is often realized using a variation of the Baugh- Wooley algorithm called the Complemented Partial Product Word Correction Al- gorithm. With this implementation, partial product bits containing a but not both, are complemented and ones are added to columns n and 2n \Gamma 1. This is equivalent to taking the two's complement of the two negative terms in Equation 2.5. The multiplication matrix for this implementation is shown in Figure 2.7. 2.2.1 Two's Complement Array Multipliers The design of an array multiplier that uses the Complemented Partial Product Word Correction Algorithm and conventional overflow detection is shown in Figure 2.8. The design shown in this figure is similar to the unsigned array multiplier design in Figure 2.2. AND gates in the left-most column are replaced by NAND gates and the last row of MFAs are replaced by Negating Modified Full Adders (NMFA). The specialized half adder (SHA) in the bottom right corner is a half adder that takes the sum and carry bits of the previous row and adds them with '1'. This cell has approximately the same area and delay as a regular half adder. The last product bit p 2n\Gamma1 is inverted to add the one in column 2n \Gamma 1. Inverting p 2n\Gamma1 has the same effect as adding one in column 2n \Gamma 1, because the carry out from this column is ignored. In Figure 2.8, the bottom two rows of cells, consisting of n XOR gates and gates, are dedicated to overflow detection. The XOR gates identify whether differs from any of the more significant product bits p n to outputs from the XOR gates are combined to determine if the overflow flag V should be set. The logic equation for the overflow detection flag is AND MHA MHA MHA MHA MHA MHA MFA MFA AND MFA MFA MFA MFA MHA MFA MFA MFA MFA MFA MFA AND MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA AND MFA MFA bb AND AND a 7 a 6 a 5 a 4 a 3 a 2 a 1 a 0 c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s FA c s FA c s FA c s AND AND NMFA NMFA NMFA NMFA NMFA NMFA NMFA FA c s FA c s FA c s Figure 2.8: Two's Complement Array Multiplication with Conventional Overflow Detection. Saturating multiplication is implemented by adding an n-bit 2-to-1 multiplexer, as shown in Figure 2.9. For two's complement multiplication, if the product over- flows, the saturated product is determined from the sign bits of A and B. If and a negative overflow has occurred and the product saturates to \Gamma2 On the other hand, if positive overflow has occurred and the product saturates to 2 then overflow has not MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA AND MFA c MFA c c s c c s MFA MFA MFA MFA MFA MFA MFA MFA MHA MHA MHA MFA c s MHA MHA MHA c AND ANDpAND AND AND a 7 a 6 a 5 a 4 a 3 a 2 a 1 a 0 c c c c c c c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c c c c c c c FA c s FA c s FA c s AND NMFA NMFA NMFA NMFA NMFA FA c s FA c FA c s MFA MFA n-bit Mux XAND Figure 2.9: Conventional Two's Complement Saturation. occurred and the saturated product is the n least significant bits. If the saturated product 2.2.2 Two's Complement Tree Multipliers Several techniques are available for implementing two's complement multiplier trees [4], [8]. Figure 2.7 shows the dot diagram of an 8 by 8 two's complement multiplier tree that uses the Complemented Partial Product Algorithm [8] and Dadda's reduction method [1]. Similar to the two's complement array multiplier, (2n \Gamma 2) of the partial product bits are inverted and ones are added to columns n and (2n \Gamma 1): Although it seems that the heights of these two columns are increased by adding ones, this does not effect adversely the design, because the most significant product bit is simply inverted and a SHA adds one in the column n. In Figure 2.10, a dot with a line above it indicates a complemented partial product bit and the circled half adder in column 8 is a (SHA) specialized half adder. An n by n two's complement Dadda tree multiplier has FAs, and a (2n \Gamma 2) bit CPA. Conventional techniques for overflow detection and saturation for two's complement tree multipliers are similar to the techniques used for two's complement array multipliers. The only difference is that tree multipliers tend to use a tree of OR gates, rather than a linear array of OR gates, when computing the overflow flag. Figure 2.10: Two's Complement Dadda Tree Multiplier. Chapter 3 Overflow Detection and Saturation for Unsigned Integer Multiplication 3.1 General Design Approach Instead of computing all 2n bits of the product, the methods proposed in this thesis only compute the n least significant product bits and have separate overflow detection logic, as shown in Figure 3.1. Carries into column n are also used in the overflow detection circuit. The main idea behind the proposed unsigned overflow detection methods is that overflow occurs if any of the partial product bits in column n to (2n \Gamma 2) are '1' or any OVERFLOW DETECTION RESULT OPERAND A OPERAND B n by n MULTIPLIER Figure 3.1: Block Diagram for Unsigned Multiplication Overflow Detection. of the carries into column n are '1'. Consequently, these ones can be detected without adding the partial products. The logic equation for unsigned overflow detection is a In this expression, V is the overflow flag, c i is the i th carry into column n, bit summations corresponds to logical ORs and bit multiplications corresponds to logical ANDs. 3.2 Unsigned Array Multipliers with Overflow Detection or Saturation Figure 3.2. shows an 8 by 8 multiplication matrix to demonstrate how the partial product bits are used to detect overflow with the proposed method. PARTIAL PRODUCTS USED FOR OVERFLOW DETECTION a b 0 a a b 0 a a a b 0 a b a b a b a b a b a b a b 1357 a 0 b 1111a b a b a b a b a b a b a b 1357 a 0 b 2222a b a b a b a b a b a b a b 1357 a b a b a b a b a ba a b 1357 a b a b a b a b a b a b a b 1357 a b a b a b a b a b a b a b 1357 a b a b a b a b a b a b a b 1357 a 0 b a a a a a Figure 3.2: 8 by 8 Unsigned Multiplication Matrix. Using Equation 3.1, with Common terms in the logic equation for overflow detection are used to reduce the hardware needed to detect overflow. An overflow detection circuit constructed using AND and OR gates is shown in the Figure 3.3 for an 8 by 8 unsigned multiplication. For an n by n multiplier, the three gates in the dashed lines are replicated times. These three gates are combined to form an overflow detection (OVD) cell. Overflow is detected using the following iterative equations. is a temporary OR bit with an initial value of and v i is a temporary overflow bit with an initial value . The are shown in Figure 3.3. An OVD cell takes a inputs and generates outputs. Each OVD cell contains one AND gate, one 2-input OR gate, and one 3- input To form an unsigned multiplier with the proposed overflow detection method, these cells are combined with an unsigned array multiplier from which the cells used to compute p n to p 2n\Gamma1 have been removed. This is shown in Figure 3.4 for an 8 by 8 unsigned array multiplier. b246 a a a a a aa 7b c c Figure 3.3: Proposed Overflow Detection Logic for An n-bit unsigned array multiplier that uses the proposed method for overflow detection requires (n 2 FAs. This corresponds to (n gates than the conventional method. The worst case delay path is indicated by the dashed line in Figure 3.4. Since a MFA has longer delay than an OVD cell, the unsigned multiplier with the proposed overflow detection logic has a delay approximately half as long as the unsigned multiplier with conventional overflow detection, shown in Figure 3.3. Unsigned saturating multiplication using the proposed method is performed by ORing the overflow bit with n least significant product bits, as shown in Figure 3.5. If the overflow bit is '1' this produces a product with n ones, which corresponds to the maximum representable unsigned number; otherwise the product is not changed. This requires n more OR gates and the worst case delay increases by one delay. AND AND AND AND AND AND MHA MHA MHA MHA MHA MHA MFA MFA MFA MHA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA AND MFAo 7v AND AND a 7 a 6 a 5 a 4 a 3 a 2 a 1 a 0 c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s MFA OVD OVD OVD OVD OVD cccc OVD c Figure 3.4: Unsigned Array Multipliers with Proposed Overflow Detection Logic. MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MFA MHA MHA MHA MHA MHA MHA MHA AND AND AND AND a 7 a 6 a 5 a 4 a 3 a 2 a 1 a 0 c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s c s MFA OVD OVD OVD OVD OVD cccc OVD c AND AND AND AND AND Figure 3.5: Unsigned Array Multipliers with Proposed Saturation Logic. 3.3 Unsigned Tree Multipliers with Overflow Detection or Saturation. For unsigned tree multipliers, the technique of using an array of OVD cells with linear delay does not work well, since the OVD cells would significantly increase the multipliers' worst case delay path. Instead, all n 2 partial product bits are generated and a tree of OR gates is used to determine if any of the partial product bits in columns n to 2n\Gamma1 or any carries into column n are one. This method is shown for an 8 by 8 multiplier in Figure 3.6, where the symbol 'o' denotes the output of a 2-input gate. Although this method requires more hardware for overflow detection than the unsigned array multiplier, the overflow detection logic has logarithmic delay and no longer contributes significantly to the critical path. An n by n unsigned array multiplier that uses this method has (n 2 HAs, and (n FAs. Since the delay of the OR gates for overflow detection is less than the delay of the partial product reduction stages, the worst case delay is equal to the delay of partial product generation, plus partial product reduction, plus an (n \Gamma 1)-bit carry-propagate addition plus one OR gate delay to include the final carry out. Saturating multiplication is performed with the same method that is used by the array multiplier. The overflow bit is OR'ed with the n least significant bits of the product. Figure Unsigned Tree Multiplier with Proposed Overflow Detection Logic. Chapter 4 Overflow and Saturation Detection for Two's Complement Integer Multiplication 4.1 General Approach The proposed method for overflow detection in two's complement multiplication detects the number of consecutive bits that are equal to the sign bit. Essentially, this method counts the number of leading zeros if the operand is positive and the number of leading ones if the operand is negative. For example, 11100101 has three leading ones and 00001001 has four leading zeros. This method works because the number of leading zeros or ones indicates the magnitude of the operand; operands with more leading zeros or ones have smaller magnitudes and therefore are less likely to cause overflow. The main issue is to determine how many of leading zeros and leading ones are needed to guarantee that overflow will occur or that overflow will not occur. A block diagram that shows the proposed approach is shown in Figure 4.1. The analysis for two's complement multiplication has three cases depending on the operands' signs; both operands positive, both operands negative, or one positive and the other negative. Overflow regions for these three cases are discussed in the following sections. Case 1: Both Operands are Positive Let ZA denote the number of leading zeros of operand A and ZB denote the number of leading zeros of operand B. Since both operands are positive n-bit integers, they have at least one and at most n leading zeros, which can be expressed as The ranges for operand A and operand B in terms of the number of the leading zeros are expressed as OVERFLOW DETECTION OPERAND A OPERAND B MULTIPLIER n by n Figure 4.1: Block Diagram of The Proposed Method for Two's Complement Multiplication with Overflow Detection. Overflow occurs for Case 1 when overflow does not occur when . Based on (4.2) the range of P is Using (4.3), overflow is guaranteed to occur when To determine the number of leading zeros in A and B that guarantee overflow, (4.4) is rewritten as Taking the base 2 logarithm of both sides gives or equivalently, Thus, if A and B together have less than n leading zeros, overflow must occur. Similarly, overflow is guaranteed not to occur when To determine the number of leading zeros in A and B that guarantee overflow does not occur, Equation 4.8 is rewritten as since \Gamma2 n\GammaZ A - \Gamma1 and \Gamma2 n\GammaZ B - \Gamma1, Consequently, overflow is guaranteed not to occur if Taking the base 2 logarithm of both sides gives or equivalently Using which is always true since ZA - 1 and ZB - 1. Therefore, 4.13 can be rewritten as Thus, overflow is guaranteed not to occur if A and B together have more than n leading zeros. cannot be directly determined whether or not overflow has occurred only by examining the number of leading zeros. Rewriting (4.3) for This problem however can be solved by further analyzing what happens when ZA n. The results obtained so far are summarized in Figure 4.2. overflow no overflow undetermined Figure 4.2: Overflow Regions for ZA To Determine whether or not overflow occurs when is necessary to calculate how many product bits are needed to represent the result when ZA n. Then by using the most significant product bits and the sign bits of the operands the overflow flag is set. If n is even, the maximum product is or equivalently If n is odd the maximum product is or equivalently Thus, when ZA can always be represented with n bits and overflow can be determined simply by examining the sign bit p n\Gamma1 . If p Otherwise, overflow does not occur. 4.1.2 Case 2: Both Operands are Negative Let I A denote the number of leading ones of operand A and I B denote the number leading zeros of operand B. Since both operands are negative n-bit integers, they have at least one and at most n leading ones. This can be expressed as The ranges for A and B in terms of leading ones are expressed as \Gamma2 n\GammaI A - A - \Gamma(1 Overflow occurs for Case 2 when overflow does not occur when . Based on (4.22) the range of P is Using (4.23), overflow is guaranteed to occur when To determine the number of leading ones in A and B that guarantee overflow occurs (4.24) is rewritten as since values that satisfy also satisfy (4.25), Taking the base 2 logarithm of both sides of (4.27) gives I A Thus, for negative integers overflow occurs when operands have less than leading ones. Using (4.23) overflow is guaranteed not to occur when Taking the base 2 logarithm of both sides of Equation 4.29 gives Thus, overflow is guaranteed not to occur when A and B together have more than leading ones. When cannot be directly determined if overflow has occurred. This is seen by using in (4.23), which gives similarly when The results so far are shown in Figure 4.3. undetermined no overflow overflow Figure 4.3: Overflow Regions for I A When I A is even, the maximum result is when n is odd, the maximum product is can be represented using only n bits, except when occurs when 4.1.3 Case 3: When The Signs of The Operands Differ Let I A denote the number of leading ones in operand A and ZB denote the number of leading zeros in operand B. Negative operand A has at least one and at most n leading zeros and positive operand B has at least one and at most n leading ones. This can be expressed as The ranges for A and B in terms of leading ones and zeros are \Gamma2 n\GammaI A - A - \Gamma(1 Overflow occurs for Case 3 when P - \Gamma2 and does not occur when P - \Gamma2 Based on (4.36) the range of P is Using Equation 4.37, overflow is guaranteed to occur when or equivalently For I A +ZB , (4.39) is always true, since 2 Using 4.37 overflow is guaranteed not to occur when \Gamma2 or equivalently For in this range. When cannot be determined whether or not overflow occurred since for The Figure 4.4 shows graphically the results obtained so far for Case 3. overflow no overflow undetermined Figure 4.4: Overflow Regions for I A + ZB . When I A is even, the minimum negative number is and when is n is odd it is For both cases, the product can be represented using only by n bits. Therefore, overflow occurs if p So far, the proposed method has been explained mathematically. In the next section implementations of the overflow logic are presented. 4.1.4 Overflow Detection Logic To allow positive and negative operands to use the same hardware for detecting leading zeros or ones, the sign bits are XNOR'ed with the remaining bits. This takes gates and is expressed logically as A logic design that detects (n \Gamma 1) or fewer leading zeros or leading ones includes gates. These AND gates are used to compute Y ba (n\Gammak)\Gamma2 (4.47) Y For 3: A preliminary overflow flag is generated, using x(i) and y(i) as where bit products correspond to logical ANDs, and bit summations correspond to logical ORs. This equation is implemented by using (n \Gamma 2) 2-input NOR gates and gates. V 1 is one when the total number of leading zeros and leading ones is less than n: In this case, overflow is guaranteed to occur. Additional logic is used to detect overflow for the undetermined regions for Cases 1-3. For Case 1 (when a is detected as For Case 2 when a neither A nor B is zero, or the n least significant bits are zero. For Case 3, (when neither A nor B is zero. This is detected as Logic Equations 4.50 through 4.54 can be realized by 9 2-input AND gates, 4 inverters and (n gates. The final overflow flag V is generated by OR'ing all of the previous flags. The overflow detection circuit requires, (2n \Gamma gates, and four inverters. An overflow detection circuit for an 8-bit two's complement multiplier is shown in Figures 4.5 and 4.6. bb a 7 a 7 a 7 a 7 a 7 a 7 a a 7 a a a a Figure 4.5: The Logic for V 1 for 8-bit Multiplication. a 7y aN LEAST SIGNIFICANT a z Figure Detection Logic for 8-bit Two's Complement Multiplication. 4.1.5 An Alternative Method An alternative method for detecting overflow in the undetermined case is, instead of generating product bits, have the multiplier generate product bits and detect the undetermined cases by checking if p n \Phi p approach is shown in Figure 4.7. This approach works since for the undetermined cases we have the following situations; only when overflow occurs. (2) Case 2: overflow occurs. The one exception is when generated and then p (3) Case 3: only when overflow occurs. For all three cases, overflow can be detected as 4.2 Two's Complement Array Multipliers with Overflow Detection or Saturation The proposed method for overflow detection for array multipliers requires half as much hardware as the conventional method. An 8-bit two's complement multiplier n by n MULTIPLIER OPERAND A DETECTION Figure 4.7: Block Diagram of the Proposed Alternative Method for Two's Complement Multiplication with Overflow Detection. with the proposed method for overflow detection is shown in Figure 4.8. The X2A cell contains one 2-input XOR gate and one 2-input AND gate, and each X3A cell contains one 3-input XOR gate and one 2-input AND gate, an X3NA contains two 2-input XOR gate and one 2-input NAND gate. A two's complemented array multiplier with the proposed overflow detection logic 3-input XOR gates, one 2-input XOR gate and four inverters. The delay for this multiplier is approximately equal to the delay of (n \Gamma plus four 2-input OR gates, plus three 3-input AND gates. The actual delay may differ according to various design decisions and the technology used. OVERFLOW DETECTION MFA MFA MFA MFA MFA MFA MFA MFA MFA b 7b MFA a 7 a 5 a 3 a 1 aa 4 MFAa 7 a 6 a 5 4 a 3 a 2 a MFA MFA MFA MHA MHA a MHA a a MHA MHA MHA AND AND AND AND AND AND c s c s c s c s c s c s c s c s c s c s c s c s c s NAND AND s s s s s s s Figure 4.8: Overflow Detection Logic for 8-bit Two's Complement Multiplication. Two's complement saturating multiplication is performed by using the V and t flags. These flags are used as inputs to an n-bit 2-to-1 multiplexer as shown in Figure 4.9. When negative overflow occurs, the result is saturated to \Gamma2 positive overflow occurs, result is saturated to 2 Adding saturation logic to the array multiplier with overflow detection only requires the addition of an inverter and an n-bit 2-to-1 multiplexor. Since V and t are already generated by the overflow detection logic, they do not require any additional hardware. The delay increases just by the delay of a 2-to-1 multiplexer plus the delay of an inverter. OVERFLOW DETECTION AND c s c s c c c MFA 6 c s s7s s MFAb 432 a 7 MFA MFA a 5 a 4 a a 2 a 1 a 7 a 6 MFA MFA MFA MFA MFA MFA MFAb MFA MFAMFA MHA MHA MHA MHA MHA AND AND AND AND AND c s c s c s c s c s c s c s c s NAND AND MFA n-bit Mux s s s s s s a 6 a 5 a 4 a 3 a 2 a 1 a 07 5 4 2 Figure 4.9: Saturating Two's Complement Array Multiplication. 4.3 Two's Complement Tree Multipliers with Overflow Detection or Saturation The alternative method is used for tree multipliers. For this method (n+1) bits of the product are computed. V 1 is computed the approach same as the first method and then the overflow flag is generated by using the logic equation as explained in Section 4.1.5. Since the detection circuit is independent of the multiplication process, only n+1 of the partial product bits needs to be generated. Consequently, the AND gates and counters that generate and reduce the partial product bits after column n of the multiplication matrix are no longer needed. The size of the carry-propagate adder is reduced to n bits since only the least significant product bits are used. These reductions are independent of the strategy used to design the tree multiplier. The reduction scheme of a Dadda multiplier that uses the alternative method is shown in Figure 4.10. A diagonal line with an x at the bottom is an 3 input XOR gate and a diagonal with a tilda on it and an x at the bottom represents a 2-input XNOR gate. The X's are used to denote that a carry output is not required. If the worst case delay is the main constraint of the custom design, the alternative design method should be considered to implement the overflow detection logic. A two's complement Dadda multiplier with proposed overflow detection has (n gates and one n-bit CPA. The worst case delay of the multiplier is also less than the conventional technique. With the alternative method,the worst case delay equals the delay of the reduction stages, plus the delay of the n adder, plus one 2-input OR gate, plus, one 2-input XOR gate, plus one 2-input AND gate. Two's complement saturating multiplication logic for the Dadda tree multiplier is similar to the logic for the array multiplier. An n-bit 2-to-1 multiplexer and an inverter are added as shown in Figure 4.9, except the partial product bits p n\Gamma2 to are not connected to the overflow detection logic. The control signals t and V for the 2-to-1 multiplexer are generated by the detection logic. The delay increases by just the delay of the inverter plus the delay of the 2-to-1 multiplexer. Figure 4.10: Dadda Dot Product Scheme after Proposed Overflow Detection. Chapter 5 Results 5.1 Area and Delay Estimates Theoretical component counts and worst case delays are given for various multipliers in Tables 5.1, 5.2 and table 5.3. In these tables, U and S denotes unsigned and signed, A and T denote array and tree multiplier, and P and C denote proposed and conventional. Table 5.1 and Table 5.2 gives the number of each component and the size of the CPA based on operand length n. Table 5.3 gives the number of each type of component on the worst case delay path. The proposed methods reduce the number of AND gates and FAs for array multipliers and reduce the number of AND gates and FAs, and the size of the CPAs for tree multipliers. The proposed methods also reduces the delays of the array and tree multiplier, since the most significant product bits are no longer calculated. Multiplier Number of Components Type INV AND NAND OR2 NOR OR3 Table 5.1: Component Counts for n-bit Multipliers with Overflow Detection I. Multiplier Number of Components Type XOR XNOR HA FA CPA Table 5.2: Component Counts for n-bit Multipliers with Overflow Detection II. Multiplier Number of Components on Worst Case Delay Path Type INV AND OR2 XOR HA FA CPA Table 5.3: Worst Case Delay for n-bit Multipliers with Overflow Detection. Table 5.4: Unsigned Array Multipliers with Overflow Detection. It is possible to reduce the amount of logic required to implement the detection circuit even further. The proposed method uses a straight forward implementation of the logic equations and structures presented in the previous chapters. Synthesis tools are used to further optimize the design. Consequently, the values shown in Table 5.1, 5.2, and Table 5.3 should be considered to be worst case values, before further optimization is performed. Gate level VHDL code for various sizes of array and Dadda tree multipliers were generated for the conventional and proposed methods for overflow detection. The VHDL code was synthesized and optimized for area using LSI Logic's 0.6 micron gate array library and the Leonardo Synthesis tool from Exemplar logic. The synthesis tool was set to a nominal operating voltage of 5.0 volts and a temperature of 25 ffi C. Area estimates are reported in equivalent gates and delay estimates are reported in nanoseconds. Table 5.4 gives area and delay estimates for unsigned array multiplier. Compared to the multipliers that use conventional overflow detection, the proposed multipliers have between 50% and 53% less area and between 41% and 42% less delay. These Table 5.5: Unsigned Dadda Tree Multipliers with Overflow Detection. Conventional Proposed Reduction n Area Delay Area Delay Area Delay Table Signed Array Multipliers with Overflow Detection. gains are mainly due to the reductions in the area and the delay of FAs used to generate the n most significant product bits. Table 5.5 gives area and delay estimates for unsigned Dadda tree multipliers. Compared to multipliers that use conventional overflow detection method, multipliers that use the proposed method have approximately 47% less area and between 23% and 28% less delay. These improvements are due to the reducing the number of FAs and reducing the size of the final carry-propagate adder from (2n \Gamma 2) to Table 5.6 gives area and delay estimates for two's complement array multipliers. Compared to the multipliers that use the conventional method, multipliers that use the proposed method require between 38% and 47% less area and between 41% and Table 5.7: Signed Dadda Tree Multipliers with Overflow Detection. Table 5.7 gives area and delay estimates for two's complement Dadda tree mul- tipliers. Compared to the multipliers that use the conventional method, multipliers that use the proposed method have between 35% and 44% less area and between 24% to 32% less delay. Chapter 6 Conclusions and Future Research 6.1 Conclusions The overflow detection and saturation methods presented in this thesis significantly reduce the area and delay of array and tree multipliers. For the multiplier sizes examined, the area is reduced by about 50% for unsigned multipliers when compared with conventional methods. The proposed methods also do not change the regularity of the multiplier structure. For the two's complement multipliers, the proposed methods are completely independent of the multipliers' internal structure. This feature provides designers increased flexibility, since they can add overflow detection logic without effecting their original design. Reduction in the multiplier hardware will also lead to reduced power dissipation. The proposed methods reduce the delay of array multipliers by about 40% to 50%. 6.2 Future Research This thesis separately presented overflow detection and saturation methods for unsigned and two's complement parallel multipliers. An important next step is to develop a single multiplier structure that can perform both unsigned and two's complement integer multiplication with overflow detection or saturation based on an input control signal. Another area for future research is to investigate techniques for further reducing the area for overflow detection in multiplier trees, without significantly impacting the delay. This research may be able to take advantage of a hybrid structure that has less delay than linear overflow detection structures and less area than overflow detection trees. Another research area is to investigate reductions in power dissipation due to the proposed techniques. It is anticipated that a significant reduction in power dissipation can be achieved due to the reduction in multiplier hardware. Methods similar to the proposed methods can be also used for other arithmetic operations that needs overflow detection, such as multiply-accumulate and squaring. --R " Some Schemes for Parellel Multipliers," "Suggestion for a Fast Multiplier," "A 40 ns 17-bit Array Multiplier," "Parallel Reduced Area Multipliers," "A reduction Scheme to Optimize The Wallace Multiplier," "A Two's Complement Parallel Array Multiplication Algorithm," "Comments on A Two's Complement Parallel Array Multiplication Algorithm," "Synthesis and Comparision of Two's Complement Parallel Multipliers," "Computer Architecture a Quantative Approach, Second Edi- tion," "Parallel Saturating Fractional Arithmetic Units," "Fixed-point Overflow Exception Detection," "Programmable High-performance IIR Filter Chip," "Overflow Indication In Two's Complement Arith- metic," "Overflow Detection in Multioperand Addition," "Zero, Sign, and Overflow Detection Schemes For Generalized Signed Arithmetic" "A Signed Binary Multiplication Technique," --TR --CTR Eyas El-Qawasmeh , Ahmed Dalalah, Revisiting integer multiplication overflow, Proceedings of the 4th WSEAS International Conference on Software Engineering, Parallel & Distributed Systems, p.1-14, February 13-15, 2005, Salzburg, Austria Eyas El-Qawasmeh , Ahmed Dalalah, Revisiting integer multiplication overflow, Proceedings of the 4th WSEAS International Conference on Software Engineering, Parallel & Distributed Systems, p.1-14, February 13-15, 2005, Salzburg, Austria

integer;computer arithmetic;saturation;two's complement;tree multipliers;array multipliers;unsigned;overflow

350566

Computing Functions of a Shared Secret.

In this work we introduce and study threshold (t-out-of-n) secret sharing schemes for families of functions ${\cal F}$. Such schemes allow any set of at least t parties to compute privately the value f(s) of a (previously distributed) secret s, for any $f\in {\cal F}$. Smaller sets of players get no more information about the secret than what follows from the value f(s). The goal is to make the shares as short as possible. Results are obtained for two different settings: we study the case when the evaluation is done on a broadcast channel without interaction, and we examine what can be gained by allowing evaluations to be done interactively via private channels.

Introduction . Suppose, for example, that we are interested in sharing a secret le among n parties in a way that will later allow any t of the parties to test whether a particular string (not known in the sharing stage) appears in this le. This test should be done without revealing the content of the whole le, or giving any other information about the le. This problem can be viewed as an extension of the traditional denition of threshold (t-out-of-n) secret sharing schemes. In the traditional denition, sets of at least t parties are allowed to reconstruct a (previously distributed) secret s, while any smaller set gets no information about the secret. We introduce a more general denition of t-out-of-n secret sharing schemes for a family of functions F . These schemes allow authorized sets of parties to compute some information about the secret and not necessarily the secret itself. More precisely, sets B of size at least t can evaluate f(s) for any function f 2 F ; any set C of size less than t knows nothing (in the information theoretic sense) about the secret prior to the evaluation of the function; and, in addition, after f(s) is computed by a set B no set C of size less than t knows anything more about the secret s than what follows from f(s). In other words, the parties in C might know the value f(s), but they know nothing more than that, although they might have heard part of the communication during the evaluation of f(s). Clearly, if we consider a family F that includes only the identity function then we get the traditional notion of secret sharing schemes. These schemes, which were introduced by Blakley [8] and Shamir [34], were the subject of a considerable amount of work (e.g., [30, 26, 28, 6, 35, 20]). They were used in many applications (e.g., [31, 5, 15, 19]) and were generalized in various ways [22, 7, 36]. Surveys are given in [35, 37]. The question of sharing many secrets simultaneously was considered (with some dierences in the denitions) by several researchers [30, 26, 21, 11, 23, 10, 24]. Simultaneous sharing of many secrets is also a special case of our setting. 1 Other Dept. of Mathematics and Computer Science, Ben-Gurion University, Beer Sheva 84105, Israel. E-mail: beimel@cs.bgu.ac.il. http://www.cs.bgu.ac.il/beimel . This work was done when the author was a Ph.D. student in the Dept. of Computer Science, Technion. y Dept. of Mathematics, Royal Holloway { University of London, Egham, Surrey TW20 OEX, U.K. E-mail: m.burmester@rhbnc.ac.uk. z Dept. of Computer Science, PO Box 4530, 206 Love Building, Florida State University, Talla- hassee, FL 32306-4530, USA. E-mail: desmedt@cs.fsu.edu. x Dept. of Computer Science, Technion, Haifa 32000, Israel. E-mail: eyalk@cs.technion.ac.il. http://www.cs.technion.ac.il/eyalk . be the secrets we want to share simultaneously. Construct the concatenated and the functions which can be evaluated are the functions f similar scenarios in which sharing is viewed as a form of encryption and the security is computationally bounded have been considered in [32, 3, 1]. Threshold cryptography [19, 18, 17] is also a special case of secret sharing for a family of functions. 2 A typical scenario of threshold cryptography is the following: Every set B of t parties should be able to sign any document such that any coalition C of less than t parties cannot sign any other document (even if the coalition C knows signatures of some documents). To achieve this goal the key is shared in such a way that every t parties can generate a signature from their shares without revealing any information on the key except the signature. Specically, assume we have a signature is the domain of messages, K is the domain of keys, and O is the domain of signatures. For every k). The previous scenario is simply sharing the key for the family Mg. These examples show that secret sharing for a family of functions is a natural primitive. Obviously one possible solution to the problem of sharing a secret for a family F is by sharing separately each of the values f(s) (for any f 2 F) using known threshold schemes. While this solution is valid, it is very ine-cient, in particular when the size of F is large. Therefore, an important goal is to realize such schemes while using \small" shares. For example, to share a single bit among n parties, the average (over the parties) length of shares is at least log(n t log n bits are su-cient [34]. The obvious solution for sharing ' bits simultaneously will require ' log n bits. By [26], it can be shown that shares of at least ' bits are necessary. We shall show that '-bit shares are also su-cient if interactive evaluation on private channels is allowed, and O(')-bit shares are su-cient if non-interactive evaluation (on a broadcast channel) is used. 3 We present an interactive scheme in which '-bit secrets are distributed using '-bit shares; this scheme allows the computation of every linear combination of the bits (and not only computation of the bits themselves). We use this scheme to construct schemes for other families of functions. The length of the shares in these schemes can be much longer than the length of the secret. An interesting family of functions that we shall consider is the family ALL of all functions of the secret. For this family, we construct a scheme in which the length of the shares is 2 ' log n (where ' is the length of the secret and 2 ' is the length of the description of a function which is evaluated). In this scheme the computation requires no interaction and can be held on a broadcast channel. If we allow interaction on private channels during the computation, then the length of the shares can be reduced to 2 ' . Note that the obvious solution of sharing each bit separately requires in this case 2 2 ' bits. Our work deals mainly with two models: the private channels model in which the computation might require a few rounds of communication; and the broadcast channel model in which the computation is non-interactive. On one hand, the broadcast model does not require secure private channels and synchronization; hence, the computation is more e-cient. On the other hand, in the private channels model, a coalition C that does not intersect the evaluating set B will know nothing about s; not even the value f(s). Interaction seems to be useful in the computation. It enables us to reduce the The secrets s i may be dependent: our model allows some information to leak provided it is no more than what follows from the evaluations of f(s). 2 The functions considered in [19, 18, 17] are, however, very limited and the scenario in [17] is restricted to computational security. 3 In fact, both results require ' to be \su-ciently large": ' log n in the interactive case and ' log n log log n in the broadcast model. length of the shares by a factor of log n in our basic scheme for linear functions. We also study ideal threshold schemes. These are schemes in which the size of the shares equals the size of the secrets 4 (see, e.g., [12, 13, 4, 33, 25]). We deal with the characterization of the families of functions F which can be evaluated by an ideal threshold scheme. For the interactive private channel model, we prove that every boolean function that can be evaluated is a linear function. For the broadcast model, we prove that F cannot contain any boolean function (for every family that contains the identity function). An example. To motivate our approach and to clarify the notion of secret sharing for a family of functions we consider a simple example, which we discuss informally. Suppose that a dealer shares a secret s among n parties and that at some time later a set of at least t parties would like to answer the question: \Is the shared secret equal to the string a?" That is, they want to evaluate the boolean function f a which is 1 if and only if a. The string a is not known when the dealer shares the secret. We describe a simple scheme which makes it possible for the parties to answer such questions. Let be the secret bit-string. The dealer chooses random strings from f0; 1g 2' , denoted by b shares each of these strings using Shamir's t-out-of-n threshold scheme [34]. Let b i;j be the share of the secret b i given to party P j . The dealer computes the sum (i.e., bitwise exclusive-or) (that is s i , the value of the i-th bit of the secret, selects whether we take b 2i or b 2i+1 ) and gives to each party. When a subset B of (at least) t parties wants to check if s = a for some a = a 1 a computes a \new" share Then, the parties in B apply Shamir's secret reconstruction procedure to compute a \new" secret from these \new" shares. Since this procedure involves only computing a linear combination of the shares, the \new" secret is simply . If the \new" secret diers from then the parties learn that s 6= a. However, they get no more information on the secret s. If the \new" secret is equal to then it is easy to see that with high probability (over the choice of must have a. In this case, the parties learn the secret and there is no additional information which should be hidden from them. Observe that in both cases the parties learn no more than what is revealed by answering the question. Connection to private computations. In our schemes we require that authorized sets of parties can compute a function of the secret without leaking any other information on the secret. This resembles the requirement of (n; k)-private protocols [5, 15] in which the set of all n parties can evaluate a function of their inputs in a way that no set of less than k parties will gain any additional information about the inputs of the other parties. Indeed, in some of our schemes a set B of size t uses a (t; t)-private protocol. However, it is not necessary to use private protocols in the computation, since the parties are allowed to leak information about their inputs (the shares) as long as they do not leak any additional information on the secret. Moreover, using private protocols does not solve the problem of e-ciently sharing a secret for all func- tions, since not all functions can be computed (t; t) privately [5, 16]. Furthermore, the parties cannot use the (t; b(t 1)=2c)-private protocols of [5] or [15] since this means that coalitions of size greater than t=2 (but still smaller than t) gain information. Organization. The rest of this paper is organized as follows. In Section 2 we discuss our model for secret sharing for families of functions and provide the denitions; denitions which are more delicate than in the case of traditional threshold schemes. 4 The size of each share is always at least the size of the secret [26]. In Section 3 we present interactive and broadcast secret sharing schemes for the family LIN of linear functions. In Section 4 we use these schemes to construct schemes for other families. In Section 5 we describe the broadcast scheme for the bit functions BIT . In Section 6 we characterize ideal schemes in the various models. Finally, in Section 7 we discuss possible extensions of our work. For completeness we give some background results in Appendices. 2. Denitions. This section contains a formal denition of secret sharing for a family of functions. We start by dening the model. We consider a system with n parties g. In addition to the parties, there is a dealer who has a secret input s. A distribution scheme is a probabilistic mapping, which the dealer applies to the secret to generate n pieces of information s are referred to as the shares . The dealer gives the share s i to party P i . Formally, Definition 2.1 (Distribution Schemes). Let S be a set of secrets, sets of shares, and R a set of random inputs. Let be a probability distribution on R. A distribution scheme over S is a mapping . The i-th coordinate s i of (s; r) is the share of P i . The pair (; ) can be thought of as a dealer who shares a secret s 2 S by choosing at random r 2 R (according to the distribution ) and then gives each party P i as a share the corresponding component of (s; r). In the scenario we consider, the dealer is only active during the initialization of the system. After this stage the parties can communicate. We next dene our communication models. Definition 2.2 (Communication Models). We consider two models of communication 1. The private channels model in which the parties communicate via a complete synchronous network of secure and reliable point-to-point communication channels. In this model a set of parties has access only to the messages sent to the parties in the set. 2. The broadcast channel model in which the parties communicate via a (public) broadcast channel. In this model a set of parties can obtain all the messages exchanged by the communicating parties. A subset of the parties may communicate in order to compute a function of their shares. We now dene how they evaluate this function. Definition 2.3 (Function Evaluation). A set of parties B U executes a protocol FB;f to evaluate a function f . At the beginning of the execution, each party P i in B has an input s i and chooses a random input r i . To evaluate f , the parties in B exchange messages as prescribed by the protocol FB;f . In each round every party in B sends messages to every other party in B. A message sent by P i is determined by s i , r i , the messages received by and the identity of the receiver. We say that a protocol FB;f evaluates f (or computes f(s)) if each party in B can always evaluate f(s) from its input and the communication it obtained. We denote by MC (hs the messages that an eavesdropping coalition C can obtain during the communication between the parties in B (this depends on the communication model). Here hs is the vector of shares of B and hr i i B the vector of random inputs of B. A protocol is non-interactive if the messages sent by each party P i depend only on its input (and not on the messages received during the execution of the protocol). Non-interactive protocols have only one-round of communication. An interactive protocol might have more than one round. In this case we require that the protocol terminates after a nite number of rounds (that is, we do not allow innite runs). The parties in the system are honest, that is, they send messages according to the protocol. A coalition C which eavesdrops on the execution of a protocol FB;f by the parties in B gains no additional information on the secret s if any information that C may get is independent of the old information. In our model we shall allow C to gain some information, but no more than what follows from the evaluation of f . The information that C gains is determined by the view of C. Definition 2.4 (The View of a Coalition). Let C U be a coalition. The view of C, denoted VIEWC , after the execution of a protocol consists of the information that C gains. In a distribution scheme the shares of C. In the evaluation of f(s) by a set of parties B U the view of C consists of: the inputs the local random inputs hr i i C\B (only the parties in C \ B are involved in the computation), and the messages that C obtains from the communication channel during the evaluation by B, i.e., MB\C (hs That is, We now dene t-out-of-n secret sharing schemes for a family of functions F . Such schemes allow any set B of at least t parties to evaluate f(s), for any f 2 F , where s is a previously distributed secret, while any set C of less than t parties gains no more information about the secret than the information inferred from f(s). We distinguish between three types of schemes depending on the way that the value f(s) is computed by B and the channels used: (1) interactive private channels schemes { where the parties in B engage in a protocol via private channels which computes non-interactive private channels schemes { where the parties in B engage in a non-interactive protocol via private channels to compute f(s); and (3) broadcast non-interactive schemes { where each party in B broadcasts a single message (which depends only on its share) on a broadcast channel (we do not consider interactive broadcast schemes in this paper). The computation of f(s) must be secure. There are two requirements to consider: (1) the usual requirement for threshold schemes, that is, before the evaluation any set C of size less than t knows nothing about the secret by viewing their shares; (2) after the evaluation of f(s) by a set B, any set C of size less than t gains no information about s that is not implied by f(s). Formally, Definition 2.5 (Secret Sharing Schemes for a Family). A t-out-of-n secret sharing scheme for a family of functions F is a distribution scheme (; ) which satises the following two conditions: Evaluation. For any set B U of size at least t and any function f 2 F the parties in B can evaluate f(s). That is, there is a protocol FB;f which given the shares of B as inputs will always output the correct value of f(s). The scheme is called non-interactive if FB;f is non-interactive, and interactive otherwise. Depending on the communication model, a broadcast channel or private channels are used. Security. Let X be a random variable on the set of secrets S and C U be any coalition of size less than t. Prior to evaluation (after the distribution of shares). For any two secrets s; s 0 2 S and any shares hs i i C (from (s; r)): After evaluation. For any f 2 F , any B U of size at least t, any secrets s; shares hs i i C , any inputs hr i i B\C , and any messages MB\C (hs from the computation of f(s) by B: We say that C gains no information that is not implied by the function f (or simply: gains no additional information) if in the computation of f(s) we have security after evaluation. A (traditional) t-out-of-n secret sharing scheme is a secret sharing scheme for the family that includes only the identity function explained in Observation 2.1 below, the family also contains all renamings of the identity function.) Remark 2.1. We only require that one function f 2 F can be evaluated securely. A more desirable property is that the scheme is reusable: any number of functions can be evaluated securely (possibly by dierent sets). All our schemes, except for the scheme in Section 5, are reusable. Remark 2.2. In Denition 2.5 we required that all coalitions C of size less than t should gain no additional information on the secret. Clearly, it is su-cient to require this only for coalitions C of size t 1, since any smaller coalition has less information on the secret, and, hence, will gain no additional information. 5 Remark 2.3. The security in Denition 2.5 is based on the requirement that the view of a coalition is the same for secrets which have the same evaluation. Alternatively we could require: That is, the probability that the secret is s, given the view of the coalition C, equals the probability of s, given the value In Appendix A we show that the two denitions are equivalent (see [14] for the analogue claim with respect to traditional secret sharing schemes). Obviously any broadcast scheme can be transformed into a scheme in which each message is sent on private channels. On the other hand, if we take a non-interactive private channels scheme and broadcast the messages then the security of the computation may be violated. A function f 0 is a renaming of f if Suppose that f 0 is a renaming of f and that f can be evaluated securely, then f 0 can also be evaluated securely. Indeed, to evaluate f 0 securely we rst evaluate f(s) securely; after this evaluation s might not be known. Nevertheless, we can nd an input y such that Observation 2.1. A secret sharing scheme enables the secure evaluation of a renaming of f if and only if it enables the secure evaluation of f . Therefore, we ignore renamings of functions. We next dene certain classes of functions over which are of particular interest. Recall the structure of GF(2 ' ). Each element a 2 GF(2 ' ) is an '-bit string a which may be represented by the polynomial a ' 1 x Addition and multiplication are as for polynomials, using the structure of GF(2) for 5 This is dierent than what happens in private multi-party computation; there a smaller set has less inputs and hence expected to learn even less. the coe-cients and reducing products modulo a polynomial p(x) of degree ', which is irreducible over GF(2). A function for every linear function f . Thus, Observation 2.2. For every linear function f , for every x 2 GF(2 ' ), and for every a 2 GF(2) it holds that Notice that we do not require that the family LIN ' to be the family of all linear functions of is the family of additive functions). Let e be the function that returns the i-th bit of x. Then, e i linear. We call the family fe the bit functions and denote it by BIT ' . We also consider the family ALL ' of all possible functions of the secret (by Observation 2.1, without loss of generality, the range of the functions in ALL ' is in GF(2 ' )). We stress that BIT ' contains only boolean functions while both ALL ' and LIN ' contain non-boolean functions as well. We will need the following claim concerning linear functions. 2.1. For every linear functions, there are constants c ' such that string Proof. Notice that b 1 . Thus, since f is linear, and by Observation 2.2, and the claim follows. 3. Schemes for the Linear Functions. 3.1. An Interactive Scheme. In this section we show that Shamir's scheme over (described in Appendix B) is also a secret sharing scheme for the family LIN ' { the family of linear functions. Theorem 3.1 (Basic Interactive Scheme). For every ' 1 there exists an interactive private channels t-out-of-n secret sharing scheme for the family LIN ' in which the secrets have length ' and the shares have length ' 0 1)eg. Proof. The dealer uses Shamir's t-out-of-n scheme over GF(2 ' 0 to distribute the shares. We show how every function f 2 LIN ' can be evaluated securely. Consider a set B of t 0 t parties with shares hs that wishes to compute f(s). By the reconstruction procedure of Shamir's scheme, for each P exist coe-cients - j such that simply the sum of the x j 's. Computing this sum is done using the following interactive protocol of Benaloh [6] (for a more detailed description of this protocol see Appendix C). For convenience of notation, we assume that g. In the rst step of the protocol each party P sends r i;j to P j 2 B. Then, each party sends y j to P 1 . Party P 1 now computes and sends it to all the other parties. By the choice of r i;t 0 , the sum thus, each party in B computes f(s) correctly. Let us now prove the security of this scheme. The security prior to evaluation follows from the security of Shamir's threshold scheme. For the security after the evaluation it is su-cient to consider a coalition C of size t 1 (by Remark 2.2). Since each x j is computed locally, and since f(s) is computed using a private protocol [6] (for a formal denition of privacy see Appendix C), the parties in C gain no information about the shares of the other parties that is not implied by f(s). That is, for any shares we have: (1) Now for a secret s and shares hs i i C with there are unique shares hs i i BnC of Shamir's scheme that are consistent with s and hs i i C . Therefore, (2) The last equation holds because the shares hs determine the secret s. By Equation (1), the probability Pr[ hr is independent of therefore of s, and by the properties of Shamir's scheme the probability is independent of hs Therefore, by (2), for every s 0 2 S such that Pr[view and the coalition C gains no information about s that is not implied by f(s). Remark 3.1. In the special cases when interaction by simply sending the messages x j . For the messages must be sent via private channels. In this case, only coalitions C B can get messages of the evaluating set B. Let g. Then, party P i gains no additional information from the message x j of P j , because P i knows and its input x i , and therefore can compute x j . For the messages may be sent via a public broadcast channel. In this case, any coalition C is a subset of B, and if C has size n 1 then gains no additional information from x j , since C can compute x j from f(s) and Remark 3.2. When 3 t n 1, the parties evaluate f(s) using an interactive private channels scheme. It can be shown that for this particular scheme interactive evaluation is essential. Otherwise, some coalitions C that intersect the evaluating set B, but are not contained in B, will gain additional information on s when they obtain the values of x j . This scheme has the advantage that it is reusable { sets coalition C does not gain any information on the secret beside the values f i (s) for every i with C \ B i 6= ;. This is because the evaluation is done using a private protocol. The private computation assures that C cannot distinguish between two vectors of shares with the same value f i (s). Hence, even if C knows some evaluations f i (s), an evaluation of another function will not reveal extra information. 3.2. A Non-interactive Broadcast Scheme. Next, we present a broadcast scheme for the family LIN ' of linear functions, in which the length of the shares is Theorem 3.2 (Basic Broadcast Scheme). For every ' 1 there exists a non-interactive broadcast t-out-of-n secret sharing scheme for the family LIN ' in which the secrets have length ' and the shares have length ' log Proof. Let q 1)e. The dealer shares each bit of the secret independently using Shamir's scheme over GF(2 q ) (this is the smallest possible eld of characteristic two for Shamir's scheme). Let ' be the secret bit-string and let s i;j 2 be the share of b j given to party P i . Then, the total length of the share of each party is ' Clearly, every bit of the secret can be securely reconstructed. To evaluate other linear functions of the secret, we use the homomorphic property of Shamir's scheme, as observed by Benaloh [6], which enables the evaluation of linear combinations of a shared secret without revealing other information on the secret. To explain how this property is used we rst observe that it is su-cient to show the result for a boolean linear function, since each coordinate of f(s), where f 2 LIN ' , is also a linear function (we use the property that this scheme is reusable). Let f be a boolean linear function and let B be a set of (at least) t parties with shares hs i;j i P i 2B;1j' that wishes to compute the bit f(s). By Claim 2.1, there are constants c . Recall that for Shamir's reconstruction procedure the secret bit b j is computed as b are appropriate coe-cients. Let x i;j be a \new" share of P i . Then, To evaluate f(s) each party P computes locally the \new" share x i and then broadcasts it. The \new" secret f(s) is just the linear combination of these \new" shares. We claim that a coalition C of size t 1 gains no information that is not implied by f(s) during this process. Indeed, the probability of the view of C is, Pr[hs The rst equality holds because the shares of C and f(s) determine the \new" shares of B and vice-versa. The second equality holds because s determines f(s). Now Pr[hs is independent of s by the security requirement prior to eval- uation. So, we get the security requirement after evaluation. Furthermore, if this process is repeated more than once and dierent linear combinations are evaluated then C will gain no information that is not implied by these linear combinations, and the parties in B can evaluate each bit of a non-boolean linear function independently. Remark 3.3. Observe that in the broadcast scheme there is no need for the set B to be given as input for the evaluation: each party P i which is available just outputs an appropriate linear combination of its shares, and its identity P i . Then, a function can be evaluated so long as there are at least t parties that agree to evaluate the function. These parties do not need to be available at the same time, or know in advance which parties would be available. 4. Schemes for Other Families of Functions. We now show how to use the basic scheme (for linear functions) to construct schemes for other families of functions. Given a family of functions, we shall construct a longer secret such that every function in the family can be evaluated as a linear function of the longer secret. Observe that any boolean function can be represented as a binary vector over GF(2) of length 2 ' whose i-th coordinate is f(i). Similarly, any can be represented as an array of ' 0 binary vectors of length 2 ' in which the j-th vector corresponds to the j-th bit function of f(x). The rank of a family of functions F is the smallest k for which there exist boolean functions such that for every function f 2 F there exists a renaming of it which each of the ' 0 vectors representing f 0 is a linear combination of f The rank of a family does not change if we add renamings of functions. So, we can assume that F contains all renamings of its functions. Theorem 4.1. Let F be any family of functions. There exists a t-out-of-n interactive private channels secret sharing scheme for F with shares of length max frank(F); log n 1g. There exists a t-out-of-n non-interactive broadcast secret sharing scheme for F with shares of length rank(F)(log Proof. Let be a basis for the vector space spanned by the functions in F . To share a secret s the dealer generates a new secret of length k (where - denotes concatenation of strings). The dealer now shares the secret E(s) using the basic scheme (either the interactive or broadcast version, depending on the communication model). Then, f i f be the function in F to be evaluated. Without loss of generality, we may assume ) is such that the ' 0 vectors representing it are spanned by Every bit of f(s) is a linear combination of f and therefore of the bits of E(s) (i.e., e i (E(s))). Since addition in GF(2 k ) is bitwise, concatenation of linear functions is a linear function too. Thus, f(s) can be computed by evaluating a linear function of E(s). Thus, Theorem 4.1 follows by Theorems 3.1 and 3.2. We demonstrate the applicability of the above construction by considering the family ALL of all possible functions of the secret (boolean and non-boolean). Corollary 4.2. There exists a t-out-of-n interactive private channels secret sharing scheme for ALL ' with shares of length max There exists a t-out-of-n non-interactive broadcast secret sharing scheme for ALL ' with shares of length Proof. By Theorem 4.1 we have to show that the rank of ALL ' is 2 ' 1. Let 1. Consider the d boolean functions f i: To evaluate a boolean function f we notice that the functions f(x) and f(x) are renamings of each other, so we can assume that a basis for ALL ' . In this case, the secret s is encoded as the vector E(s) of length d in which the s-th coordinate is 1 and all the other coordinates are zero (with the exception Notice that the length of the secret is ', while the length of the shares is (2 ' 1). However, there are 2 2 ' 1 dierent boolean functions with domain of cardinality 2 ' such that every function is not a renaming of another function (leaving aside non-boolean functions). Therefore, our scheme is signicantly better than the naive scheme in which we share every function (up to renaming) separately. Also the length of E(s) for ALL ' must be at least log so the representation for E(s) that we use is the best possible for the family ALL ' in this particular scheme. It remains an interesting open question whether there exists a better scheme for this family, or whether one can prove that this scheme is optimal. 5. Non-interactive Broadcast Scheme for the Bit Family. In this section we present a non-interactive broadcast scheme for the family of bit functions, whose shares are of length O(') (compared to O(' log n) of the scheme from Theorem 3.2). Theorem 5.1. Let ' 195 n log log n). There exists a non-interactive broadcast t-out-of-n secret sharing scheme for the family BIT ' in which the length of the secrets is ' and the length of the shares is O('). We rst present a meta scheme (Section 5.1). Then we show a possible implementation of the meta scheme that satises the conditions of the Theorem. 5.1. A Meta Scheme. Let k and h be integers (to be xed later) such that 1)e and the secret s has length ' kh. We view the secret as a binary matrix with k rows and h columns and denote its (i; entry by s[i; j]. We construct, in a way to be specied below, a new binary matrix H with 3k rows and h columns. Then we share every row of H using Shamir's t-out-of-n scheme over GF(2 h ). Since the length of every row is h dlog(n 1)e, Shamir's scheme is ideal and every party gets 3k shares of length h. The distribution stage of this scheme is illustrated in Fig. 1. When a set B of cardinality (at least) t wants to reconstruct the bit s[i; j] of the secret, the parties in B reconstruct a subset T i;j of rows (which depends only on and not on B). We will guarantee that these rows do not give any additional information on the secret. Each party Shamir Fig. 1. An illustration of the meta scheme of the non-interactive broadcast scheme for the bit family. More specically, for every 1 i k and 1 j h, we x a set T i;j (independently of the secret). The j-th column of H is constructed independently, and depends only on the j-th column of the secret. It is chosen uniformly at random among the column vectors such that: To reconstruct s[i; j] it is enough to reconstruct the T i;j -rows of H , and compute the sum (modulo 2) of the j-th bit of the reconstructed rows. (Actually, every party can sum the shares of the T i;j -rows of H , and then the parties reconstruct the secret from these shares. The j-th bit of the reconstructed secret equals to s[i; j].) That is, the message of a party in a reconstructing set consists of the shares corresponding to the T i;j -rows of H . The existence of a matrix H satisfying Equation (3), and the security requirement after reconstruction depend on the choice of the sets T i;j . On the other hand, independently of the choice of the sets T i;j , any coalition of size less than t prior to any reconstruction does not have any information on the rows of H (by the properties of Shamir's scheme), and, hence, does not have any information on the secret. That is, the meta scheme is secure prior to the evaluation. 5.2. Implementing the Meta Scheme. We show how to construct sets T i;j such that the security requirement after reconstruction will hold. Let R 1 3kg be a collection of dierent sets of size k. In particular, no set is contained in another set (R 1 h is a Sperner family [2]). For The number of sets of cardinality k which are contained in fk xing k 4 su-ces. We require that h dlog hk ', and choose the smallest possible h. The length of the shares (i.e., 3h dlog he) is ('). The following Claims are useful for proving the security of a reconstruction: 5.1. For every secret s, the number of matrices H that satisfy Equation (3) for all j (1 j h) is at least 1 and is independent of s. Proof. Set H [i; Equation (3). To show that the number of matrices satisfying Equation (3) is independent of s, notice that H is a solution of a non-homogeneous system of linear equations. Since the system has at least one solution, the number of such matrices is the number of solutions to the homogeneous linear system of equations, where every To show that no coalition gains any additional information, we rst consider the case in which the coalition knows only the T i;j -rows of H . In this case, the coalition can reconstruct s[i; j]. We prove that the coalition does not gain any information on the other bits of the secret (i.e., bits s[i 5.2. Let s be a secret and x the T i;j -rows of H such that The number of matrices H that satisfy Equation (3) for all j (1 j h) is at least 1 and is independent of s. Proof. We rst prove that there exists a matrix H satisfying the requirements. Since the columns of H are constructed independently, it is enough to prove the existence of every column j 0 . Only the j 0 -th column of the T i;j -rows of H , and the -th column of s in uence the j 0 -th column of H , hence, while considering the j 0 -th column we can ignore the rest of the columns. We rst consider the j-th column of H (i.e., we can assign H [i 0 ; j] a value as follows: for every i 0 such that k (this is arbitrary); for every i 0 such that 1 i 0 k set The case j 0 6= j is similar except that we have to be careful about there is an element . We can assign H [i a value as follows: for every i 0 such that k We have shown that there exists a matrix as required. By the same arguments as in the proof of Claim 5.1, the number of possible matrices given a secret s, is independent of s. We are now ready to prove the security requirement (note that this scheme is not reusable). 5.3. The above implementation of the meta scheme is secure. Proof. Let B be any set that evaluates a bit s[i; j] of the secret, and C be any coalition. The security requirement before reconstruction is easy (this was discussed at the end of Section 5.1). Suppose that C has obtained the messages sent by the parties in B. That is, the parties in C know the shares of the parties of B corresponding to the T i;j -rows of H . Therefore, the only information that they gain is the T i;j -rows of H . Then, given the shares of the coalition and all the messages that were sent, every matrix H that agrees with these rows is possible. We need to prove that, given two secrets in which the (i; j)-th bit is the same, and a matrix H in which only the are xed: the probability that H was constructed from each secret is equally likely. By Claim 5.1 and Claim 5.2 this probability only depends on s[i; j]. 6. Characterization of Families with Ideal Schemes. In this section we consider ideal secret sharing schemes for a family F . We say that two secrets s; s 0 are distinguishable by F if there exists a function f 2 F for which f(s) 6= f(s 0 ). A secret sharing scheme for F is ideal if the cardinality of the domain of shares equals the cardinality of the domain of distinguishable secrets. These are the shortest shares possible by [26]. The following theorem gives several impossibility results for ideal schemes for a family F which contains the bit functions. Theorem 6.1 (Characterization Theorem). Suppose that there is an ideal interactive t-out-of-n secret sharing scheme for a family of functions F such that BIT ' F . Then, any boolean function f 2 F is linear. ng. Suppose that there is an ideal non-interactive t-out-of-n secret sharing scheme for a family of functions F such that BIT ' F . Then, F LIN ' . (3) Let 3 t n 1. In an ideal t-out-of-n secret sharing the evaluation of every non-constant boolean function requires interaction on private channels. 2. In an ideal non-interactive 2-out-of-n secret sharing every non-constant boolean function cannot be evaluated via a broadcast channel. It follows that for families F , with BIT ' F , if interaction is allowed then every boolean function in F is linear. If interaction is not allowed, then either the functions in F are linear. In particular when must be used. In the Section 6.1 we will discuss some basic properties of ideal schemes that will be needed for the proof of this Theorem. In Section 6.2 we prove (1) and (2), and in Section 6.3 we prove (3) and (4). 6.1. Properties of Ideal Schemes. In this section we discuss some simple properties of ideal schemes which are useful in the sequel. Proposition 6.2 ([26]). Fix the shares of any t 1 parties in an ideal t-out-of-n secret sharing scheme. Then, the share of any other party P is a permutation of the secret. In particular all shares are possible for P . In our proof, we use the following proposition regarding private functions, that is functions that can be computed without revealing any other information on the inputs. formal denition of privacy the reader is referred to Appendix C.) Bivariate boolean private functions were characterized by Chor and Kushilevitz [16]: Proposition 6.3 ([16]). Let A 1 ; A 2 be nonempty sets and f : A 1 A 2 ! f0; 1g be an arbitrary boolean function. Then, f can be computed privately if and only if there exist boolean functions such that for every holds that f(x; 6.1. Let f be a boolean function that can be evaluated securely in an ideal 2-out-of-2 secret sharing scheme. Then, there exist boolean functions f 1 and f 2 such that are the shares of P 1 and P 2 respectively. Proof. The function f is evaluated from x and y, i.e., there is some function f 0 such that Proposition 6.2 any information that a party gets on the share of the other party is translated to information on the secret. That is, the parties must compute f 0 (x; y) in such a way that each party receives only information implied by f 0 (x; y). In other words, they compute the boolean function f 0 privately. By Proposition 6.3 there exist f 1 and f 2 such that Consider an ideal t-out-of-n secret sharing scheme for F . Without loss of gener- ality, we may assume that all the secrets are distinguishable by F . So, the parties can reconstruct the secret from their shares. Denote the reconstructing function by h(s We next prove an important property of the messages sent in a non-interactive protocol that evaluates f . 6.2. Let f be a function that can be securely evaluated in an ideal t-out- of-n secret sharing scheme without interaction. Without loss of generality, we may assume that all the messages sent by P i , while holding the share s i , to the other parties are identical, and equal to | {z } | {z } Proof. It is su-cient to prove the claim for P 1 . Suppose that the shares of the parties are s Consider a possible message of P 1 while the set computes the value f(s). Party P 1 might toss coins, so if there are several messages choose the rst one lexicographically. When P 1 holds two shares s 1 and s 0 1 such that f(h(s 1 the messages of P 1 to P i have to be dierent, otherwise P i will compute an incorrect value in one of the cases. On the other hand, for two shares s 1 and s 0 1 such that f(h(s 1 the messages sent by P 1 to the parties in fP have to be the same in both cases, otherwise the coalition fP can distinguish between the secrets loss of generality, the messages sent by P 1 while holding the share s 1 are identical and equal to 6.2. Proofs of (1) and (2) of Theorem 6.1. The proofs of Items (1) and (2) are similar, and have two stages. We consider the following two-party distribution scheme over GF(2 ' ), denoted XOR. Given a secret s, the dealer chooses at random The share of the rst party is x, and the share of the second party is x. In the rst stage (Section 6.2.1) we characterize the functions that can be evaluated from the shares of XOR. In the second stage (Section 6.2.2) we show that if there exists an ideal t-out-of-n secret sharing scheme for F , then XOR is a secret sharing scheme for F . The combination of the two stages implies Items (1) and (2). 6.2.1. Characterizing the Functions which can be Evaluated with XOR. We rst characterize the functions that can be evaluated with XOR without interac- tion. It can be seen that every linear function of the secret can be evaluated with XOR (the details are the same as in Remark 3.1). We prove that no other functions can be evaluated securely without interaction. Then we characterize the boolean functions that can be evaluated with XOR with interaction and show that these functions are exactly the boolean linear functions. To prove the characterization without interaction, we will prove that f(s) can be computed from f(x) and f(y) (where x and y are the shares). In the next claim we prove that every function with this property is a renaming of a linear function. 6.3. function. If there exists a function g such that f(x is a renaming of a linear function. Proof. Let f(0)g. We rst prove that The \only if" direction follows Similarly the \if" direction follows from the following simple equations: That is, if In other words the set X 0 is a linear space over the eld GF(2). Now consider a linear transformation whose null space is X 0 (that is, f is a linear space, such linear transformations exists. We claim that f is a renaming of f 0 , i.e., Equation (4), only if (by the denition of f 0 ). Since f 0 is linear, f only if 6.4. Let f be any function that can be evaluated without interaction with the scheme XOR. Then, f is a renaming of a linear function. Proof. Assume, without loss of generality, that f : consider the function of x dened as min fy which is a renaming of f ). By Claim 6.3 it su-ces to prove that f(x + y) can be computed from f(x) and f(y). Denote the share of P 1 by x and the share of P 2 by y. Recall that y. By 6.2 the message sent by P 2 while holding a share y to P 1 is f(y), and the message sent by P 1 while holding a share x to P 2 is f(x). Hence, P 1 can compute y) from x and f(y). Moreover, for every two shares x 1 and x 2 held by P 1 such that every share y held by P 2 , party P 1 must compute the same value of f(s), since P 2 receives the same message from P 1 in both cases, and therefore computes the same value of f(s). Hence, P 1 can compute f(x + y) from f(x) and f(y). The computed function is the desired function g with f(x By Claim 6.3 this implies that f is a renaming of a linear function. We now prove a similar claim for interactive evaluations. However, this claim applies only to boolean functions. 6.5. Let f be a boolean function that can be evaluated interactively with the scheme XOR. Then, f is a renaming of a linear function. Proof. Since XOR is an ideal scheme, Claim 6.1 implies that there exists boolean functions In particular there exists a bit b such that is a linear function which is a renaming of f . The exact family of functions which can be evaluated interactively with XOR consists of the two-argument functions for which can be computed privately from x and y (as characterized in [29]). 6.2.2. Reduction to XOR. In this section we prove that if there exists an ideal t-out-of-n secret sharing scheme for a family F , then there exists an ideal 2-out-of-2 secret sharing scheme for F . We then prove that this implies that XOR is a secret sharing scheme for F . 6.6. If there exists an interactive (respectively non-interactive) ideal t- out-of-n secret sharing scheme for F , then there exists an interactive (respectively non-interactive) ideal 2-out-of-2 secret sharing scheme for F . Proof. Let be a set of parties. We ignore the shares distributed to parties not in B. Therefore, we have an ideal t-out-of-t secret sharing scheme for F . Let hs (xed) vector of shares that is dealt to B with positive probability. To share a secret s the dealer now generates a random vector of shares of s (according to the scheme) that agrees with s respectively, and gives the rst two components of this vector to P 1 and P 2 . Parties P 1 and P 2 know the shares of the other parties (as they are xed) and therefore P 1 , say, can simulate the parties P in the protocol which evaluates the function f 2 F . Hence, P 1 and P 2 can evaluate f(s) from their shares and the messages they exchange. On the other hand, P 1 has no more information than the information known to the coalition of cardinality t 1 in the t-out-of-t scheme. So, P 1 does not gain additional information on s. Similar arguments hold for P 2 . This implies that this scheme is a 2-out-of-2 secret sharing scheme for F . 6.7. Let BIT ' F . If there exists an interactive (respectively non- ideal t-out-of-n secret sharing scheme for F , then XOR is an interactive (respectively non-interactive) secret sharing scheme for F . Proof. By Claim 6.6 we can assume that there exists an ideal 2-out-of-2 secret sharing scheme for F , say . We transform into XOR in a way that every function that can be evaluated in can also be evaluated in XOR. 6.1 implies that for every j, where 1 j ', there exist boolean functions 2 such that e j (y). For as the concatenation of the values of these functions, that is m i (x) 4 every secret s and random input r of the dealer Therefore, the scheme 0 dened by 0 (s; is a distribution scheme equivalent to XOR. We still have to show that the two parties can evaluate securely every function in F with 0 . We rst prove that m 1 is invertible. Clearly, the two parties can reconstruct the secret in 0 , while every single party knows nothing about the secret. Therefore, 0 is a 2-out-of-2 secret sharing scheme and, by [26], the cardinality of the domain of shares is at least the cardinality of the domain of secrets. Thus, the cardinality of the range of m 1 is at least as large as 2 ' . The domain of m 1 , which is the set of shares of P 1 , has cardinality 2 ' . Therefore, m 1 is a bijection and, hence, invertible. So, P 1 can reconstruct the share x from m 1 (x). Similarly, P 2 can reconstruct the share y from m 2 (y). This implies that the parties P 1 and P 2 , while holding m 1 respectively, can evaluate every function f 2 F . We have proved that every function can be evaluated with 0 which is equivalent to XOR. Furthermore, if the evaluation of the original scheme required no interaction, then also the evaluation with the XOR scheme requires no interaction. 6.3. Proofs of (3) and (4) of Theorem 6.1. Next we prove Item (4). Namely, in any ideal non-interactive t-out-of-n secret sharing scheme (2 t n 1) evaluation of a boolean function requires private channels. The proof is by contradiction. Suppose that there is an ideal non-interactive t-out-of-n scheme in which the evaluation can be held on a broadcast channel. Then, by the same arguments as in Claim 6.6, since t n 1, there is an ideal 2-out-of-3 scheme with the same property. Our rst claim is that, without loss of generality, in the evaluation of f(s) every party sends a one bit message such that the sum of the messages is f(s). 6.8. Assume there exists an ideal 2-out-of-3 secret sharing scheme in which a non-constant boolean function f can be securely evaluated without interaction via a broadcast channel. Then, P 1 and P 2 can securely compute f(s) by sending one Proof. By Claim 6.2 we can assume that the message of P 1 while holding the share s 1 is the bit Similarly, we assume that the message of P 2 while holding the share s 2 is Furthermore, the value of f(s) is determined by these two messages. There are two values for f(s) and two values for each message. Since each party knows nothing about f(s), a change of any message will result in a change of f(s). The only boolean functions with two binary variables satisfying this requirement are addition modulo 2, i.e., or loss of generality, assume that P 1 sends the message the claim follows. 6.9. Consider an ideal non-interactive 2-out-of-3 secret sharing scheme for a family F which contains a non-constant boolean function f . Then, f cannot be evaluated via a broadcast channel. Proof. Fix the share of P 3 to be z = 0. By Proposition 6.2, the share x of P 1 is a permutation of the secret. Assume, without loss of generality, that this share is equal to the secret. (Of course P 1 does not know that z = 0 and does not know the secret.) Since P 3 does not know f(s) although z = 0, the value of f(s) is not xed, and the message that P 1 sends to P 2 while P 1 and P 2 evaluate f(s) cannot be constant will not be able to evaluate f(s)). Furthermore, for any two shares held by P 1 such that the messages of P 1 to P 2 should be the same (otherwise P 3 will distinguish between the two secrets). Without loss of generality, the message m 1 which P 1 sends to P 2 is f(x). By Claim 6.8, the message that P 2 sends to P 1 is constant and P 1 will not be able to evaluate f(s), which leads to a contradiction. Hence, f(s) cannot be evaluated on a broadcast channel. We conclude this section by proving Item (3). That is, for 3 t n 1, the evaluation requires interaction via private channels. Again, we assume towards contradiction that there exists an ideal non-interactive t-out-of-n scheme for the family F , and construct an ideal non-interactive 3-out-of-4 scheme for the family F . We show that this contradicts Claim 6.9. 6.10. Assume there is an ideal 3-out-of-4 secret sharing scheme for a family F in which a function f 2 F can be evaluated via private channels without interaction. Then, there is an ideal 2-out-of-3 secret sharing scheme in which f can be evaluated via a broadcast channel. Proof. By Claim 6.2 we may assume that the messages sent during the evaluation by one party to the other two parties are identical. In particular, if fP evaluates a function, then P i knows the messages that P To construct a 2-out-of-3 secret sharing scheme, we x a share u for P 4 , and share the secret s with a random vector of shares such that the share of P 4 is u. When P i and P j want to evaluate f(s), where 1 i < j 3, then P i simulates P 4 , and all messages and broadcasted. After this evaluation, every party P k , where k 2 f1; 2; 3g, will have the same information as the coalition fP had in the original scheme. Hence, P k does not gain any information that is not implied by f(s). That is, we get a 2-out-of-3 secret sharing scheme for F . 7. Discussion. In this section we consider possible extensions of our work. So far, we considered only threshold secret sharing schemes. In [22] a general notion of secret sharing schemes for arbitrary collections of reconstructing sets is dened. That is, we are given a collection A of sets of parties called an access structure and require that any set in A can reconstruct the secret, while any set not in A does not know anything about the secret. Clearly, secret sharing schemes exist only for monotone collections. Conversely, it is known that for every monotone collection there exists a secret sharing scheme [22]. More e-cient schemes for general access structures were presented (e.g., in [7, 36]). However, the length of the shares in these schemes can be exponential in the number of parties (i.e., of length '2 (n) where n is the number of parties in the system and ' is the length of the secret). Our denition of secret sharing for a family F of functions can be naturally generalized for an arbitrary access structure. To construct such schemes, observe that the schemes of [22, 7, 36] are \linear": the share of each party is a vector of elements over some eld, and every set in A reconstructs the secret using a linear combination of elements in their shares. Thus, if we share every bit of the secret independently, we can evaluate every linear function of the secret without any interaction (the details are as in Theorem 3.2). This implies that for every access structure, there exists a scheme for the family ALL ' in which the length of the shares is O(2 ' 2 n ). However, if the access structure has a more e-cient linear scheme for sharing a single bit then the length of the shares can be shorter (but at least 2 ' ). Another possible extension of the denition is by allowing a weaker \non-perfect" denition of security. Such extensions in the context of traditional secret sharing schemes were considered, e.g., in [9]. As remarked, there are several related works on simultaneous secret sharing. For example, in [23] each set of cardinality at least t should be able to reconstruct some of the secrets (but not all). In [11] each set B of cardinality at least t is able to reconstruct any of the distributed secrets. 6 The security requirement considered in [11, 24, 10] is somewhat weak: after revealing one of the secrets, limited information about other secrets may be leaked. Our scheme for the family of functions ALL ' uses shares of size 2 ' . We can construct an e-cient scheme if we relax the security requirement as follows: every set of size at least t can evaluate any function f(s) in a way that every coalition of at most b(t 1)=2c parties gains no information on the secret (but larger coalitions may get some information). In this case, we can use Shamir's t-out-of-n scheme to distribute the secret and the interactive protocol of [5] or [15] to securely compute f(s). In this scenario, the length of the share is the same as the length of the secret. However, for most functions f , the length of the communication in the evaluation of f is exponential in '. 6 In fact [11] deals with more general access structures and not only threshold schemes. Acknowledgment . The authors would like to thank the referees for several helpful comments. --R On hiding information from an oracle Combinatorial Theory Hiding instances in multioracle queries Universally ideal secret sharing schemes Completeness theorems for noncrypto- graphic fault-tolerant distributed computations Keeping shares of a secret secret in Advances in Cryptology - CRYPTO '88 The security of ramp schemes Some ideal secret sharing schemes On the classi Some improved bounds on the information rate of perfect secret sharing schemes How to share a function securely in Advances in Cryptology - AUSCRYPT '92 Shared generation of authenticators and signatures Communication complexity of secure computation sharing schemes realizing general access structure in Advances On secret sharing systems Generalized linear threshold scheme Privacy and communication complexity Communications of the ACM On data banks and privacy homomor- phisms How to share a secret An introduction to shared secret and/or shared control and their application The geometry of shared secret schemes An explication of secret sharing schemes --TR --CTR Christian S. Collberg , Clark Thomborson, Watermarking, tamper-proffing, and obfuscation: tools for software protection, IEEE Transactions on Software Engineering, v.28 n.8, p.735-746, August 2002

secret sharing;private computations;private channels;interaction;broadcast channel

350571

The Online Transportation Problem.

We study the online transportation problem under the assumption that the adversary has only half as many servers at each site as the online algorithm. We show that the GREEDY algorithm is $\Theta( {\rm min}(m, \lg C))$-competitive under this assumption, where m is the number of server sites and C is the total number of servers. We then present an algorithm BALANCE, which is a simple modification of the GREEDY algorithm, that is, O(1)-competitive under this assumption.

Introduction We consider the natural online version of the well-known transportation problem [2, 5]. The initial setting consists of a collection of server sites in a metric space M. Each server site s j has a positive integral capacity c j . The online algorithm A sees over time a sequence ng of requests for service, with each request being a point in M. In response to the request r i , A must select a site s oe(i) to service r i . The cost for this assignment is the distance in the metric space between s oe(i) and r i . Each site s j can service at most c j requests. The dilemma faced by the online algorithm A is that, at the time of the request r i , A is not aware of the location of the future requests. The goal for the online algorithm is to minimize 1 cost to service the requests. Note that this is equivalent to minimizing the total cost For concreteness, consider the following two examples of online transportation problems. In the fire station problem, the site s j is a fire station that contains c j fire crews. Each request is a fire that must be handled by a fire crew. The problem is to assign the crews to the fire so as to minimize the average distance traveled to get to a fire. In the school assignment problem, the site s j is a school that can has a capacity of c j students. Each request is a new student who moves into the school district. The problem is to assign the children to a kalyan@cs.pitt.edu, Computer Science Dept., University of Pittsburgh, Pittsburgh, PA 15260, Supported in part by NSF under grant CCR-9202158. y kirk@cs.pitt.edu, Computer Science Dept., University of Pittsburgh, Pittsburgh, PA 15260, Supported in part by NSF under grant CCR-9209283. school so as to minimize the average distance traveled by the children to reach their schools. The standard measure of "goodness" of an online algorithm is the competitive ratio. For the online transportation problem, the competitive ratio for an online algorithm A is the supremum over all possible instances I, of A(I)=OPT (I), where A(I) is the total cost of the assignment made by A, and OPT (I) is the total cost of the minimum cost assignment for instance I. The standard way to interpret the competitive ratio is as a payoff of a game played by the online algorithm A against an all powerful adversary that specifies the requests, and services them in the optimal way. Note that the instance I specifies the metric space as well as the values of each s j , c j , and r i . In [1, 3] the online assignment problem, a special case of online transportation in which each capacity c In [1], it was shown that the competitive ratio of the intuitively appealing greedy algorithm, which assigns the nearest available server site to the new request, has a competitive ratio of In [1, 3], it was shown that the optimal deterministic competitive ratio is 1. The algorithm that achieves this competitive ratio requires a shortest augmenting path computation for each request. These results illustrate some shortfalls of using competitive analysis, namely: ffl The achievable competitive ratios often grow quickly with input size, and would seem to overly pessimistic for "normal" inputs. ffl The algorithm that achieves the optimal competitive ratio is often unnecessarily complicated for "normal" inputs. ffl The poor competitive ratio of an intuitive greedy algorithm may not reflect the fact that it may perform reasonably well on "normal" inputs. In situations where competitive analysis suffers such shortcomings, it is important to find alternate ways to to identify online algorithms that would work well in practice. In this paper, we adopt a modified version of competitive analysis that we call the weak adversary model. Generally speaking, in this model the adversary is given slightly less resources than the online algorithm. The intuition is that for "normal" inputs, one might expect that the performance of an offline algorithm would not degrade significantly if its resources were slightly reduced. Hence, if we can prove that an online algorithm is competitive against an adversary with slightly less resources, then one might argue that the online algorithm will be competitive against an equivalently equipped offline algorithm on "normal" inputs. One can also view this weakening of adversary as measuring the additional resources required by the online algorithm to offset the decrease in performance due to the online nature of the problem. In the case of the transportation problem we compare the online algorithm with c i servers at s i to an offline line algorithm with a servers at s i (we assume that c i is even). Given an instance I of the online transportation problem with n requests, I 0 be the same instance with each capacity c i replaced by a i . We then say the halfopt-competitive ratio of an online algorithm A is the supremum over all instances I, with of the ratio A(I)=OPT (I 0 ). We assume that the online algorithm has twice as many servers as the adversary because this is the least advantage that we can give to the online algorithm without annulling our analysis techniques. In this paper we present the following results. In section 3, we show that the halfopt-competitive ratio of the greedy algorithm is \Theta(min(m; log C)), where is the sum of the capacities. If the server capacity of each site is constant, then the halfopt-competitive ratio is logarithmic in m, a significant improvement over the exponential bound on the traditional competitive ratio. In section 4, we describe the algorithm Balance, which is a simple modification of the greedy algorithm, and has a halfopt-competitive ratio that is O(1). Recall that the traditional competitive ratio of every deterministic online algorithm is m). We now summarize related results. The weakened adversary model was introduced in [6] in the context of studying paging. This model has also been used to study variants of the k-server problem, a generalization of the paging problem (see for example [8]). References to other other suggested variants of competitive analysis can be found in [4]. Further ancillary results on online assignment, which are not directly related to the results in this paper, can be found in [1]. In [7], the average competitive ratio for the greedy algorithm in the online assignment problem is studied under the assumption that the metric space is the Euclidean plane and the points are uniformly distributed in a unit square. The offline transportation problem can be solved in polynomial-time[2, 5]. Preliminaries In this section we introduce some definitions, facts, and concepts that are common to the remaining sections. We generally begin by assuming the simplifying condition that the online capacity c i of each server site is two. We will think of s i as containing two online servers s 1 i that move to service requests. We also think of s i as containing one adversary server s a i . We assume, without loss of generality, that the adversary services request r i with s a i . We use s oe(i) to denote the site that the online algorithm uses to service request r i . We define a weighted bipartite graph E), which we call the response graph, by including an online edge (r and an adversary edge (r request r i . The weight of each edge (r G is the distance d(r the r i and s j in the underlying metric space. In figure 1, the online edges are the solid edges, the adversary edges are the dashed edges, the server sites are the filled circles, and the requests are the question marks. The notation shown in figure 1 will be used throughout this paper. request vertex that is in a cycle in G. Let T be the connected component of G \Gamma (s contains r i . Then T is a tree. Proof: Consider a breadth first search of T starting from r i . Such a search divides T into levels L where the vertices in level L k are k hops from r a s a s b r c r s r r s s d d e e f f Figure 1: An Example Connected Component of the Response Graph r i . The vertices in the odd levels are server vertices, and the vertices in even levels are request vertices. Edges that go from an even level to an odd level are adversary edges, and edges that go from an odd level to an even level are online edges. Assume to reach a contradiction that C is a cycle in T . Let y be the vertex in C that is in the highest level L c (i.e., largest c), and let x and z be the two vertices adjacent to y in C. Note that it may be the case that x = z. Now it must be the case that x and z are in L c\Gamma1 or we would get a contradiction to one of the bipartitieness of G, the definition of x, or the definitions of the levels. If c is odd, then we get a contradiction to the fact that the adversary has only one server per site. If c is even, we get a contradiction to the fact that the online algorithm only uses one server to service each request. Let T be a tree as described in lemma 1. If we root the tree T at r i then T has the following structure. For each request r j 2 T , the one child of r j is s j . If r j is not the root, then the server site s oe(j) is the parent of r j in T . The leaves of are server sites with no incident online edges. We denote the total cost of the adversary edges in a tree T by OPT (T ), that is, OPT Analogously, we define ON Note that ON (T ) includes the cost of the online edge incident to the root of T , even though this edge is not in T . For a vertex x 2 T , we define the leaf distance ld(x) to be the minimum over all leaves s j in T of the distance between x and s j . If a server at site s j serviced the root r i of T , then ld(s x is a node in T , we define T (x) to the the subtree of T rooted at x. In this paper, log means the logarithm base 2. 3 Analysis of the Greedy Algorithm We begin with the upper bound on the competitive ratio for the algorithm Greedy, which uses the nearest available server to service each request. We first assume that the online capacity of each server site is two, and then show how to extend the proof to the general case. Theorem 2 The halfopt-competitive ratio of Greedy for online transportation, with and two online servers per site, is at most 2 log m. In order to prove this theorem, we will divide the response graph G into edge disjoint rooted trees, T l . For each such tree, we will establish the competitive bound independently. Our construction yields trees (T j 's) that satisfy the following tree invariants: 1. Each nonleaf server site s i in T j has two incident online edges in T j . 2. Each leaf of T j is a server site that had an unused server at the time of each request in T j . Using the following iterative procedure to construct the trees. Tree Construction Procedure: Assume that trees T constructed. We explain how to construct T j in our next iteration. During this construction we will modify G. The root of T j is the most latest request r ff(j) not included in a previous tree T . The online edge incident to r ff(j) is removed from G. Let L be the collection of server vertices s i such that s i is reachable from r ff(j) , and s i currently has at most one incident online edge in G. Note that an s i 2 L might have originally had two incident online edges if one or both of these edge lead to the root of one of the trees T T j be the edges on paths in G from r ff(j) to the server vertices in L. Note that by lemma 1 there is a unique path from r ff(j) to each vertex in L. It is not hard to see that T j satisfies the tree invariants. The edges and request vertices in are then removed from G, and we proceed to our next iteration to construct contains edges. We now fix a particular tree, say simplicity drop the j superscript. Lemma 3 For each request r Proof: The proof is by induction on the number k of request nodes in the induced tree T (r i ). If then the child s i of r i is a leaf, and T (r i ) consists of one adversary edge (r by the definition of Greedy. Now suppose k ? 1. If s i is not a leaf then it has two children in T (r i ), say r a and r b (see figure 1). By induction, it must be the case that ld(r a OPT (T (r a )), d(s OPT (T (r b )). Therefore by induction and the triangle inequality, The fact that d(s since Greedy would only assign s oe(i) to r i if d(s Lemma 4 Let r i be a request in T , and let k be the number of request vertices in Proof: The proof is by induction on k. Observe that according to our tree construction, each server node has either two children or none. Therefore, k must be odd. For by the definition of Greedy. Now consider the case 3. Let the two children of s i in T (r i ) be r a and r b . Note that ON (T (r and that By the definition of Greedy, d(s lemma 3, d(s Hence, by substitution, ON (T (r i Now consider the case k ? 3. Once again let the two children of s i in T (r i ) be r a and r b . Notice that ON (T (r By equation 1, Hence, ON (T (r i loss of generality, that OPT (T (r a y be the number of request nodes in T (r a ). We now break the proof into cases. In the first case, we assume that both T (r a ) and T (r b ) consist of more than one request vertex. Hence, 3 - y 4. By induction ON (T (r a and ON (T (r b Hence, by substituting into equation 2 we get ON (T (r i in order to show ON (T (r i OPT (T (r i )) it is sufficient to show 2x log y In turn it is sufficient to show that x log y Let f(x; x be the left hand side of this inequality. Notice that f(x; y) is linear in x. Hence the maximum of f(x; y) must occur at the boundary, that is, at the point or the point Inequality 3 follows immediately for we have to find the value of y that maximizes f(w=2; 1). The derivative of f(w=2; y) with respect to y is w y k\Gamma1\Gammay )). Hence, one can see that the maximum f(w=2; y) occurs at 1)=2. We now must show that which by algebraic simplification is equivalent to log(k \Gamma 1) - log k. We now consider the case that T (r a ) contains only one request. So By induction ON (T (r a by substituting into equation 2 we get ON (T (r i In order to show that ON (T (r i is sufficient to show Since the left hand side is linear in x, we need only consider The case immediately. If are left with showing k. This is equivalent to log(2 log k, or which one can verify holds for k - 3. We now consider the case that T (r b ) contains only one request. So 2. By induction ON (T (r a by substituting into equation 2 we get ON (T (r i In order to show that ON (T (r i is sufficient to show that Since the left hand side is linear in x, one need only verify that the inequality holds at the boundaries Proof: (of Theorem 2.) Applying lemma 4 to each tree T i , we get the desired result. We now extend the result to the case that the online capacities are larger than two. Recall that is the total online capacity. Theorem 5 The halfopt-competitive ratio of Greedy for online transportation is O(min(m; log C)). Proof: The upper bound of O(log C) is immediate by theorem 2 if we conceptually split a server site with c i online servers into c i =2 sites with 2 arbitrary online servers and 1 arbitrary adversary server. To see the O(m) bound we need to be more careful about how we split the server sites up. Assume that the tree construction procedure just constructed a tree T k . We perform some pruning of sT k , if necessary, before we proceed to construct T k+1 . If no root-to-leaf path in T k passes through two server sites that are at the same location, then the number of vertices in T k is O(2 m ). Hence, the O(m) bound follows from lemma 4. If T k contains root-to-leaf paths that pass through two server sites that are at the same location, we show how to modify T k to remove such paths. Assume that T k contains a root-to-leaf path that first passes through s i and then passes through s j , where s i and s j are at the same location. We modify T k by making server vertex s j the child of r i in T k . See figure 2. Note that may remove edges and vertices originally below s i from T k . We repeat this process until T k has no root to leaf path passing through two server sites at the same location. Notice that the resulting tree T k still satisfies the tree invariants. Now we start the construction of T k+1 . Figure 2: The original T k on the left, and the new T k on the right We now prove an asymptotically matching lower bound for the halfopt- competitive ratio for Greedy. Theorem 6 The halfopt-competitive ratio of Greedy for the transportation problem is \Omega\Gamma/15 (m; log C)). Proof: Assume without loss of generality that C). We embed m server sites on the real line. The server site s 1 is located at the point \Gamma1. The server site s i m) is located at 2 1. The online algorithm has servers at s i , while the adversary has a servers at s i . Thus the online algorithm has a total of servers, while the adversary has a total of servers. The requests occur in m batches. The first batch consists of 2 m\Gamma1 requests at the point 0. The ith batch (2 - i - m) contains requests that occur at 2 the location of s i . Greedy responds to batch m) by answering each request in batch i with a server at site s i+1 , thus depleting site s i+1 . Greedy responds to the mth batch by moving one server from s 1 . Thus the total online cost is m2 . By using the servers in s i to handle batch i (1 - i - m) it is possible to obtain a total cost of 2 4 The Algorithm Balance In this section we present an algorithm, Balance, with a halfopt-competitive ratio of O(1). Algorithm Balance: At each site s h we classify half of the servers as primary and half of the servers as secondary. Let c ? 5 2 be some constant. Define the pseudo-distance from a request r i to a primary server at site s j to be d(s and the pseudo-distance from r i to a secondary server at site s j to be c \Delta d(s Balance services each request r i with an arbitrary server with minimal pseudo-distance from r i . Our goal is now to show that the halfopt-competitive ratio of Balance for online transportation, with two online servers per site, is O(1). We first break the response graph G into disjoint trees. Let C l be the connected components of the response graph G. By lemma 1 each connected component of the response graph contains a unique cycle. Let r ff(i) be the most recent request in the cycle in C i . Let T j be the tree that is C j minus the online edge incident to r ff(j) , and we set the root of T j to be r ff(i) . Each such tree T j satisfies the following two tree invariants: 1. Each nonleaf server site s incident online edge in T , had the secondary server available just before the time of each request 2. Each leaf of T j is a server site s i that had both of its servers available just before the time of each request in T j . We now fix a particular tree, say simplicity drop the superscript j. In order to show that the halfopt-competitive ratio of Balance is O(1) it is sufficient to show that ON (T Definition 7 Let s i be a generic server site in T . We say that the primary server child s a of s i is the server site that the adversary used to service the request serviced by s 1 , and the secondary server child s b of s i is the server site that the adversary used to service the request serviced by s 2 . The server parent s p of s i is the server site used by Balance to service r i . The site s i is a double if it has two server children, and otherwise s i is a single. Lemma 8 If Balance uses a server at site s p to handle a request r and Proof: Observe that there is a path in T from r i to some leaf s k with total length at most d(r Balance didn't use s k to service r i . If s by the triangle inequality. Now Lemma 9 Assume that Balance uses a secondary server s 2 p to handle a request that is not the root of T . Then and Proof: Observe that there is a path in T from r i to some leaf s k with total length at most d(r Balance didn't use the primary server at s k to service r i . Now Fact 11 For all nonnegative reals x and y, and for all c ? 1, Proof: Suppose 2x - (1 + 1=c)y. It suffices to show that min(2x; (1 shows that (1 On the other hand, suppose 1=c)y. It suffices to show that c+1 x). Simple algebra shows that Lemma 12 Assume that s i is a double server site with server children s a and s b . Then Proof: Note that ld(s i by lemma 8, and that ld(s i 9. The lemma then follows by fact 11. local if either s i is a single, or one of s i 's server children is a single. Otherwise, s i is global. For convenience, we call a request r i local if s i is local. Otherwise r i is global. For a server site s i we use z(s p ) to denote We now break the accounting of the online edges in T into cases. We first show that for every local server s i , the cost for Balance to serve r i is O(z(s p )). local r i Further for each parent server s p of a local server s i Proof: We break the proof into two cases. In the first case assume that s i is a single. It must be the case that d(s Balance didn't use a server at s i to handle r i . Hence, single local s i single local s i Further In the second case assume that s i is double, and one of s i 's server children is a single. For simplicity assume that s a is a single; one can verify that the following argument also holds if s b is a single. Since s a is a single we can apply the analysis from the previous case to get that d(s Balance serviced r i with a server at s p instead of the unused server at s a we get that Hence, by substitution, Thus X double local s i double local s i double local s i Furthermore, The result then follows. Now we consider the online cost of servicing the global r i 's. Observe that by lemma 8 it is the case that global r i 2T global r i 2T global s i 2T We now will show that global s i 2T Definition 15 Let s i be a global server site in T . We define S(s i ) to be the set of global server sites in the subtree of T rooted at s i with the property that, for any server site s j 2 S(s i ), there are no global server sites in the unique path from s i to s j in T . We define LC(s i ) to be the sum of the cost of the offline edges the subtree rooted at s i with the property that the unique path from s i to r j in T does not pass through a global server site. To give an alternative explanation of LC(s i ) consider pruning T at the global server vertices, which results in a collection of trees rooted at global server vertices. Then LC(s i ) is the total cost of the offline edges in the tree rooted at Lemma 16 For any global server site s i , Proof: If the server children s a and s b of s i are both global then Hence, the result follows in this case since fl - Now assume that at least one server child, say s a , of s i is local. Note that s a and s b must be a double, since s i is global. Further assume for the moment that s b of s i is global. Let s c and s d be the two server children of s a . Then using equation 5 we get that In this case we continue to expand ld(s c ) and ld(s d ) using the general expansion method described below. Now consider the case that both s a and s b are local. Let s e and s f be the server children of s b . Then using equation 5 we get that In this case we continue to expand ld(s c ), ld(s d ), ld(s e ) and ld(s f ) using the general expansion method described below. We now describe the general expansion method. We expand each ld(s j ) using equations 4 and 6. A term of the form ld(s j ) is expanded according to the following rules: 1. If s j is a leaf in T then ld(s j ) is set to 0. 2. If s j has a local server child s k , then ld(s j ) is set to which is valid by equation 6. Observe that since s k is local it will be subsequently expanded. 3. If none of s j 's server children are local then ld(s j ) is set to 2(d(r is an arbitrary server child of s k . This is valid by equation 4. Note that in this case the term ld(s k ) is not expanded again since The that appear in this general expansion process are all included in LC(s i ). Hence, we get for some ff and fi. Since each offline edge appears in this general expansion at most twice, and in each case the coefficient is at most 1), we can conclude that ff - 1). Note that in rule 2, the coefficient in front of the ld(s j ) term before the expansion is the same as the coefficient in front of the ld(s k ) term after expansion. Since the coefficient in front of each ld term is fl 2 before the application of any of the general expansion rules, and the only way that it can change is by application of rule 3, which is a terminal expansion, each coefficient on a ld(s j ) term when general expansion terminates is at most 2fl 2 . The result then follows. Lemma global s i 2T Proof: First observe that if s i and s j are two different global server sites, then there is no common offline cost in the sums LC(s i ) and LC(s j ). As a consequence we get X global s i 2T Therefore, it suffices to show that global s i 2T global s i 2T Applying lemma 16 we have that global s i 2T global s i 2T global s i 2T we get that global s i 2T global s i 2T Now substituting back, we get global s i 2T global s i 2T The result then follows. Theorem The halfopt-competitive ratio of Balance, with two online servers per site, in the online transportation problem is Proof: The total online cost is (c 2 +c+1)OPT (T ) from lemma 14, plus OPT (T ), plus We now assume an arbitrary number of online servers per site. Theorem 19 The halfopt-competitive ratio of Greedy for the online transportation problem is Proof: This follows immediately by conceptually splitting each server site s i into sites. 5 Conclusion The most obvious avenue for further investigation is to determine the competitive ratio in the weakened adversary model when the adversary's capacity is more than half of the online capacity. It seems that some new techniques will be needed in this case since the response graph no longer has the treelike property from lemma 1 that was so critical in our proofs. --R "Online weighted matching" Algorithms for Network Programming "On-line algorithms for weighted matchings and stable marriages" "Beyond competitive analysis" Networks and Matroids "Amortized efficiency of list update and paging rules" "Average performance of a greedy algorithm for the on-line minimum matching problem on Euclidean space" "The k-server dual and loose competitiveness for paging," --TR --CTR Adam Meyerson , Akash Nanavati , Laura Poplawski, Randomized online algorithms for minimum metric bipartite matching, Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, p.954-959, January 22-26, 2006, Miami, Florida Wun-Tat Chan , Tak-Wah Lam , Hing-Fung Ting , Wai-Ha Wong, A unified analysis of hot video schedulers, Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, May 19-21, 2002, Montreal, Quebec, Canada

matching;online algorithms;competitive analysis

350710

Superlinear Convergence of an Interior-Point Method Despite Dependent Constraints.

We show that an interior-point method for monotone variational inequalities exhibits superlinear convergence provided that all the standard assumptions hold except for the well-known assumption that the Jacobian of the active constraints has full rank at the solution. We show that superlinear convergence occurs even when the constant-rank condition on the Jacobian assumed in an earlier work does not hold.

Introduction . We consider the following monotone variational inequality over a closed convex set C ae Find z 2 C such that (z (1) and the set C is defined by the following algebraic inequality: . The mapping \Phi is assumed to be C 1 (continuously differentiable) and monotone; that is, while each component function g i (\Delta) of g(\Delta) is convex and twice continuously differentiable By introducing g(\Delta) explicitly into the problem (1), we obtain the following mixed nonlinear complementarity (NCP) problem: Find the vector triple (z; -; y) 2 IR n+2m such that y \Gammag(z) (2) is the C 1 function defined by It is well known [3] that, under suitable conditions on g such as the Slater constraint qualification, z solves (1) if and only if there exists a multiplier - such that (z; -) solves (2). To show superlinear (local) convergence in methods for nonlinear programs, on eusually makes several assumptions with regard to the solution point. Until recently, these assumptions included (local) uniqueness of the solution (z; -; y). This uniqueness condition was relaxed somewhat in [6] to allow for several multipliers - corresponding Department of Mathematics, The University of Melbourne, Parkville, Victoria 3052, Australia. The work of this author was supported by the Australian Research Council. y Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, U.S.A. This work was supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W-31-109-Eng-38. to a locally unique solution z of (1), by introducing a constant rank condition on the gradients of the constraints g i that are active at z . The point of this article is to show that superlinear convergence holds in the previous setting [6] even when the constant rank condition does not hold. This result lends theoretical support to our numerical observations [6, Section 7]. Moreover, we believe that the superlinear convergence result can be shown for other interior-point methods whose search directions are asymptotically the same as the pure Newton (affine-scaling) direction defined below (6). Briefly stated, the assumptions we make to obtain the superlinear results are as follows: monotonicity and differentiability of the mapping from (z; -) to (f(z; -); \Gammag(z)), such that the partial derivative with respect to z is Lipschitz near z ; a positive definiteness condition to ensure invertibility of the linear system that is solved at each iteration of the interior-point method; the Slater constraint qualification on g; existence of a strictly complementary solution; and a second-order condition that guarantees local uniqueness of the solution z of (1). A formal statement of these assumptions and further details are given in Section 2.2. Superlinear convergence has been proved for other methods for nonlinear programming without the strict complementarity as- sumption, but these results typically require the Jacobian of active constraints to have full rank (see Pang [5], Bonnans [1], and Facchinei, Fischer, and Kanzow [2]). Possibly the best known application of (1) is the convex programming problem defined by min z OE(z) subject to z 2 C; DOE. It is easy to show that the formulation (2),(3) is equivalent to the standard Karush-Kuhn-Tucker (KKT) conditions for (4). If a constraint qualification holds, then solutions of (4) correspond, via Lagrange multipliers, to solutions of (2)-(3) and, in addition, solutions of (1) and coincide. We consider the solution of (1) by the interior-point algorithm of Ralph and Wright [6], which is in turn a natural extension of the safe-step/fast-step algorithm of Wright [7] for monotone linear complementarity problems. The algorithm is based on a restatement of the problem (2) as a set of constrained nonlinear equations, as \Gamma\LambdaY e5 =4 r f (z; -) r g (z; y) where the residuals r f and r g are defined in an obvious way. All iterates (z satisfy the positivity conditions strictly; that is, (- The interior-point algorithm can be viewed as a modified Newton's method applied to the equality conditions in (5), in which search directions and step lengths are chosen to maintain the positivity condition on (-; y). Near a solution, the algorithm takes steps along the pure Newton direction defined by4 D z f Dg T 0 \Delta- r g (z; y) The solution (\Deltaz; \Delta-; \Deltay) of this system is also known as the affine-scaling direction. The duality measure defined by is used frequently in our analysis as a measure of nonoptimality and infeasibility. To extend the superlinear convergence result of [6] without a constant rank condition on the active constraint Jacobian, we show that the affine-scaling step defined by (6) has size O(-). Hence, the superlinearity result can be extended to most algorithms that take near-unit steps along directions that are asymptotically the same as the affine-scaling direction. Since we are extending our work in [6], much of the analysis in that earlier paper, much of the analysis in the earlier work carries over without modification to the present case, and we omit many of the details here. We focus instead on the main technical result needed to prove fast local convergence-the estimate (\Deltaz; \Delta-; the affine-scaling step-and restate just enough of the earlier material to make the current note self-contained. 2. The Algorithm. In this section, we review the notation, assumptions, and the statement of the algorithm from Ralph and Wright [6]. We also state the main global and superlinear convergence results, which differ from the corresponding theorems in [6] only in the absence of the constant rank assumption. 2.1. Notation and Terminology. We use S to denote the solution set for (2), and S z;- to denote its projection onto its first For a particular z to be defined in Assumption 4, we define We can partition f1; into basic and nonbasic index sets B and N such that for all solutions (z The solution (z strictly complementary if - for all We use -N and -B to denote the subvectors of - that correspond to the index sets N and B, respectively. Similarly, we use DgB (z) to denote the jBj \Theta n row submatrix of Dg(z) corresponding to B. Finally, if we do not specify the arguments for functions g, Dg, f , and so on, they are understood to be the appropriate components of the current point (z; -; y). The notation Dg refers to Dg(z ). 2.2. Assumptions. Here we give a formal statement of the assumptions needed for global and superlinear convergence. Some motivation is given here, but we refer the reader to the earlier paper [6] for further details. The first assumption ensures that the mapping f defined by (3) is monotone with respect to z and therefore that the mapping (z; -) ! (f(z; -); \Gammag(z)) is monotone. Assumption 1. \Phi : and each component function The second assumption requires positive definiteness of a certain matrix projec- tion, to ensure that the coefficient matrix of the Newton-like system to be solved for each step in the interior-point algorithm is nonsingular (see (11)). Assumption 2. The two-sided projection of the matrix D z f(z; onto ker Dg(z) is positive definite for all z 2 IR n and - that is, for any basis Z of ker Dg(z), the matrix Z T D z f(z; -)Z is invertible. Note that this assumption is trivially satisfied when the nonnegativity condition z - 0 is incorporated in the constraint function g(\Delta). We assume, too, that the Slater condition holds for the constraint function g. Assumption 3. There is a vector - z 2 C such that g(-z) ! 0. Next, we assume the existence (but not uniqueness) of a strictly complementary solution. Assumption 4. There is a strictly complementary solution (z The strict complementarity condition is essential for superlinear convergence in a number of contexts besides NCP and nonlinear programming. See, for example Wright [8, Chapter 7] for an analysis of linear programming and Monteiro and Wright [4] for asymptotic properties of interior-point methods for monotone linear complementarity problems. Next, we make a smoothness assumption on \Phi and g in the neighborhood of the first component z of the strictly complementary solution from Assumption 4. (We show in [6, Lemma 4.2] that, under this assumption, z is the first component of all solutions.) Assumption 5. The matrix-valued functions D\Phi and D 2 g i , are Lipschitz continuous in a neighborhood of z . Finally, we make an invertibility assumption on the projection of the Hessian onto the kernel of the active constraint Jacobian. This assumption is essentially a second-order sufficient condition for optimality. Assumption 6. Let z be defined as in Assumption 4, and let B, S z;- and S - be defined as in Section 2. Then for each - 2 S - , the two-sided projection of D z f(z ; -) onto ker(Dg In the statements of our results, we refer to a set of "standing assumptions," which we define as follows: Standing Assumptions: Assumptions 1-6, together with an assumption that the algorithm of Ralph and Wright [6] applied to the problem (2) generates an infinite sequence f(z with a limit point. Along with Assumptions 1-6, the superlinear convergence result in Ralph and Wright [6] requires a constant rank constraint qualification to hold. To be specific, the analysis of that paper requires the existence of an open neighborhood U of z such that for all matrix sequences fH k g ae fDgB (z) T index sets J ae we have that However, in the analysis of [6], this assumption is not invoked until Section 5.4, so we are justified in reusing many results from earlier sections of that paper here. Indeed, we also reuse results from later sections of [6] by applying them to constant matrices (which certainly satisfy the constant rank condition). The algorithm makes use of a family of defined for positive parameters and fi as follows: In particular, the kth iterate (z belongs to \Omega\Gamma chooses the sequences ffl k g and ffi k g to satisfy Given the notation it is easy to see that Since all iterates (z belong to \Omega\Gamma and since the residual norms kr f k and kr g k are bounded in terms of - for vectors in this set, we are justified in using - alone as an indicator of progress, rather than a merit function that also takes account of the residual norms. We assume that the sequence of iterates has a limit point, which we denote by By [6, Theorem 3.2], we have that (z ; -; y S. We are particularly interested in points in\Omega that lie close to this limit point, so we define the near-solution neighborhood S(ffi) by 2.3. The Algorithm. The major computational operation in the algorithm is the repeated solution of 2m-dimensional linear systems of the form4 D z f Dg T 0 \Delta- r g (z; y) where the centering parameter ~ oe lies in the range [0; 1 2 ]. These equations are simply the Newton equations for the nonlinear system of equality conditions from (2), except for the ~ oe term. The algorithm searches along the direction (\Deltaz; \Delta-; \Deltay) obtained from (11). At each iteration, the algorithm performs a fast step along a direction obtained by solving (6) (or, equivalently, (11) with ~ We choose the neighborhood\Omega k+1 to be strictly larger than\Omega k (by appropriate choice of fl k+1 and fi k+1 ), thereby allowing a nontrivial step ff k to be taken along this direction without leaving\Omega k+1 . If the fast step achieves at least a certain fixed decrease in -, it is accepted as the new iterate. Otherwise, we reset\Omega k+1 /\Omega k and defne a safe step by solving (11) with ~ oe chosen in the range [-oe; 1) for some constant - We perform a backtracking line search along this direction, stopping when we identify a value of ff k that achieves a "sufficient decrease" in - without leaving the set\Omega k+1 . The algorithm is parametrized by the following quantities whose roles are explained more fully in [6]. where exp(\Delta) is the exponential function. The constants fi min and fl max are related to the starting point (z The main algorithm is as follows. terminate with solution (z else Although we may calculate both a fast step and a safe step in the same iteration, the coefficient matrix in (11) is the same for both steps, so the coefficient matrix is factored only once. The safe-step procedure is defined as follows. choose ~ oe 2 [-oe; 1], ff solve (11) to find (\Deltaz; \Delta-; \Deltay); choose ff to be the first element in the sequence ff such that the following conditions are satisfied: return (z(ff); -(ff); y(ff)). The fast step routine is described next. solve (11) with ~ to find (\Deltaz; \Delta-; \Deltay); set ~ define choose ff to be the first element in the sequence ff such that the following conditions are satisfied: return (z(ff); -(ff); y(ff)). 2.4. Convergence of the Algorithm. The algorithm converges globally according to the following theorem. Theorem 2.1. (Ralph and Wright [6, Theorem 3.2]) Suppose that Assumptions 1 and 2 hold. Then either limit points of f(z belong to S. Here, however, our focus is on the following local superlinear convergence theorem. It is simply a restatement of [6, Theorem 3.3] without the constant rank condition on the active constraint Jacobian matrix [6, Assumption 7]. Theorem 2.2. Suppose that Assumptions 1, 2, 3, 4, 5, and 6 are satisfied and that the sequence f(z is infinite, with a limit point (z ; -; y S. Then the algorithm eventually always takes fast steps, and (i) the sequence f- k g converges superlinearly to zero with Q-order at least 1 and (ii) the sequence f(z converges superlinearly to (z ; -; y ) with R-order at least 1 - . The proof of this result follows that of the earlier paper in all respects except for the estimate for the affine-scaling step calculated from (6). The remainder of this section is devoted to proving that this estimate holds under the given assumptions. 3. An O(-) Estimate for the Affine-Scaling Step. Our strategy for proving the estimate (12) for the step (6) is based on a partitioning of the right-hand side in (6). The following vectors are useful in defining the partition. (13a) (13c) (13d) where z is defined in Assumption 4 and (z ; -) is the projection of the current point (z; -) onto the set S z;- of (z; -) solution components. The right-hand side of (6) can be partitioned as4 r f r g \Gamma\LambdaY e5 =4 j f \Gamma\LambdaY e5 +4 ffl f We define a corresponding splitting of the affine-scaling step: the following linear systems:4 D z f (Dg) T 0 We define a third variant on (6) as follows:4 D z f (Dg \Gamma(Dg c \Deltaz c \Delta- \Deltay7 5 =4 - and split the step ( c \Deltaz; c \Delta-; c \Deltay) as \Deltaz; c \Delta-; c t; ~ u; ~ v) v) and ( ~ t \Gamma(Dg ~ u \Gamma(Dg ~ Because of Assumption 2, the matrices in (15), (16), (17), (19), and (20) are all invertible, so all these systems have unique solutions. Our basic strategy for proving the estimate (12) is as follows. From [6, Section 5.3], we have without assuming the constant rank condition that positive constant. The constant rank assumption is, however, needed in [6] to prove that the other step component (t; u; v) is also O(-). In this article, we obtain the same estimate without the constant rank assumption, by proving that \Deltaz; c \Delta-; c \Deltay for all (z; -; y) 2 S(ffi). Our first result, proved in the earlier paper [6], collects some bounds that are useful throughout this section. Lemma 3.1. [6, Lemma 5.1] Suppose that the standing assumptions hold. Then there is a constant C 1 such that the following bounds hold for all (z; -; y) 2 S(1): (22a) (22c) Lemma 3.1 implies that the limit point (z ; - defined in (9) has The second result is as follows. Lemma 3.2. (cf. [6, Lemma 5.2]) Suppose that the standing assumptions are satisfied. Then there are constants - such that for all (z; -; y) 2 \Deltaz; c \Delta-; c \Deltay) of (17) and ( ~ t; ~ u; ~ v) of (19) satisfy and respectively. Proof. We claim first that the right-hand-side components of (17) and (19) are O(-). From [6, Equation (78)], we have that for some positive constants C 2;1 and 1). By Lipschitz continuity (Assump- tion 5), the definitions (8) and (10), and the fact that f(z ; - -) is defined in (13), there are constants where L denotes the Lipschitz constant of Assumption 5. (The radius ffi 3 is chosen so that lies inside the neighborhood of Assumption 5.) For the second right-hand side component, we have simply that after a possible adjustment of C 2;2 . For the remaining right-hand-side component in (17), we have trivially that Consider now the system (17). As in the proof of [6, Lemma 5.2], we have that eliminating the c \Deltay component we obtain \Deltaz c \Delta- We reduce the system further by eliminating the vector c \Delta- N to obtain (D z f) (Dg \Gamma(Dg \Deltaz c One can easily see that the right-hand-side vector in this expression is O(-), because of (27), (28) and the bounds (22), which imply that y B , -N , and N are all O(-) for (z; -; y) 2 By using the estimate \Gamma1 3.1 again) and recalling the notation for the limit point of the sequence, we have for (z; -; y) 2 S(ffi 3 ) that D z f(z ; -) (Dg \Gamma(Dg \Deltaz c f O(-k c O(-k c denotes the right-hand-side vector in (29). By partitioning c \Deltaz into its components in ker Dg B and ran (Dg we have from Assumption 6 that c \Deltaz is bounded in norm by the size of the right-hand side in (30). Hence, there is a constant C 2;3 such that for all (z; -; y) 2 S(ffi 3 ). By choosing - enough that C 2;3 - :5 for the result (24) follows from some simple manipulation of the inequality above. The proof of (25) is similar. The next result and others following make use of the positive diagonal matrix D defined by From Lemma 3.1, there is a constant C 3 such that for all (z; -; y) 2 S(1). Lemma 3.3. (cf. [6, Lemma 5.3]) Suppose that the standing assumptions are satisfied. Then for - defined in Lemma 3.2, there is C 4 ? 0 such that for all (z; -; y) 2 S( - Proof. First, let - ffi be defined in Lemma 3.2, but adjusted if necessary to ensure that The proof closely follows that of [6, Lemma 5.3], but we spell out the details here because the analytical techniques are also needed in a later result (Theorem 3.8). Recall the splitting (18) of the step ( c \Deltaz; c \Delta-; c \Deltay) into components ( ~ t; ~ defined by (19) and (20), respectively. By multiplying the last block row in (20) by \Gamma1=2 Y \Gamma1=2 and using (31), we find that e: Using (20) again, we obtain (D z f) ~ since D z f is positive semidefinite by Assumption 1. Hence, by taking inner products of both sides in (35), we obtain and therefore For ( ~ t; ~ u; ~ v), the third block row in (19) implies that Therefore, we have \Gamma(Dg ) ~ (D z f) ~ where again we have used monotonicity of D z f . Define the constant C 4;1 as From (25), (27), (32), and (34), we have for (z; -; y) 2 S( - ffi ) that From (28) and (32), we have By substituting the last two bounds into (37), we obtain It follows from this inequality by a standard argument that for some constant C 4;2 depending only on C 4;1 , and - . By combining this bound with (18) and (36), we obtain \Delta-k and the first part of (33) follows if we define C the second part of (33) follows likewise. Bounds on some of the components of c \Delta- and c \Deltay follow easily from Lemma 3.3. Theorem 3.4. (cf. [6, Theorem 5.4]) Suppose that the standing assumptions are satisfied. Then there are positive constants - ffi and C 5 such that \Delta- \Deltay Proof. Let - ffi be as defined in Lemma 3.3. From the definition (31) and the bounds (33), we have c for any i 2 N . Hence, by using (22), we obtain min which proves that k c \Delta- for an obvious choice of C 5 . The bound on c \Deltay B is derived similarly. Lemma 3.5. (cf. [6, Lemma 5.10]) Let ; 6= J ae B and ; 6= K ae N . If the two-sided projection of D z f(z; -) onto ker Dg B is positive definite, then for t 2 IR n and -J 2 IR jJ j , we have that (D z f) (Dg \GammaDg if and only if In addition, we have that dimker (D z f) (Dg \GammaDg 0 \GammaI \DeltaK Proof. This result differs from [6, Lemma 5.10] only in that z replaces z as the argument of Dg(\Delta). The proof is essentially unchanged. By Assumptions 5 and 6, the two-sided projection of D z f(z; -) onto the kernel of (Dg positive definite for all (z; -; y) sufficiently close to the limit point (z ; -; y ) defined in (9). It follows from Lemma 3.5 and (23) that the set ae- D z f(z; -) (Dg \Gamma(Dg oe has constant column rank for some - Theorem 3.6. Suppose that the standing assumptions hold. Then there is a positive constant ~ ffi such that for all (z; -; y) 2 S( ~ ffi ), we have that ( c \Deltaz; c \Deltay N ) is the solution of the following convex quadratic program: min subject to \Gamma(Dg uB \Delta- N I \DeltaB c \Deltay B Moreover, there is a constant C 6 such that \Deltay \Deltay B )k: Proof. The value ~ ffi from (39) and - ffi from Theorem 3.4, suffices to prove this result. The technique of proof is by now familiar (it follows the proof of [6, Theorem 5.12] closely), and we omit the details. At this point, we have proved the first estimate in (21), as we summarize in the following theorem. Theorem 3.7. Suppose that the standing assumptions hold. Then there are constants ~ 7 such that for any (z; -; y) 2 S( ~ ffi ) we have \Deltaz; c \Delta-; c Proof. Let ~ ffi be as defined in Theorem 3.6. From Theorem 3.4, (27), and (28), we have for (z; -; y) 2 S( ~ \Deltay Hence, from (41) we have also that \Deltay and it follows from (24) that k c Our last result is concerned with the second estimate in (21) involving the relationship between (t; u; v) and ( c \Deltaz; c \Delta-; c \Deltay). Theorem 3.8. Suppose that the standing assumptions hold. Then there are positive constants ffi ? 0 and C 8 such that \Deltay for all (z; -; y) 2 S(ffi). Proof. By taking differences of (15) and (17), we obtain4 D z f (Dg) T 0 c c c \Deltay \Delta- \Deltaz We have from (26), Lipschitz continuity of Dg(\Delta) (Assumption 5), and Theorem 3.7 that there is a radius ffi 4 2 (0; ~ \Delta- \Deltaz3 The remainder of the proof follows that of [6, Lemma 5.7]. By applying the technique used in Lemma 3.2 to the system (42), and using the estimate (43), we have that there are constants Next, we note that the technique used in the second half of the proof of Lemma 3.3 can be used to prove that there is \Deltay where D is the diagonal scaling matrix defined in (31). Modifications are needed only to account for the different right-hand side estimate (43) and the different estimate (44) of k c we omit the details. From (32) and (45), it follows immediately that \Deltay The final estimate for ( c obtained by substituting these expressions into (44). Corollary 3.9. Suppose that the standing assumptions hold. Then there are constants 9 such that the affine-scaling step defined by (6) satisfies Proof. We have from Theorems 3.7 and 3.8 that (t; u; defined as in Theorem 3.8. Moreover, it follows directly from [6, Section 5.3] that possibly after some adjustment of ffi . Hence, the result follows from (14). 4. Conclusions. The result proved here explains the numerical experience reported in Section 7 of Ralph and Wright [6], in which the convergence behavior of our test problems seemed to be the same regardless of whether the active constraint Jacobian satisfied the constant rank condition. We speculated in [6] about possible relaxation of the constant rank condition and have verified in this article that, in fact, this condition can be dispensed with altogether. Our results are possibly the first proofs of superlinear convergence in nonlinear programming without multiplier nondegeneracy or uniqueness. --R Local study of Newton type algorithms for constrained problems On the accurate identification of active constraints A survey of theory Local convergence of interior-point algorithms for degenerate monotone LCP Convergence of splitting and Newton methods for complementarity problems: An application of some sensitivity results Superlinear convergence of an interior-point method for monotone variational inequalities --TR --CTR Hiroshi Yamashita , Hiroshi Yabe, Quadratic Convergence of a Primal-Dual Interior Point Method for Degenerate Nonlinear Optimization Problems, Computational Optimization and Applications, v.31 n.2, p.123-143, June 2005 Lus N. Vicente , Stephen J. Wright, Local Convergence of a Primal-Dual Method for Degenerate Nonlinear Programming, Computational Optimization and Applications, v.22 n.3, p.311-328, September 2002 Huang , Defeng Sun , Gongyun Zhao, A Smoothing Newton-Type Algorithm of Stronger Convergence for the Quadratically Constrained Convex Quadratic Programming, Computational Optimization and Applications, v.35 n.2, p.199-237, October 2006

interior-point method;superlinear convergence;monotone variational inequalities

350715

An Inexact Hybrid Generalized Proximal Point Algorithm and Some New Results on the Theory of Bregman Functions.

We present a new Bregman-function-based algorithm which is a modification of the generalized proximal point method for solving the variational inequality problem with a maximal monotone operator. The principal advantage of the presented algorithm is that it allows a more constructive error tolerance criterion in solving the proximal point subproblems. Furthermore, we eliminate the assumption of pseudomonotonicity which was, until now, standard in proving convergence for paramonotone operators. Thus we obtain a convergence result which is new even for exact generalized proximal point methods. Finally, we present some new results on the theory of Bregman functions. For example, we show that the standard assumption of convergence consistency is a consequence of the other properties of Bregman functions, and is therefore superfluous.

Introduction In this paper, we are concerned with proximal point algorithms for solving the variational inequality problem. Specifically, we consider the methods which are based on Bregman distance regularization. Our objective is two- fold. First of all, we develop a hybrid algorithm based on inexact solution of proximal subproblems. The important new feature of the proposed method is that the error tolerance criterion imposed on inexact subproblem solution is constructive and easily implementable for a wide range of applications. Sec- ond, we obtain a number of new results on the theory of Bregman functions and on the convergence of related proximal point methods. In particular, we show that one of the standard assumptions on the Bregman function (con- vergence consistency), as well as one of the standard assumptions on the operator defining the problem (pseudomonotonicity, in the paramonotone operator case), are extraneous. Given an operator T on R n (point-to-set, in general) and a closed convex subset C of R n , the associated variational inequality problem [12], from now on VIP(T; C), is to find a pair x and v such that where h\Delta; \Deltai stands for the usual inner product in R n . The operator stands for the family of subsets of R n , is monotone if for any x; y 2 R n and any u 2 T (x), v 2 T (y). T is maximal monotone if it is monotone and its graph G(T contained in the graph of any other monotone operator. Throughout this paper we assume that T is maximal monotone. It is well known that VIP(T; C) is closely related to the problem of finding a zero of a maximal monotone operator - Recall that we assume that T is maximal monotone. Therefore, (2) is a particular case of VIP(T; C) for On the other hand, define NC as the normal cone operator, that is NC The operator T +NC is monotone and x solves VIP(T; C) (with some v 2 only if Additionally, if the relative interiors of C and of the domain of T intersect, then T +NC is maximal monotone [31], and the above inclusion is a particular case of (2), i.e., the problem of finding a zero of a maximal monotone operator. Hence, in this case, VIP(T; C) can be solved using the classical proximal point method for finding a zero of the operator - . The proximal point method was introduced by Martinet [26] and further developed by Rockafellar [34]. Some other relevant papers on this method, its applications and modifications, are [27, 33, 3, 29, 25, 17, 18, 15]; see [24] for a survey. The classical proximal point algorithm generates a sequence fx k g by solving a sequence of proximal subproblems. The iterate x k+1 is the solution of regularization parameter. For the method to be imple- mentable, it is important to handle approximate solutions of subproblems. This consideration gives rise to the inexact version of the method [34], which can be written as where e k+1 is the associated error term. To guarantee convergence, it is typically assumed that (see, for example, [34, 8])X Note that even though the proximal subproblems are better conditioned than the original problem, structurally they are as difficult to solve. This observation motivates the development of the "nonlinear" or "generalized" proximal point method [16, 13, 11, 19, 23, 22, 20, 6]. In the generalized proximal point method, x k+1 is obtained solving the generalized proximal point subproblem The function f is the Bregman function [2], namely it is strictly convex, differentiable in the interior of C and its gradient is divergent on the boundary of C (f also has to satisfy some additional technical conditions, which we shall discuss in Section 2). All information about the feasible set C is embedded in the function f , which is both a regularization and a penalization term. Properties of f (discussed in Section 2) ensure that solutions of subproblems belong to the interior of C without any explicit consideration of constraints. The advantage of the generalized proximal point method is that the subproblems are essentially unconstrained. For example, if VIP(T; C) is the classical nonlinear complementarity problem [28], then a reasonable choice of f gives proximal subproblems which are (unconstrained!) systems of nonlinear equa- tions. By contrast, subproblems given by the classical proximal algorithm are themselves nonlinear complementarity problems, which are structurally considerably more difficult to solve than systems of equations. We refer the reader to [6] for a detailed example. As in the case of the classical method, implementable versions of the generalized proximal point algorithm must take into consideration inexact solution of subproblems: In [14], it was established that if exists and is finite , (3) then the generated sequence converges to a solution (provided it exists) under basically the same assumptions that are needed for the convergence of the exact method. Other inexact generalized proximal algorithms are [7, 23, 41]. However, the approach of [14] is the simplest and the easiest to use in practical computation (see the discussion in [14]). Still, the error criterion given by (3) is not totally satisfactory. Obviously, there exist many error sequences that satisfy the first relation in (3), and it is not very clear which e k should be considered acceptable for each specific iteration k. In this sense, criterion (3) is not quite constructive. The second relation in (3) is even somewhat more problematic. In this paper, we present a hybrid generalized proximal-based algorithm which employs a more constructive error criterion than (3). Our method is completely implementable when the gradient of f is easily invertible, which is a common case for many important applications. The inexact solution is used to obtain the new iterate in a way very similar to Bregman generalized projections. When the error is zero, our algorithm coincides with the generalized proximal point method. However, for nonzero error, it is different from the inexact method of [14] described above. Our new method is motivated by [40], where a constructive error tolerance was introduced for the classical proximal point method. This approach has already proved to be very useful in a number of applications [38, 37, 35, 39, 36]. Besides the algorithm, we also present a theoretical result which is new even for exact methods. In particular, we prove convergence of the method for paramonotone operators, without the previously used assumption of pseu- domonotonicity (paramonotone operators were introduced in [4, 5], see also [9, 21]; we shall state this definition in Section 3, together with the definition of pseudomonotonicity). It is important to note that the subgradient of a proper closed convex function is paramonotone, but need not be pseudomono- tone. Hence, among other things, our result unifies the proof of convergence for paramonotone operators and for minimization. We also remove the condition of convergence consistency which has been used to characterize Bregman functions, proving it to be a consequence of the other properties. This work is organized as follows. In Section 2, we discuss Bregman functions and derive some new results on their properties. In Section 3, the error tolerance to be used is formally defined, the new algorithm is described and the convergence result is stated. Section 4 contains convergence analysis. A few words about our notation are in order. Given a (convex) set A, ri(A) will denote the relative interior, - A will denote the closure, int(A) will denote the interior, and bdry(A) will denote the boundary of A. For an operator T , Dom(T ) stands for its domain, i.e., all points x 2 R n such that Bregman Function and Bregman Distance Given a convex function f on R n , finite at x; y 2 R n and differentiable at y, the Bregman distance [2] between x and y, determined by f , is Note that, by the convexity of f , the Bregman distance is always nonnegative. We mention here the recent article [1] as one good reference on Bregman functions and their properties. Definition 2.1 Given S, a convex open subset of R n , we say that is a Bregman function with zone S if 1. f is strictly convex and continuous in - 2. f is continuously differentiable in S, 3. for any x 2 - S and ff 2 R, the right partial level set fy is bounded, 4. If fy k g is a sequence in S converging to y, then lim Some remarks are in order regarding this definition. In addition to the above four items, there is one more standard requirement for Bregman function, namely Convergence Consistency : If fx k g ae - S is bounded, fy k g ae S converges to y, and also converges to y. This requirement has been imposed in all previous studies of Bregman functions and related algorithms [10, 11, 13, 19, 9, 14, 1, 22, 20, 6]. In what follows, we shall establish that convergence consistency holds automatically as a consequence of Definition 2.1 (we shall actually prove a stronger result). The original definition of a Bregman function also requires the left partial level sets to be bounded for any y 2 S. However, it has been already observed that this condition is not needed to prove convergence of proximal methods (e.g., [14]). And it is known that this boundedness condition is extraneous regardless, since it is also a consequence of Definition 2.1 (e.g., see [1]). Indeed, observe that for any y, the level set L 0 (0; so it is nonempty and bounded. Also Definition 2.1 implies that D f (\Delta; y) is a proper closed convex function. Because this function has one level set which in nonempty and bounded, it follows that all of its level sets are bounded (i.e., L 0 (ff; y) is bounded for every ff) [32, Corollary 8.7.1]. To prove convergence consistency using the properties given in Definition 2.1, we start with the following results. Lemma 2.2 (The Restricted Triangular Inequality) Let f be a convex function satisfying items 1 and 2 of Definition 2.1. If x 2 - and w is a proper convex combination of x and y, i.e., with Proof. We have that Since rf is monotone, Taking into account that w latter relation yields Therefore Lemma 2.3 Let f be a convex function satisfying items 1 and 2 of Definition 2.1. If fx k g is a sequence in - S converging to x, fy k g is a sequence in S converging to y and y 6= x, then lim inf Proof. Define is a sequence in S converging to z S. By the convexity of f , it follows that for all k: Therefore Letting k !1 we obtain Using the strict convexity of f and the hypothesis x 6= y, the desired result follows. We are now ready to prove a result which is actually stronger than the property of convergence consistency discussed above. This result will be crucial for strengthening convergence properties of proximal point methods, carried out in this paper. Theorem 2.4 Let f be a convex function satisfying items 1 and 2 of Definition 2.1. If fx k g is a sequence in - fy k g is a sequence in S, lim and one of the sequences (fx k g or fy k g) converges, then the other also converges to the same limit. Proof. Suppose, by contradiction, that one of the sequences converges and the other does not converge or does not converge to the same limit. Then there exist some " ? 0 and a subsequence of indices fk j g satisfying Suppose first that fy k g converges and lim i.e., ~ x j is a proper convex combination of x k j and y k j . Using Lemma 2.2 we conclude that D f (~x which implies that lim fy k j g converges, it follows that f~x j g is bounded and there exists a subsequence f~x j i g converging to some ~ x. Therefore we have the following set of relations which is in contradiction with Lemma 2.3. If we assume that the sequence fx k g converges, then reversing the roles of and fy k g in the argument above, we reach a contradiction with Lemma 2.3 in exactly the same manner. It is easy to see that Convergence Consistency is an immediate consequence of Theorem 2.4. We next state a well-known result which is widely used in the analysis of generalized proximal point methods. Lemma 2.5 (Three-Point Lemma)[11] Let f be a Bregman function with zone S as in Definition 2.1. For any holds that In the sequel, we shall use the following consequence of Lemma 2.5, which can be obtained by subtracting the three-point inequalities written with and s; x; z. Corollary 2.6 (Four-Point Lemma) Let f be a Bregman function with zone S as in Definition 2.1. For any holds that 3 The Inexact Generalized Proximal Point Method We start with some assumptions which are standard in the study and development of Bregman-function-based algorithms. Suppose C, the feasible set of VIP(T; C), has nonempty interior, and we have chosen f , an associated Bregman function with zone int(C). We also assume that so that T +N C is maximal monotone [31]. The solution set of VIP(T; C) is We assume this set to be nonempty, since this is the more interesting case. In principle, following standard analysis, results regarding unboundedness of the iterates can be obtained for the case when no solution exists. Additionally, we need the assumptions which guarantee that proximal subproblem solutions exist and belong to the interior of C. H1 For any x 2 int(C) and c ? 0, the generalized proximal subproblem has a solution. H2 For any x 2 int(C), if fy k g is a sequence in int(C) and lim then lim A simple sufficient condition for H1 is that the image of rf is the whole space R n (see [6, Proposition 3]). Assumption H2 is called boundary coer- civeness and it is the key concept in the context of proximal point methods for constrained problems for the following reason. It is clear from Definition 2.1 that if f is a Bregman function with zone int(C) and P is any open sub-set of int(C), then f is also a Bregman function with zone P , which means that one cannot recover C from f . Therefore in order to use the Bregman distance D f for penalization purposes, f has to possess an additional prop- erty. In particular, f should contain information about C. This is precisely the role of H2 because it implies divergence of rf on bdry(C), which makes C defined by f : Divergence of rf also implies that the proximal subproblems cannot have solutions on the boundary of C. We refer the readers to [9, 6] for further details on boundary coercive Bregman functions. Note also that boundary coerciveness is equivalent to f being essentially smooth on int(C) [1, Theorem 4.5 (i)]. It is further worth to note that if the domain of rf is the interior of C, and the image of rf is R n , then H1 and H2 hold automatically (see [6, Proposition 3] and [9, Proposition 7]). We are now ready to describe our error tolerance criterion. Take any consider the proximal subproblem which is to find a pair (y; v) satisfying the proximal system The latter is in turn equivalent to Therefore, an approximate solution of (5) (or (6) or (7)) should satisfy We next formally define the concept of inexact solutions of (6), taking the approach of (8). Definition 3.1 Let x 2 int(C), c ? 0 and oe 2 [0; 1). We say that a pair (y; v) is an inexact solution with tolerance oe of the proximal subproblem (6) if and z, the solution of equation satisfies Note that from (4) (which is a consequence of H2), it follows that Note that equivalently z is given by Therefore z, and hence D f (y; z), are easily computable from x; y and v whenever rf is explicitly invertible. In that case it is trivial to check whether a given pair (y; v) is an admissible approximate solution in the sense of Definition 3.1: it is enough to obtain z verify if our algorithm is based on this test, it is most easy to implement when rf is explicitly invertible. We point out that this case covers a wide range of important applications. For example, Bregman functions with this property are readily available when the feasible set C is an orthant, a polyhedron, a box, or a ball (see [9]). Another important observation is that for oe = 0, we have that Hence, the only point which satisfies Definition 3.1 for precisely the exact solution of the proximal subproblem. Therefore our view of inexact solution of generalized proximal subproblems is quite natural. We note, in the passing, that it is motivated by the approach developed in [40] for the classical ("linear") proximal point method. In that case, Definition 3.1 (albeit slightly modified) is equivalent to saying that the subproblems are solved within fixed relative error tolerance (see also [37]). Such an approach seems to be computationally more realistic/constructive than the common summable-error-type requirements. Regarding the existence of inexact solutions, the situation is clearly even easier than for exact methods. Since we are supposing that the generalized proximal problem (5) has always an exact solution in int(C), this problem will certainly always have (possibly many) inexact solutions (y; v) satisfying also y 2 C. Now we can formally state our inexact generalized proximal method. Algorithm 1 Inexact Generalized Proximal Method. Initialization: Choose some c ? 0, and the error tolerance parameter oe 2 [0; 1). Choose some x Iteration k: Choose the regularization parameter c k - c, and find (y an inexact solution with tolerance oe of satisfying repeat. We have already discussed the possibility of solving inexactly (9) with condition (10). Another important observation is that since for subproblem solution coincides with the exact one, in that case Algorithm 1 produces the same iterates as the standard exact generalized proximal method. Hence, all our convergence results (some of them are new!) apply also to the exact method. For oe 6= 0 however, there is no direct relation between the iterates of Algorithm 1 and considered in [14]. The advantage of our approach is that it allows an attractive constructive stopping criterion (given by Definition 3.1) for approximate solution of subproblems (at least, when rf is invertible). Under our hypothesis, Algorithm 1 is well-defined. From now on, fx k g and f(y k ; v k )g are sequences generated by Algorithm 1. Therefore, by the construction of Algorithm 1 and by Definition 3.1, for all k it holds that We now state our main convergence result. First, recall that a maximal monotone operator T is paramonotone ([4, 5], see also [9, 21]) if Some examples of paramonotone operators are subdifferentials of proper closed convex functions, and strictly monotone maximal monotone operators Theorem 3.2 Suppose that VIP(T; C) has solutions and one of the following two conditions holds : 1. 2. T is paramonotone. Then the sequence fx k g converges to a solution of VIP(T; C). Thus we establish convergence of our inexact algorithm under assumptions which are even weaker than the ones that have been used, until now, for exact algorithms. Specifically, in the paramonotone case, we get rid of the "pseudomonotonicity" assumption on T [6] which can be stated as follows: Take any sequence fy k g ae Dom(T ) converging to y and any sequence for each x 2 Dom(T ) there exists and element (y) such that Until now, this (or some other, related) technical assumption was employed in the analysis of all generalized proximal methods (e.g., [14, 7, 6]). Among other things, this resulted in splitting the proof of convergence for the case of minimization and for paramonotone operators (the subdifferential of a convex function is paramonotone, but it need not satisfy the above condition). And of course, the additional requirement of pseudomonotonicity makes the convergence result for paramonotone operators weaker. Since for the tolerance parameter our Algorithm 1 reduces to the exact generalized proximal method, Theorem 3.2 also constitutes a new convergence result for the standard setting of exact proximal algorithms. We note that the stronger than convergence consistency property of Bregman functions established in this paper is crucial for obtaining this new result. To obtain this stronger result, the proof will be somewhat more involved than the usual, and some auxiliary analysis will be needed. However, we think that this is worthwhile since it allows us to remove some (rather awkward) additional assumptions. Convergence Analysis Given sequences fx k g, fy k g and fv k g generated by Algorithm 1, define ~ X as all points x 2 C for which the index set is finite. For x 2 ~ X, define k(x) as the smallest integer such that 0: Of course, the set ~ X and the application k(\Delta) depend on the particular sequences generated by the algorithm. These definitions will facilitate the subsequent analysis. Note that, by monotonicity of T , and in fact, Lemma 4.1 For any s 2 ~ X and k - k(s), it holds that Proof. Take s 2 ~ X and k - k(s). Using Lemma 2.6, we get By (14) and (15), we further obtain which proves the first inequality in (16). Since the Bregman distance is always nonnegative and oe 2 [0; 1), we have The last inequality in (16) follows directly from the hypothesis s 2 ~ k(s) and the respective definitions. As an immediate consequence, we obtain that the sequence fD f (-x; x k )g is decreasing for any - x 2 X . Corollary 4.2 If the sequence fx k g has an accumulation point - x 2 X then the whole sequence converges to - x. Proof. Suppose that some subsequence fx k j g converges to - x 2 X . Using Defintion 2.1 (item 4), we conclude that lim Since the whole sequence fD f (-x; x k )g is decreasing and it has a convergent subsequence, it follows that it converges : lim Now the desired result follows from Theorem 2.4. Corollary 4.3 Suppose that ~ ;. Then the following statements hold: 1. The sequence fx k g is bounded ; 2. 3. For any s 2 ~ 4. The sequence fy k g is bounded . Proof. Take some s 2 ~ X. From Lemma 4.1 it follows that for all k greater than k(s), D f (s; x k Therefore, D f (s; x k ) is bounded and from Definition 2.1 (item 3), it follows that fx k g is bounded. By Lemma 4.1, it follows that for any r 2 N Therefore Since r is arbitrary and the terms of both summations are nonnegative, (recall the definition of k(s)), it follows that we can take the limit as r !1 in both sides of the latter relation. Taking further into account that fc k g is bounded away from zero, the second and third assertions of the Corollary easily follow. As consequences, we also obtain that lim and lim X: (18) Suppose now that fy k g is unbounded. Then there exists a pair of subsequences and fy k j g such that fx k j g converges but fy k j g diverges. How- ever, by (17) and Theorem 2.4, fy k j g must converge (to the same limit as which contradicts the assumption. Hence, fy k g is bounded. The next proposition establishes the first part of Theorem 3.2, namely the convergence of the inexact generalized proximal algorithm in the case when Proposition 4.4 If ~ converges to some x 2 int(C) which is a solution of VIP(T; C). Proof. By Corollary 4.3, it follows that fx k g is bounded, so it has some accumulation point - x 2 C, and for some subsequence fx k j g, lim Take any - x 2 ~ and, by H2, lim it follows that lim But the latter is impossible because D f (-x; x k ) is a decreasing sequence, at least for k - k(-x) (by Lemma 4.1). Hence, Next, we prove that - x is a solution of VIP(T; C). By (17), we have that lim Because, by (15), D f (y lim converges to - Theorem 2.4 and (19) imply that fy k j g also converges to - x. Applying Theorem 2.4 once again, this time with (20), we conclude that fx k j +1 g also converges to - x. Since - and rf is continuous in int(C), we therefore conclude that lim using (14) we get lim Now the fact that fy k j together with the maximality of T , implies that 0 2 T (-x). Thus we have a subsequence fx k j g converging to - . By Corollary 4.2, the whole sequence fx k g converges to - x. We proceed to analyze the case when T is paramonotone. By (18), we already know that if s the limit with respect to v k (for example, using the technical assumption of pseudomonotonicity stated above), then we could conclude that 0 - h-v; - x is an accumulation point of fx k g (hence also of fy k g), and - v 2 T (-x); v s 2 T (s). By paramonotonicity, it follows that v s 2 T (-x). Now by monotonicity, we further obtain that for any x 2 C which means that - However, in the absence of the assumption of pseudomonotonicity one cannot use this well-established line of argument. To overcome the difficulty resulting from the impossibility of directly passing onto the limit as was done above, we shall need some auxiliary constructions. Let A be the affine hull of the domain of T . Then there exists some V , a subspace of R n , such that for any x 2 Dom(T ). Denote by P the orthogonal projection onto V , and for each k define The idea is to show the following key facts: k g has an accumulation point : With these facts in hand, we could pass onto the limit in a manner similar to the above, and complete the proof. First, note that This can be verified rather easily: if x 62 Dom(T ) then both sets in (21) are empty, so it is enough to consider x then so that By monotonicity of T , for any z 2 Dom(T ) and any w 2 T (z), it holds that holds that Therefore which implies that u 2 T (x) by the maximality of T . Since also follows that u 2 T Lemma 4.5 If ~ some subsequence of fu k g is bounded. Proof. We assumed that Dom(T and let P be the projection operator onto V discussed above. In particular, Furthermore, the operator - defined by is maximal monotone as an operator on the space V (this can be easily verified using the maximal monotonicity of T on R n ). We also have that T is bounded around zero [30]. So, P ffi T is bounded around - x, i.e., there exist some r ? 0 and M - 0 such that Since ~ X. Therefore, by the definition of ~ X, there exists an infinite subsequence of indices fk j g such that Note that u holds that for each j, Then for each j there exists - Furthermore, where the first inequality is by the monotonicity of T , and the second is by (22). Using further the Cauchy-Schwarz and triangular inequalities, we obtain Since the sequence fy k g is bounded (Corollary 4.3, item 4), it follows that We conclude the analysis by establishing the second part of Theorem 3.2. Proposition 4.6 Suppose X 6= ; and T is paramonotone. Then fx k g converges to some - Proof. If ~ then the conclusion follows from Proposition 4.4. Suppose now that ~ By Lemma 4.5, it follows that some subsequence of fu k g is bounded. Since X, from Corollary 4.3 it follows that the whole sequence fx k g is bounded. Hence, there exist two subsequences fx k j g, fu k j g which both converge lim lim Recall from the proof of Lemma 4.5 that u 4.3 (item 2), we have that lim Therefore, by Theorem 2.4, lim and by the maximality of T . Take now some s 2 X . There exists some v s 2 T such that for all x 2 C. Therefore, using also the monotonicity of T , Note that for any x 2 Dom(T ) Taking passing onto the limit as j !1, (18) implies that Together with (23), this implies that Using now the paramonotonicity of T , we conclude that Finally, for any x 2 C, we obtain xi 0: Therefore - we have a subsequence fx k j g converging to - from Corollary 4.2 it follows that the whole sequence fx k g converges to - x. --R Legendre functions and the method of random Bregman projections. The relaxation method of finding the common points of convex sets and its application to the solution of problems in convex programming. Produits infinis de r'esolvantes. An iterative solution of a variational inequality for certain monotone operators in a Hilbert space. Corrigendum to A generalized proximal point algorithm for the variational inequality problem in a Hilbert space. Enlargement of monotone operators with applications to variational inequalities. A variable metric proximal point algorithm for monotone operators. An interior point method with Bregman functions for the variational inequality problem with para- monotone operators The proximal minimization algorithm with D-functions Convergence analysis of proximal-like optimization algorithm using Bregman functions Variational Inequalities and Complementarity Problems Nonlinear proximal point algorithms using Bregman func- tions Approximate iterations in Bregman-function-based proximal algorithms On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone op- erators Multiplicative iterative algorithms for convex pro- gramming Finite termination of the proximal point algorithm. New proximal point algorithms for convex minimization. On some properties of generalized proximal point methods for quadratic and linear programming. On some properties of generalized proximal point methods for the variational inequality problem. On some properties of paramonotone operators. Proximal minimization methods with generalized Bregman functions. The proximal algorithm. Asymptotic convergence analysis of the proximal point algorithm. Regularisation d'inequations variationelles par approximations successives. Proximit'e et dualit'e dans un espace Hilbertien. Complementarity problems. Weak convergence theorems for nonexpansive mappings in Banach spaces. Local boundedness of nonlinear monotone operators. On the maximality of sums of nonlinear monotone operators. Convex Analysis. Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Monotone operators and the proximal point algorithm. A truly globally convergent Newton-type method for the monotone nonlinear complementarity problem A comparison of rates of convergence of two inexact proximal point algorithms. Forcing strong convergence of proximal point iterations in a Hilbert space A globally convergent inexact Newton method for systems of monotone equations. A hybrid approximate extragradient- proximal point algorithm using the enlargement of a maximal monotone operator A hybrid projection - proximal point algorithm Convergence of proximal-like algorithms --TR --CTR Lev M. Bregman , Yair Censor , Simeon Reich , Yael Zepkowitz-Malachi, Finding the projection of a point onto the intersection of convex sets via projections onto half-spaces, Journal of Approximation Theory, v.124 n.2, p.194-218, October

maximal monotone operator;bregman function;variational inequality;proximal point method

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Performance Evaluation of Conservative Algorithms in Parallel Simulation Languages.

AbstractParallel discrete event simulation with conservative synchronization algorithms has been used as a high performance alternative to sequential simulation. In this paper, we examine the performance of a set of parallel conservative algorithms that have been implemented in the Maisie parallel simulation language. The algorithms include the asynchronous null message algorithm, the synchronous conditional event algorithm, and a new hybrid algorithm called Accelerated Null Message that combines features from the preceding algorithms. The performance of the algorithms is compared using the Ideal Simulation Protocol. This protocol provides a tight lower bound on the execution time of a simulation model on a given architecture and serves as a useful base to compare the synchronization overheads of the different algorithms. The performance of the algorithms is compared as a function of various model characteristics that include model connectivity, computation granularity, load balance, and lookahead.

Introduction Parallel discrete event simulation (PDES) refers to the execution of a discrete event simulation program on parallel or distributed computers. Several algorithms have been developed to synchronize the execution of PDES models, and a number of studies have attempted to evaluate the performance of these algorithms on a variety of benchmarks. A survey of many existing simulation protocols and their performance studies on various benchmarks appears in [9, 11]. A number of parallel simulation environments have also been developed that provide the modeler with a set of constructs to facilitate design of PDES models[4]. One of these is Maisie[3], a parallel simulation language that has been implemented on both shared and distributed memory parallel computers. Maisie was designed to separate the model from the underlying synchronization protocol, sequential or parallel, that is used for its execution. Efficient sequential and parallel optimistic execution of Maisie models have been described previously[2]. In this paper, we evaluate the performance of a set of conservative algorithms that have been implemented in Maisie. The set of algorithms include the null message algorithm[6], the conditional event[7] algorithm, and a new conservative algorithm called the Accelerated Null Message (ANM) algorithm that combines the preceding two approaches. Unlike previous performance studies which use speedup or throughput as their metric of comparison, we use efficiency as our primary metric. We define the efficiency of a protocol using the notion of the Ideal Simulation Protocol or ISP, as introduced in this paper. The performance of a parallel simulation model depends on a variety of factors which include partitioning of the model among the processors, the communication overheads of the parallel platform including both hardware and software overheads, and the overheads of the parallel synchronization algorithm. When a parallel model fails to yield expected performance benefits, the analyst has few tools with which to ascertain the underlying cause. For instance, it is difficult to determine whether the problem is due to inherent lack of parallelism in the model or due to large overheads in the implementation of the synchronization protocol. The Ideal Simulation Protocol offers a partial solution to this problem. ISP allows an analyst to experimentally identify the maximum parallelism that exists in a parallel implementation of a simulation model, assuming that the synchronization overhead is zero. In other words, for a specific decomposition of a model on a given parallel ar- chitecture, it is possible to compute the percentage degradation in performance that is due to the simulation algorithm, which directly translates into a measure of the relative efficiency of the synchronization scheme. Thus, ISP may be used to compute the efficiency of a given synchronization algorithm and provide a suitable reference point to compare the performance of different algorithms, including conservative, optimistic, and adaptive techniques. Previous work has relied on theoretical critical path analyses to compute lower bounds on model execution times. These bounds can be very approximate because they ignore all overheads, including architectural and system overheads over which the parallel simulation algorithm has no control. The remainder of the paper is organized as follows: the next section describes the conservative algorithms that have been used for the performance study reported in this paper. Section 3 describes the Ideal Simulation Protocol and its use in separating protocol-dependent and independent over- heads. Section 4 describes implementation issues of synchronization algorithms, including language level constructs to support conservative algorithms. Section 5 presents performance comparisons of the three conservative algorithms with the lower bound prediction of ISP. Related work in the area is described in Section 6. Section 7 is the conclusion. Conservative Algorithms In parallel discrete event simulation, the physical system is typically viewed as a collection of physical processes (PPs). The simulation model consists of a collection of Logical Processes (LPs), each of which simulates one or more PPs. The LPs do not share any state variables. The state of an LP is changed via messages which correspond to the events in the physical system. In this section, we assume that each LP knows the identity of the LPs that it can communicate with. For any LP p , we use the terms dest-set p and source-set p to respectively refer to the set of LPs to which LP p sends messages and from which it receives messages. The causality constraint of a simulation model is normally enforced in teh simulation algorithm by ensuring that all messages to a Logical Process (LP) are processed in an increasing timestamp order. Distributed simulation algorithms are broadly classified into conservative and optimistic based on how they enforce this. Conservative algorithms achieve this by not allowing an LP to process a message with timestamp t until it can ensure that the LP will not receive any other message with a timestamp lower than t. Optimistic algorithms, on the other hand, allow events to be potentially processed out of timestamp order. Causality errors are corrected by rollbacks and re-computations. In this paper, we focus on the conservative algorithms. In order to simplify the description of the algorithms, we define the following terms. Each term is defined for an LP p at physical time r. We assume that the communication channels are FIFO. Earliest Input Time (EIT p (r)): Lower bound on the (logical) timestamp of any message that LP p may receive in the interval (r; 1). Earliest Output Time (EOT p (r)): Lower bound on the timestamp of any message that LP p may send in the interval (r; 1). Earliest Conditional Output Time (ECOT p (r)): Lower bound on the timestamp of any message that LP p may send in the interval (r; 1) assuming that LP p will not receive any more messages in that interval. ffl Lookahead (la p (r)): Lower bound on the duration after which the LP will send a message to another LP. The value of EOT and ECOT for a given LP depends on its EIT , the unprocessed messages in its input queue, and its lookahead. Figure 1 illustrates the computation of EOT and ECOT for an LP that models a FIFO server. The server is assumed to have a minimum service time of one time unit, which is also its lookahead. The three scenarios in the figure respectively represent the contents of the input message queue for the LP at three different points in its execution; messages Simulation time (a) Simulation time (b) Simulation time (c) Figure 1: The computation of EOT and ECOT: The LP is modeling a server with a minimum service time of one time unit. Shaded messages have been processed by the LP. already processed by the LP are shown as shaded boxes. Let T next refer to the earliest timestamp of a message in the input queue. In (a), since T next both EIT and ECOT are equal to T next plus the minimum service time. In (b), EOT is equal to plus the minimum service time, because the LP may still receive messages with a timestamp less than T next = 40, but no smaller than EIT . However, the ECOT is equal to T next plus minimum service time because if the LP does not receive any more messages, its earliest next output will be in response to processing the message with timestamp T next . In (c), T next = 1 because there are no unprocessed messages. Therefore, ECOT is equal to 1. Various conservative algorithms differ in how they compute the value of EIT for each LP in the system. By definition, in a conservative algorithm, at physical time r, LP p can only process messages with timestamp - EIT p (r). Therefore, the performance of a conservative algorithm depends on how efficiently and accurately each LP can compute the value of its EIT. In the following sections, we discuss how three different algorithms compute EIT. 2.1 Null Message Algorithm The most common method used to advance EIT is via the use of null messages. A sufficient condition for this is for an LP to send a null message to every LP in its dest set, whenever there is a change in its EOT. Each LP computes its EIT as the minimum of the most recent EOT received from every LP in its source-set. Note that a change in the EIT of an LP typically implies that its EOT will also advance. A number of techniques have been proposed to reduce the frequency with which null messages are transmitted: for instance, null messages may be piggy-backed with regular messages or they may be sent only when an LP is blocked, rather than whenever there is a change in its EOT. The null message algorithm requires that the simulation model not contain zero-delay cycles: if the model contains a cycle, the cycle must include at least one LP with positive lookahead (i.e. if the LP accepts a message with timestamp t, any message generated by the LP must have a timestamp that is strictly greater than t)[13]. 2.2 Conditional Event Algorithm In the conditional event algorithm [7], the LPs alternate between an EIT computation phase, and an event processing phase. For the sake of simplicity, we first consider a synchronous version which ensures that no messages are in transit when the LPs reach the EIT computation phase. In such a state, the value of EIT for an LP p is equal to the minimum of ECOT over all the LPs in the transitive closure 1 of source-set p . The algorithm can be made asynchronous by defining the EIT to be the minimum of all ECOT values for the LPs in the transitive closure of the source-set and the timestamps of all the messages in transit from these LPs. Note that this definition of EIT is the same as the definition of Global Virtual Time (GVT) in optimistic algorithms. Hence, any of the GVT computation algorithms [5] can be used. Details of one such algorithm are described in section 4.4. 2.3 Accelerated Null Message Algorithm We superimpose the null message protocol on an asynchronous conditional event protocol which allows the null message protocol to perform unhindered. The EIT for any LP is computed as the maximum of the values computed by the two algorithms. This method has the potential of combining the efficiency of the null message algorithm in presence of good lookahead with the ability of conditional event algorithm to execute even without lookahead - a scenario in which null message algorithm alone will deadlock. In models with poor lookahead where it may take many Transitive closure of source-set in many applications is almost the same as the set of all LPs in the system. rounds of null messages to sufficiently advance the EIT of an LP, ANM could directly compute the earliest global event considerably faster. Message piggy-backing is used extensively to reduce the number of synchronization messages, and the global ECOT computation at a node is initiated only when the node is otherwise blocked. Most experimental performance studies of parallel simulations have used speedup or throughput (i.e., number of events executed per unit time ) as the performance metric. While both metrics are appropriate for evaluating benefits from parallel simulation, they do not shed any light on the efficiency of a simulation protocol. A number of factors affect the execution time of a parallel simulation model. We classify the factors into two categories - protocol-independent factors and protocol-specific factors. The former refer to the hardware and software characteristics of the simulation engine, like the computation speed of the processor, communication latency, and cost of a context switch, together with model characteristics, like partitioning and LP to processor mapping, that determine the inherent parallelism in the model. The specific simulation protocol that is used to execute a model has relatively little control on teh overhead contributed by these factors. In contrast, the overhead due to the protocol-specific factors does depend on the specific simulation protocol that is used for the execution of a model. For conservative protocols, these may include the overhead of processing and propagating null messages[6] or conditional event messages[7], and the idle time of an LP that is blocked because its EIT has not yet advanced sufficiently to allow it to process a message that is available in its input queue. In the case of optimistic protocols, the protocol-specific overheads may include state saving, rollback, and the Global Virtual Time (GVT) computation costs. A separation of the cost of the protocol-specific factors from the total execution time of a parallel simulation model will lend considerable insight into its performance. For instance, it will allow the analyst to isolate a model where performance may be poor due to lack of inherent parallelism in the model (something that is not under the control of the protocol) from one where the performance may be poor due to a plethora of null messages (which may be addressed by using appropriate optimizations to reduce their count). In the past, critical path analyses have been used to prove theoretical lower bounds and related properties[12, 1] of a parallel simulation model. However, such analyses do not include the cost of many protocol-independent factors in the computation of the critical path time. Excluding these overheads, which are not contributed by the simulation protocol, means that the computed bound is typically a very loose lower bound on the execution time of the model, and is of relatively little practical utility to the simulationist either in improving the performance of a given model or in measuringthe efficiency of a given protocol. We introduce the notion of an Ideal Simulation Protocol (ISP), that is used to experimentally compute a tight lower bound on the execution time of a simulation model on a given architecture. Although ISP is based on the notion of critical path, it computes the parallel execution time by actually executing the model on the parallel architecture. The resulting lower bound predicted by ISP is realistic because it includes overheads due to protocol-independent factors that must be incurred by any simulation protocol that is used for its execution, and assumes that the synchronization overheads are zero. The primary idea behind ISP is simple: the model is executed once, and a trace of the messages accepted by each LP is collected locally by the LP. In a subsequent execution, an LP may simply use the trace to locally deduce when it is safe to process an incoming message; no synchronization protocol is necessary. As no synchronization protocol is used to execute the model using ISP, the measured execution time will not include any protocol-specific overheads. However, unlike the critical path analyses, the ISP-predicted bound will include the cost of all the protocol-independent factors that are required in the parallel execution of a model. Given this lower bound, it is easy to measure the efficiency of a protocol as described in the next section. Besides serving as a reference point for computing the efficiency of a given protocol, a representative pre-simulation with ISP can yield a realistic prediction of the available parallelism in a simulation model. If the speedup potential is found to be low, the user can modify the model, which may include changing the partitioning of the system, changing the assignment of LPs to processors, or even moving to a different parallel platform, in order to improve the parallelism in the model. Implementation Issues The performance experiments were executed using the Maisie simulation language. Each algorithm, including ISP, was implemented in the Maisie runtime system. A programmer develops a Maisie model and selects among the available simulation algorithms as a command line option. This separation of the algorithm from the simulation model permits a more consistent comparison of the protocols than one where the algorithms are implemented directly into the applictaion. In this section, we briefly describe the Maisie language and the specific constructs that have been provided to support the design of parallel conservative simulations. The section also describes the primary implementation issues for each algorithm. 4.1 Maisie Simulation Environment Maisie [3] is a C-based parallel simulation language, where each program is a collection of entity definitions and C functions. An entity definition (or an entity type) describes a class of objects. An instance, henceforth referred to simply as an entity, represents a specific LP in the model; entities may be created dynamically and recursively. The events in the physical system are modeled by message communications among the corresponding entities. Entities communicate with each other using buffered message passing. Every entity has a unique message buffer; asynchronous send and receive primitives are provided to deposit and remove messages from the buffer respectively. Each message carries a timestamp which corresponds to the simulation time of the corresponding event. Specific simulation constructs provided by Maisie are similar to those provided by other process-oriented simulation languages. In addition, Maisie provides constructs to specify the dynamic communication topology of a model and the lookahead properties of each entity. These constructs were used to investigate the impact of different communictaion topologies and lookahead patterns on the performance of each of the conservative algorithms described earlier. Figure 2 is a Maisie model for an FCFS server. This piece of code constructs the tandem queue system described in Section 5, and is used, with minor changes, in the experiments described there. The entity "Server" defines a message type "Job" which is used to simulate the requests for service to the server. The body of the entity consists of an unbounded number of executions of the wait statement (line 12). Execution of this statement causes the entity to block until it receives a message of the specified type (which in this case is "Job"). On receiving a message of this type, the entity suspends itself for "JobServiceTime" simulation time units to simulate servicing the job by executing the hold statement (line 13). Subsequently it forwards the job to the adjacent server identified by variable "NextServer," using the invoke statement (line 14). The performance of many conservative algorithms depends on the connectivity of the model. In the absence of any information, the algorithm must assume that every entity belongs to the source- set and dest-set of every other entity. However, Maisie provides a set of constructs that may be used by an entity to dynamically add or remove elements from these sets: add source(), del source(), add dest(), and del dest(). In Figure 2, the server entity specifies its connectivity to the other entities using the add source() and add dest() functions (line 8-9). Lookahead is another important determinant of performance for conservative algorithms. Maisie provides a pre-defined function setlookahead(), that allows the user to specify its current lookahead to the run-time system. The run-time system uses this information to compute a better estimate of EOT and ECOT then may be feasible otherwise. For instance, when the server is idle, it is possible clocktype MeanTime; 4 ename NextServer; 6 message Jobfg; 8 add source(PrevServer); 9 add dest(NextServer); 14 invoke NextServer with Jobfg; Figure 2: Maisie entity code for "First Come First Served" (FCFS) server with user defined lookahead for the server to precompute the service time of the next job (line 15), which becomes its lookahead and is transmitted to the run-time system using the function setlookahead() in (line 11) before it suspends itself. 4.2 Entity Scheduling If multiple entities are mapped to a single processor, a scheduling strategy must be implemented to choose one among a number of entities that may be eligible for execution at a given time. An entity is eligible to be executed if its input queue contains a message with a timestamp that is lower than the current EIT of the entity. In the Maisie runtime, once an entity is scheduled, it is allowed to execute as long as it remains eligible. This helps reduce the overall context switching overhead since more messages are executed each time the entity is scheduled but may result in the starvation of other entities mapped on the same processor. This, in turn, may cause blocking of other processors that might be waiting for messages from these entities, resulting in sub-optimal performance. Note, however, that without prior knowledge of the exact event dependencies, no scheduling scheme can be optimal. An alternative scheduling strategy, used in the Global Event List algorithm, is to schedule events across all entities mapped to a processor in the order of their timestamps. We examine and discuss the overheads caused by these two scheduling strategies in Section 5. 4.3 Null Message Algorithm One of the tunable parameters for any null message scheme is the frequency with which null messages are transmitted. Different alternatives include eager null messages - an LP sends null messages to all successors as soon as its EOT changes, lazy null messages - an LP sends null messages to all successors only when it is idle, or demand driven - an LP sends null message to a destination only when the destination demands to know the value of its EOT. The performance of different null message transmission schemes including eager, lazy, and demand driven schemes have been discussed in [17]. The experiments found that demand driven schemes performed poorly, whereas the lazy null message scheme combined with eager event send- ing, (i.e. regular messages sent as soon as they are generated, as is the case in our runtime), was found to marginally outperform other schemes. We use the lazy null message scheme in our experi- ments. This has the advantage of reducing the overall number of null messages because several null messages may be replaced by the latest message. However, the delay in sending null messages may delay the processing of real messages at the receiver. 4.4 Conditional Event Computation For the conditional event and the ANM protocol, it is necessary to periodically compute the earliest conditional event in the model. As discussed in section 2.3, an asynchronous algorithm allows the earliest event time to be computed without the need to freeze the computation at each node. In our runtime system, this computation takes place in phases. At the start of a new phase i, the jth processor computes its is the ECOT value for the phase i, and M ij is the smallest timestamp that is sent by the processor in the phase accounts for the messages in transit; in the worst case none of the messages sent by the processor in the phase may have been received and accounted for in E i of other processors. Thus, the global ECOT for the phase i is calculated as the following: min where n is the number of processors. Messages sent during phases do not need to be considered because a new phase starts only when all messages from the previous phase have been received. This, along with the FIFO communication assumption, implies that E ij takes into account all messages sent to the processor by any processor during phase 2. During the phase i, every processor sends an ECOT message which contains the minimum of its can compute the global ECOT to be the minimum of ECOT messages from all the processors. Note that the ECOT message needs to carry the phase identifier with it. But, since a new phase does not start until the previous one has finished, a boolean flag is sufficient to store the phase identifier. 4.5 ISP The Ideal Simulation Protocol (ISP) has been implemented as one of the available simulation protocols in the Maisie environment. The implementation uses the same data structures, entity scheduling strategy, message sending and receiving schemes as the conservative algorithms discussed earlier. The primary overhead in the execution of a model with ISP is the time for reading and matching the event trace. The implementation minimizes this overhead as follows: The entire trace is read into an array at the beginning of the simulation in order to exclude the time of reading the trace file during execution. To reduce the matching time, the trace is stored simply as a sequence of the unique numeric identifiers that are assigned to each message by the runtime system to ensure FIFO operation of the input queue. Thus the only 'synchronization' overhead in executing a simulation model with ISP is the time required to execute a bounded number of numeric comparison operations for each incoming message. This cost is clearly negligible when compared with other overheads of parallel simulation and the resulting time is an excellent lower bound on the exection time for a parallel simulation. 5 Experiments and Results The programs used for the experiments were written in Maisie and contain additional directives for parallel simulation, i.e. (a) explicit assignment of Maisie entities to processors, (b) code to create the source and destination sets for each entity, and (c) specification of lookaheads. All experimental measurements were taken on a SPARCserver 1000 with 8 SuperSPARC processors, and 512M of shared main memory. We selected the Closed Queuing Networks (CQN), a widely used benchmark, to evaluate parallel simulation algorithms. This benchmark was used because it is easily reproducible and it allows the communication topology and lookahead, the two primary determinants of performance of the conservative protocols, to be modified in a controlled manner. The CQN model can be characterized by the following parameters: ffl The type of servers and classes of jobs: Two separate versions of this model were used: CQNF, where each server is FIFO and CQNP, where each server is a the results for the CQNP experiments Switch Server Queue Figure 3: A Closed Queuing Network(N ffl The number of switches and tandem queues (N ) and the number of servers in each tandem queue (Q). ffl Number of jobs that are initially assigned to each switch: J=N . ffl Service time: The service time for a job is sampled from a shifted-exponential distribution with mean value of S unit time (indicating a mean value of ffl Topology: When a job arrives at a switch, the switch routes the job to any one of the N tandem queues with equal probability after 1 time unit. After servicing a job, each server forwards the job to the next server in the queue; the last server in the queue must send the job back to its unique associated switch. ffl The total simulation time: H Each switch and server is programmed as a separate entity. Thus, the CQN model has a total of entities. The entities corresponding to a switch and its associated tandem queue servers are always mapped to the same processor. The shifted-exponential distribution is chosen as the service time in our experiments so that the minimum lookahead for every entity is non-zero, thus preventing a potential deadlock situation with the null message algorithm. The topology of the network, for 4, is shown in Figure 3. The performance of parallel algorithms is typically presented by computing speedup, where Parallel execution time on Pprocessors However, as we will describe in the next section, the sequential execution time of a model using the standard Global Event List (GEL) algorithm[13] does not provide the lower bound for single processor execution and can in fact be slower than a parallel conservative algorithm. For this reason, the speedup metric, as defined above might show super-linear behavior for some models or become saturated if the model does not have enough parallelism. In this paper, we suggest computing efficiency using ISP which does provide a tight lower bound, and then compare the conservative protocols with respect to their efficiency relative to the execution time with ISP. The efficiency of a protocol S on architecture A is defined as follows: using ISP on A Execution time using Protocol S on A The efficiency of each protocol identifies the fraction of the execution time that is devoted to synchronization-related factors or the protocol-specific operations in the execution of a model. We begin our experiments (section 5.1) with comparing the execution time of each of the five protocols: GEL(global event list), NULL(null message), COND(conditional event), ANM, and ISP on a single processor. This configuration is instructive because it identifies the unique set of 'sequen- tial' overheads incurred by each protocol on one processor. In subsequent sections, we investigate the impact of modifications in various model characteristics like model connectivity (section 5.2), lookahead properties (section 5.3), and computation granularity (section 5.4). 5.1 Simulation of a CQNF Model on One Processor Figure 4 shows the execution times for a CQNF model with 100; 000 on one processor of SPARCserver 1000. As seen in the figure, the execution times of the protocols differ significantly even on one processor. The computation granularity associated with each event in our model is very small. This makes the protocol-specific overheads associated with an event relatively large, allowing us to clearly compare these overheads for the various protocols, which is the primary purpose of the study. In Figure 4, ISP has the best execution time which can be regarded as a very close approximation of the pure computation cost for the model. We first compare the ISP and GEL protocols. The major difference between the two protocols is the entity scheduling strategy. As described in Section 4.5, ISP employs the same entity scheduling scheme as the conservative protocols, where each entity removes and processes as many safe messages from its input queue as possible, before being swapped out. For ISP, this means that once an entity has been scheduled for execution, it is swapped out only when the next message in its local trace is not available in the input queue for that entity. In contrast, the GEL algorithm schedules entities in strict order of message timestamps, which can cause numerous context switches. Table 1 shows the total number of context switches for each protocol for this configuration. The number of context switches for ISP is approximately 15 times ISP GEL NULL COND ANM Execution Time Protocols Figure 4: Execution times of a CQNF model on one processor Table 1: Number of context switches, null messages and global ECOT computations in one processor executions [\Theta1; 000] ISP GEL NULL COND ANM Context Switches 85 1,257 393 1,149 914 Global Computation The number of regular messages processed in the simulation is 1,193 [\Theta1; 000]. less than that required by the GEL algorithm. Considering that the total messages processed during the execution was about 1,200,000 messages, the GEL protocol does almost one context switch per message 2 . ISP and conservative protocols also use a more efficient data structure to order the entities than the GEL algorithm. Whereas GEL must use an ordered data structure like the splay tree to sort entities in the order of the earliest timestamps of messages in their input queues, the other protocols use a simple unordered list because the entities are scheduled on the basis of safe messages. Thus, the performance difference between the ISP and GEL protocols is primarily due to the differences in their costs for context switching and entity queue management. We next compare the performance of ISP and the three conservative protocols. First, ISP has fewer context switches than the other protocols because the number of messages that can be processed by an entity using the conservative protocols is bounded by its EIT, whereas ISP has no such upper bound. The differences in the number of context switches among the three conservative protocols is due to differences in their computation of EIT as well as due to the additional context switches that may be required to process synchronization messages. For instance, the COND protocol 2 Since the Maisie runtime system has a main thread which does not correspond to any entity, the number of context switches can be larger than the number of messages. has three times the number of context switches than the NULL protocol. This is because the COND protocol has to switch out all the entities on every global ECOT computation, which globally updates the EIT value of each entity, while the NULL protocol does not require a context switch to process every null message. The CQNF model used in this experiment consists of 128 entities, and each switch entity has a lookahead of 1 time unit, thus the window size between two global ECOT computations is small. As shown in Table 1, the COND protocol had approximately 49,000 global computations, which averages to one global computation for every advance of 2 units in simulation time. Even though the NULL protocol uses few context switches, its execution time is significantly larger than ISP because of the large number of null messages (1732,000 from Table 1) that must be processed and the resulting updates to EIT. Although the ANM protocol incurs overhead due to both null messages and global ECOT computations, both the number of null messages and global ECOT computations are significantly less than those for the NULL and COND protocols. As a result, its performance lies somewhere between that of the NULL and COND protocols. 5.2 Communication Topology In Section 4.1, we introduced Maisie constructs to alter the default connectivity of each entity. As the next experiment, we examine the impact of communication topology on the performance of parallel conservative simulations. For each protocol, the CQNF model was executed with different values of the total number of serevrs were kept constant among the three configurations, by keeping N (Q 128. The other parameters for the model were set to 000. The total number of entities (N (Q + 1)) is kept constant among the different configurations to also keep the size of the model constant. As N increases, the connectivity of the model increases dramatically because each of the N switch entities has N outgoing channels while each of the Q server entities in each tandem queue has only one outgoing channel. For instance, the total number of links is 368, 1120, and 4160 when respectively. We first measured the speedup with the ISP protocol, where the speedup was calculated with respect to the one processor ISP execution. As seen in Figure 5, the speedup for ISP is relatively independent of N , which implies that the inherent parallelism available in each of the three configurations is not affected significantly by the number of links. 3 Figure 6 to 8 show the efficiency of the three conservative protocols with 64g. As 3 The small reduction in speedup with N=64, is due to the dramatic increase in the number of context switches for this configuration as seen from Table 2. Number of Processors Figure 5: CQNF: Speedup achieved by ISP0.20.61 Efficiency Number of Processors NULL COND Figure CQNF: Efficiency of each protocol Efficiency Number of Processors NULL COND Figure 7: CQNF: Efficiency of each protocol Efficiency Number of Processors NULL COND Figure 8: CQNF: Efficiency of each protocol Null Message Number of Processors Figure 9: CQNF: NMR in the NULL protocol135 Window Size Number of Processors Figure 10: CQNF: Window size in the COND protocol seen from the three graphs, the NULL and ANM protocols have similar efficiencies except for the sequential case, while the COND protocol performs quite differently. We examine the performance of each protocol in the following subsections. 5.2.1 Null Message Protocol For the NULL protocol, the EIT value for an entity is calculated using EOT values from its incoming channels, i.e., from entities in its source-set. When the number of incoming channels to an entity is small, its EIT value can be advanced by a few null messages. Thus, the efficiency of the NULL protocol is relatively high in case of low connectivity, as shown in Figure 6. However, with high connectivity, the protocol requires a large number of null messages and the performance degrades significantly. The null message ratio (NMR) is a commonlyused performance metric for this protocol. It is defined as the ratio of the number of null messages to the number of regular messages processed in the model. As seen in Figure 9, the NMR for low and increases substantially for when the connectivity of the model becomes dense. As an entity must send null messages even to other entities on the same processor, NMR for each configuration does not change with the number of processors. However, the efficiency decreases as the number of processors is increased because null messages to remote processors are more expensive than the processor-local null messages. 5.2.2 Conditional Event Protocol Unlike the other consrvative protocols, the performance of the COND protocol is not affected by the communictaion topology of the model because the cost of the global ECOT computation is independent of the topology. Thus, in Figures 6 to 8, the efficiency of the COND protocol does not degrade; rather it improves as N goes from 32 to 64. The reason for this is that relative to the COND protocol, the performance of ISP degrades more causing the relative efficiency of the COND protocol to improve! This effect can be explained by looking at Table 2, which presents the number of context switches incurred in the one processor execution for each of the three model configurations with each protocol. As seen in the table, this measure increases dramatically for ISP as N increases from 32 to 64, while the COND protocol already has one context switch per message with and this does not change significantly as N increases. The average window size is another commonly used metric for this protocol, which is defined as: Number of synchronizations Figure shows the average window size for each of the three experiments with the COND protocol. Table 2: Number of context switches in one processor executions [\Theta1; 000]. Regular Messages ISP GEL NULL COND ANM As seen from the figure, the average window size in the parallel implementations is close to 1, which implies that the protocol requires one global computation for every simulation clock tick. The one processor execution has a window size of 2. The smaller window size for the parallel implementation is caused by the need to flush messages that may be in transit while computing the global ECOT as described in Section 4.4. However, the value of the window size (W ) does not appear to have a strong correlation with the performance of the COND protocol: first, although W drops by 50% as we go from 1 to 2 processors, the efficiency does not drop as dramatically. Second, although the efficiency of the protocol continues to decrease as the number of processors (P ) increases in Figures 6 Figure shows that W does not change dramatically for P ? 2. The performance degradation of this protocol with number of processors may be explained as follows: first, as each processor broadcasts its ECOT messages, each global ECOT computation requires messages in a P processor execution (P ? 1). Second, the idle time spent by an LP while waiting for the completion of each global ECOT computation also grows significantly as increases. 5.2.3 ANM Protocol The efficiency of the ANM protocol shown in Figure 6 to 8 is close to that of the NULL protocol except for the one processor execution. Figure 11 and 12 show the NMR and the average window size respectively. The ANM protocol has almost the same NMR values as the NULL protocol. The average window sizes vary widely for all the models although the window size becomes narrower as N increases. This occurs because of the following reasons: ffl ECOT computation is initiated only when no entity on a processor has any messages to process. As N increases, since every processor must send one ECOT message for one global ECOT computation, the EIT updates by ECOT messages can easily defer unless every processor has few entities to schedule. ffl Even if a global ECOT calculation is unhindered, it may have no effect if null messages have already advanced EIT values. In such cases, the number of null messages is not decreased by the global ECOT computations, and the ANM protocol will have higher overheads than the NULL protocol. Therefore, the efficiencies of the ANM protocol in Figure 6 to 8 are close to the ones of the NULL protocol except for the one processor executions. In the models with are rarely sent out since null messages can efficiently advance the EIT value of each entity and the situation where all the processors have no entity to schedule is unlikely. However, for 64, the performance of the ANM protocol is expected to be closer to that of the COND protocol since the EIT updates by ECOT messages are more efficient than the updates by null messages. As seen in Figure 12, when ECOT messages are in fact sent more frequently (the average window size is approximately 7) when 64. The inefficiency in this case arises because of the large number of null messages sent by each entity; hence the duration when the processor would be idle in the COND protocol, is filled up with the processing time of null messages. This can be seen by comparing Figure 9 and 11 in which both protocols have almost the same NMR. Considering that the ANM protocol has more overhead caused by the ECOT computation, the reason that the efficiency is the same as the NULL protocol is that the ECOT messages actually improve the advancement of EIT a little, but the improvement makes up only for the overhead of ECOT computations. In the experiments in this section, the global ECOT computations in the ANM protocol does not have sufficient impact to improve its performance. However, all the models in these experiments have good lookahead values, thus null messages can efficiently update EIT values of the entities. In the next section, we examine the cases where the simulation models have poor lookahead and compare the performance of the ANM and NULL protocols in this case. 5.3 Lookahead 5.3.1 No Precomputation of Service Times It is well known that the performance of conservative protocols depends largely on the lookahead value of the model. The lookahead for a stochastic server can be improved by precomputing service times[14]. In this section, we examine the effects of lookahead in the CQNF models by not precomputing the service times; instead the lookahead is set to one time unit, which is the lower bound of the service time generated by the shifted exponential distribution. With this change, all the entities have the lookahead value of one time unit, which is the worst lookahead value of our experiments. Figures 13 to 15 show the effect of this change where we plot p, the performance degradation that results from this change as a function of the number of processors. Note that the performance of ISP cannot be affected by the lookahead value because it does not use lookahead for the simulation run. Thus, the ISP performance is identical to the result shown in Figure 5. The fractional performance degradation is calculated as follows: precomputation of service times Execution time with precomputation of service times \Gamma 1 Interestingly, the figures show that the performance degradation is negative in some cases, which indicates that some executions actually become faster when lookahead is reduced. For the NULL and ANM protocols, this is attributed to the choice of the scheduling strategy discussed in Section 4.2; although this strategy reduces context switching overheads, it may also cause starvation of other entities. An entity that is scheduled for a long period blocks the progress of other entities on the same processor, which may also block other entities on remote processors and thus increase the overall idle time. Note that the performance improvement with poorer lookahead occurs primarily in the configuration with where the communication topology is not dense, and the number of jobs available at each server is typically greater than 1, which may lead to long scheduling cycles. For the COND protocol, the duration of the window size determines how long each entity can be scheduled before a context switch. A longer window implies fewer ECOT computations which may also lead to increased blocking of other entities. Thus, in some cases, poorer lookahead may force frequent ECOT messages and thus improve performance. As this effect is not related to the communication topology, the relative performance of the COND protocol with and without lookahead is almost the same for all three values of N (Figures 13 to 15). The performance of the NULL and ANM protocols degrade as the connectivity of the model becomes dense, while the performance of the COND protocol does not change significantly as discussed above. In particular, for the performance of the COND protocol does not degrade, although it already outperforms the other two protocols in the experiment with better lookahead Figure 8). This indicates that the COND protocol has significant advantage over the null message based protocols in the case where the simulation model has high connectivity or poor lookahead. Another interesting observation is that the performance of the ANM protocol does not degrade as much as that of the NULL protocol in Figure 15. This behavior may be understood from Figure 16 which shows the relative increase in the number of null messages and global ECOT computations between the poor lookahead experiment considered in this section and the good lookahead scenario of Section 5.2. As seen in Figure 16, for the model with poor lookahead, the ANM protocol suppresses the increase of null messages by increasing the number of global ECOT computations. This result shows the capability of the ANM protocol to implicitly switch the EIT computation base to ECOT messages when the EIT values cannot be calculated efficiently with null messages. Null Message Number of Processors Figure 11: CQNF: NMR in the ANM protocol501502503504501 2 3 4 5 6 7 8 Window Size Number of Processors Figure 12: CQNF: Window size in the ANM protocol0.20.61 Performance Degradation Number of Processors NULL COND Figure 13: CQNF: Performance with low lookahead Performance Degradation Number of Processors NULL COND Figure 14: CQNF: Performance with low lookahead Performance Degradation Number of Processors NULL COND Figure 15: CQNF: Performance with low lookahead CQNF with good lookahead =Number of Processors Number of global computations in COND Number of global computations in ANM Number of null messages in NULL Number of null messages in ANM Figure Increase of null messages and global computations with low lookahead 5.3.2 CQN with Priority Servers (CQNP) In Section 5.3.1, we examined the impact of lookahead by explicitly reducing the lookahead value for each server entity. In this section, we vary the lookahead by using priority servers rather than FIFO servers in the tandem queues. We assume that each job can be in one of two priority classes: low or high, where a low priority job can be preempted by a high priority job. In this case, precomputation of service time cannot always imrove the lookahead because when a low priority job is in service, it may be interrupted at any time so the lookahead can at most be the remaining service time for the low priority job. Precomputed service times are useful only when the server is idle. The experiments described in this section investigate the impact of this variability in the lookahead of the priority server on the performance of each of the three protocols. Figure 17 shows the speedup achieved by ISP in the CQN models with priority servers (CQNP), where 25% of the jobs (256) are assumed to have a high priority. In comparison with Figure 5, ISP achieves slightly better performance because the events in priority servers have slightly higher computation granularity than those in the FCFS servers. The impact of computation granularity on protocol performance is explored further in the next section. Since the ISP performance does not depend on the lookahead of the model, no performance degradation was expected or observed for this protocol. Figures to 20 show the efficiency of each protocol for 64g. The performance of the COND protocol with the priority servers is very similar to that with the FCFS servers (Figures 6 to 8), while the performance of the other two protocols is considerably worse with the priority servers. Even though the connectivity is the same, the COND protocol outperforms the other two protocols for which shows that the COND protocol is not only connectivity insensitive but also less lookahead dependent than the null message based protocols. 5.4 Computation Granularity One of the good characteristics of the CQN models for the evaluation of parallel simulation protocols is its fine computation granularity. Computation granularity is a protocol-independent factor, but it changes the breakdown of the costs required for different operations. For a model with high computation granularity, the percentage contribution of the protocol-dependent factors to the execution time will be sufficiently small so as to produce relatively high efficiency for all the simulation protocols. We examine this by inserting a synthetic computation fragment containing 1,000,000 operations which is executed for every incoming message at each server. Figure 21 shows the impact of the computation granularity on speedup using ISP for the CQNF model with 32. The Number of Processors Figure 17: CQNP: Speedup achieved by ISP0.20.61 Efficiency Number of Processors NULL COND Figure CQNP: Efficiency of each protocol Efficiency Number of Processors NULL COND Figure 19: CQNP: Efficiency of each protocol Efficiency Number of Processors NULL COND Figure 20: CQNP: Efficiency of each protocol Number of Processors Original With Computation Fragment Figure 21: CQNF: Speedup with synthetic computation fragment Efficiency Number of Processors NULL COND Figure 22: CQNF: Efficiency of each protocol with synthetic computation fragment simulation horizon H is reduced to 1,000 as the sequential execution time for the model with the dummy computation fragment is 1,200 times longer than the original model. In the figure, the performance improvement with ISP is close to linear. The performance degrades a little when the number of processors is 5, 6 or 7 because the decomposition of the model among the processors leads to an unbalanced workload, which is one of the protocol-independent factors in parallel simulation. The improvement in speedup with larger computation granularity can be attributed to an improvement in the ratio of the event processing time to the other protocol-independent factors such as communication latency and context switch times. Figure 22 shows the impact of increasing the computation granularity on the efficiency of the three conservative protocols. Clearly, the additional computation diminishes the protocol-dependent overheads in the NULL and the ANM protocols, and the performance of these two protocols is very close to that of ISP. The performance of the COND protocol, however, gets worse as the number of processors increases, and the efficiency is only 54% with 8 processors. As described in 5.2.2, this degradation is due to the idle time that is spent by a processor while it is waiting for the completion of each global computation. This implies that even if the COND protocol employs a very efficient method in global ECOT computation, and the overhead for it becomes close to zero, the protocol can only achieve half the speedup of the ISP execution with 8 processors because of the idle time that must be spent waiting for the global ECOT computation to complete. This result shows the inherent synchronousness of the COND protocol, although our implementation of the algorithm is asynchronous. 6 Related Work Prior work on evaluating performance of discrete-event models with conservative protocols have used analytical and simulation models, as well as experimental studies. Performance of the null message deadlock avoidance algorithm [6] using queuing networks and synthetic benchmarks has been studied by Fujimoto [10]. Reed et al [16] have studied the performance of the deadlock avoidance algorithm and the deadlock detection and recovery algorithm on shared memory architecture. Chandy and Sherman [7] describe the conditional event algorithm and study its performance using queuing networks. Their implementation of the conditional event algorithm is synchronous (i.e. all LPs carry out local computations followed by a global computation), and the performance studies carried out in the paper assume a network with very high number of jobs. The paper does not compare the performance of the conditional event protocol with others. Nicol[15] describes the overheads of a synchronous algorithm similar to the conditional event algorithm studied in this paper. A mathematical model is constructed to qualify the overhead due to various protocol-specific factors. However, for analytical tractability, the paper ignores some costs for protocol-independent operations or simplification of the communication topology of the physical system. Thus, the results are useful for qualitative comparisons but do not provide the tight upper bound on potential performance that can be derived using ISP. The effect of lookahead on the performance of conservative protocols was studied by Fujimoto [10]. Nicol [14] introduced the idea of precomputing the service time to improve the lookahead. Cota and Sargent [8] have described the use of graphical representation of a process in automatically computing its lookahead. These performance studies are carried out with simulation models specific to their experiments. The performance study presented in this paper differs in two important respects: first, we developed and used a new metric - efficiency with respect to the Ideal Simulation protocol, which allows the protocol-specific overheads to be separated cleanly from other overheads that are not directly contributed by a simulation protocol. Second, the implementation of each of the algorithms was separated from the model which allows the performance comparison to be more consistent than if the algorithm is implemented directly in the simulation model. Conclusions An important goal of parallel simulation research is to facilitate its use by the discrete-event simulation community. Maisie is a simulation language that separates the simulation model from the specific algorithm (sequential or parallel) that is used to execute the model. Transparent sequential and optimistic implementations of Maisie have been developed and described previously [2]. This paper studied the performance of a variety of conservative algorithms that have been implemented in Maisie. The three algorithms that were studied include the null message algorithm, the conditional event algorithm, and a new algorithm called the accelerated null message algorithm that combines the preceding approaches. Maisie models were developed for standard queuing network benchmarks. Various configurations of the model were executed using the three different algorithms. The results of the performance study may be summarized as follows: ffl The Ideal Simulation Protocol (ISP) provides a suitable basis to compare the performance of conservative protocols as it clearly separates the protocol dependent and independent factors that affect the performance of a given synchronization protocol. It gives a realistic lower bound on the execution time of a simulation model for a given partitioning and architecture. Existing metrics like speedup and throughput indicate what the performance is but do not provide additional insight into how it could be improved. Other metrics like NMR (Null Message Ratio) and the window size are useful for the characterization of the protocol dependent overheads for the null message and conditional event protocols respectively, but they cannot be used among the diverse set of simulation protocols. ISP, on the other hand, can be used by an analyst to compare the protocol-dependent overheads even between conservative and optimistic protocols. ffl The null message algorithm exploited good parallelism in models with dense connectivity. However, the performance of this algorithm is very sensitive to the communication topology and lookahead characteristics of the model, and thus is not appropriate for models which have high connectivity or low lookahead values. ffl Because of its insensitivity to the communication topology of the model, the conditional event protocol outperforms the of null message based protocols when the LPs are highly coupled. Also, it performs very well for models with poor lookahead. Although the algorithmic syn- chronousness degrades its performance in parallel executions with a high number of processors, the performance of this protocol is very stable throughout our experiments. With faster global ECOT computations, it may achieve a better performance. Since the conditional event algorithm also has a good characteristic of not requiring positive lookahead values; this algorithm may also be used to simulate models which have zero lookahead cycles. ffl The performance of the ANM protocol is between those of the null message and the conditional event protocols for one processor execution, as expected. The performance in parallel executions, however, is very close to that of the null message protocol since small EIT advancements by the null messages defer the global ECOT computation. Thus, it performs worse than the conditional event protocol when the simulation model has high connectivity or very poor lookahead. However, compared to the null message protocol in such cases, the ANM protocol prevents an explosion in the number of null messages with frequent global ECOT computa- tions, and performs better than the null message protocol. This shows that the protocol has the capability for adaptively and implicitly switching its execution mode from the null message based synchronization to the conditional event based one for better performance. Since this algorithm also inherits the characteristic of the conditional event algorithm that can simulate the models with zero lookahead cycles, it is best suited for models whose properties are unknown. Future work includes the use of ISP for performance evaluation of various algorithms such as optimistic algorithms and synchronous algorithms together with the algorithms examined in this paper, using additional applications from different disciplines. Acknowledgments The authors wish to thank Professor Iyer and the three reviewers for their valuable comments and suggestions that resulted in significnat improvements to the presentation. We also acknowledge the assistance of past and present team members in the Parallel Computing Laboratory for their role in developing the Maisie software. Maisie may be obtained by annonymous ftp from http://may.cs.ucla.edu/projects/maisie. --R Stability of event synchronisation in distributed discrete event sim- ulation A unifying framework for distributed simulations. Maisie: A language for design of efficient discrete-event simulations Language support for parallel discrete-event simulations Global virtual time algorithms. Asynchronous distributed simulation via a sequence of parallel computations. The conditional event approach to distributed simulation. A framework for automatic lookahead computation in conservative distributed simulation. Parallel discrete event simulation. Performance measurements of distributed simulation strategies. State of the art in parallel simulation. Understanding the Limits of Optimistic and Conservative Parallel Simulation. Distributed discrete-event simulation Parallel discrete event simulation of fcfs stochastic queueing networks. The cost of conservative synchronization in parallel discrete event simulations. Parallel discrete event simulation: A shared memory approach. Variants of the chandy-misra-bryant distributed simulation al- gorithm --TR --CTR Alfred Park , Richard M. Fujimoto , Kalyan S. Perumalla, Conservative synchronization of large-scale network simulations, Proceedings of the eighteenth workshop on Parallel and distributed simulation, May 16-19, 2004, Kufstein, Austria Jinsheng Xu , Moon Jung Chung, Predicting the Performance of Synchronous Discrete Event Simulation, IEEE Transactions on Parallel and Distributed Systems, v.15 n.12, p.1130-1137, December 2004 Moo-Kyoung Chung , Chong-Min Kyung, Improving Lookahead in Parallel Multiprocessor Simulation Using Dynamic Execution Path Prediction, Proceedings of the 20th Workshop on Principles of Advanced and Distributed Simulation, p.11-18, May 24-26, 2006

lookahead;parallel simulation languages;discrete-event simulation;algorithmic efficiency;parallel and distributed simulation;conservative algorithms

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Modeling and Performance Comparison of Reliability Strategies for Distributed Video Servers.

AbstractLarge scale video servers are typically based on disk arrays that comprise multiple nodes and many hard disks. Due to the large number of components, disk arrays are susceptible to disk and node failures that can affect the server reliability. Therefore, fault tolerance must be already addressed in the design of the video server. For fault tolerance, we consider parity-based as well as mirroring-based techniques with various distribution granularities of the redundant data. We identify several reliability schemes and compare them in terms of the server reliability and per stream cost. To compute the server reliability, we use continuous time Markov chains that are evaluated using the SHARPE software package. Our study covers independent disk failures and dependent component failures. We propose a new mirroring scheme called Grouped One-to-One scheme that achieves the highest reliability among all schemes considered. The results of this paper indicate that dividing the server into independent groups achieves the best compromise between the server reliability and the cost per stream. We further find that the smaller the group size, the better the trade-off between a high server reliability and a low per stream cost.

Introduction 1.1 Server Design Issues Many multimedia applications such as online news, interactive television, and video-on-demand require large video servers that are capable of transmitting video data to thousands of users. In contrary to traditional file systems, video servers are subject to real-time constraints that impact the storage, retrieval, and delivery of video data. Furthermore, video servers must support very high disk bandwidth for data retrieval in order to serve a large number of video streams simultaneously. The most attractive approach for implementing a video server relies on disk arrays that (i) achieve high I/O performance and high storage capacity, (ii) can gradually grow in size, and (iii) are very cost efficient. Unfortunately, large disk arrays are vulnerable to disk failures, which results in poor reliability for the video server. A challenging task is therefore to design video servers that provide not only high performance but also high reliability. The video server considered in this paper is composed of many disk arrays, which are also referred to as server nodes. Each server node comprises a set of magnetic disk drives as illustrated in Figure 1 and is directly attached to the network. A video to store is divided into many blocks and the blocks are distributed among all disks and nodes of the video server in a round robin fashion. All server nodes are identical, each node containing the same number of disks D n . The total number of server disks D is then denotes the number of server nodes. Client Client Client Client Network Node Node Node Figure 1: Video Server Architecture A client that consumes a video from the server is connected to all server nodes and is served once every time interval called the service round. A video block that is received at the client during service round i is consumed during service round i+1. Further, since the transfer rate of a single disk is much higher than the stream playback rate, each disk can serve many streams during one service round. In order to efficiently retrieve multiple blocks from a disk during one service round, the video server applies the well known SCAN algorithm that optimizes the seek overhead by reordering the service of the requests. A very important decision in a video server design concerns the way data is distributed (striped) over its disks. To avoid hot spots, each video is partitioned into video blocks that are distributed over all disks of the server as already mentioned. Based on the way data are retrieved from disks, the literature distinguishes between the Fine-Grained (FGS) and the Coarse-Grained (CGS) Striping algorithms. FGS retrieves for one stream (client) multiple, typically small, blocks from many disks during a single service round. A typical example of FGS is RAID3 as defined by Katz et al. [1]. Derivations of FGS include the streaming RAID of Tobagi et al. [2], the staggered-group scheme of Muntz et al. [3], and the configuration planner scheme of Ghandeharizadeh et al. [4]. The main drawback of FGS is that it suffers from large buffer requirements that are proportional to the number of disks in the server [5, 6]. CGS retrieves for one stream one large video block from a single disk during a single service round. During the next service round, the next video block is retrieved from possibly a different disk. RAID5 is the classical example of CGS. Oezden et al. [7, 6] showed that CGS results in higher throughput than FGS for the same amount of resources (see also Vin et al. [8], Beadle et al. [9], and our contribution [5]). Accordingly, we will adopt CGS to store original video data on the video server. The paper is organized as follows. Section 2 classifies reliability schemes based on (i) whether mirroring or parity is used and (ii) the distribution granularity of redundant data. Related work is discussed at the end of the section. Section 3 studies reliability modeling for the reliability schemes considered. The reliability modeling is based on Continuous Time Markov Chains (CTMC) and concerns the case of independent disk failures as well as the case of dependent component failures. We focus in section 4 on the server performance, where we compare the per stream cost for the different reliability schemes. Section 5 emphasizes the trade-off between the server reliability and the per stream cost and studies the effect of varying the group size on the server reliability and the per stream cost. The results of section 5 lead to the conclusions of this paper, which are presented in section 6. 1.2 Our Contribution In the context of video servers, reliability has been addressed previously either by applying parity-based techniques (RAID2-6), e.g. [10, 8, 6, 11, 5], or by applying mirroring-based techniques (RAID1), e.g. [12, 13, 14]. However, all of the following aspects have not been considered together: ffl Comparison of several parity-based and mirroring-based techniques under consideration of both, the video server performance and cost issues. Our cost analysis concerns the storage and the buffering costs to achieve a given server throughput. ffl Reliability modeling based on the distribution granularity of redundant data in order to evaluate the server reliability for each scheme considered. We will perform a detailed reliability modeling that incorporates the case of independent disk failures and the case of dependent component failures. ffl Performance, cost, and reliability trade-offs of different parity-based as well as mirroring-based techniques. We will study the effect of varying the group size on the server reliability and the per stream cost and determine the best value of the group size for each technique. We use CGS to store/retrieve original video blocks, since it outperforms FGS in terms of the server throughput. Adding fault-tolerance within a video server implies the storage of redundant (replicated/parity) blocks. What remains to be decided is how redundant data is going to be stored/retrieved on/from the server. For mirroring, we limit ourselves to interleaved declustering schemes [15], where original data and replicated data are spread over all disks of the server. For parity, we limit ourselves to RAID5-like schemes, where parity blocks are evenly distributed over all server disks. We retain these schemes since they distribute the load uniformly among all server components. Additionally, we consider the case where only original blocks of a video are used during normal operation mode. During disk failure mode, replicated/parity blocks are needed to reconstruct lost original blocks that are stored on the failed component. Mirroring (also called consists in storing copies of the original data on the server disks. The main disadvantage of mirroring schemes is the 100% storage overhead due to storing a complete copy of the original data. Reliability based on parity consists in storing parity data in addition to original data (RAID2-6). When a disk failure occurs, parity blocks together with the remaining original data are used to reconstruct failed original blocks. The RAID5 [18] parity scheme requires one parity block for each (D \Gamma 1) original blocks, where D is the total number of server disks. The (D \Gamma 1) original blocks and the one parity block constitute a parity group. Although the additional storage volume is small for parity-based reliability, the server needs additional resources in terms of I/O bandwidth or main memory when working in disk failure mode. In fact, in the worst case, the whole parity group must be retrieved and temporarily kept in the buffer to reconstruct a lost block. In [5], we have distinguished between the second read strategy and the buffering strategy. The second read strategy doubles the I/O bandwidth requirement [19], whereas the buffering strategy increases the buffer requirement as compared to the failure free mode. We will restrict our discussion to the buffering strategy, since it achieves about twice as much throughput as the second read strategy [5, 20]. We will see that the buffering strategy becomes more attractive in terms of server performance (lower buffer requirements) and also regarding the server reliability when the size of the parity group decreases. 2.1 Classification of Reliability Schemes Reliability schemes differ in the technique (parity/mirroring) and in the distribution granularity of redundant data. We define below the distribution granularity of redundant data. ffl For the parity technique, the distribution granularity of redundant data is determined by whether the parity group comprises All (D) or Some (D c ) disks of the server. For the latter case, we assume that the server is partitioned into independent groups and that all groups are the same size, each of them containing D c disks. Let C denote the number of groups in the server ffl For the mirroring technique, the distribution granularity of redundant data has two different aspects: - The first aspect concerns whether the original blocks of one disk are replicated on One, Some (D c ), or All remaining (D \Gamma 1) disks of the server. - The second aspect concerns how a single original block is replicated. Two ways are distinguished. The first way replicates one original block entirely into one replicated block [13], which we call entire block replication. The second way partitions one original block into many sub-blocks and stores each sub-block on a different disk [14], which we call sub-block replication. We will show later on that the distinction between entire block and sub-block replication is decisive in terms of server performance (throughput and per stream cost). Table classifies mirroring and parity schemes based on their distribution granularity. We use the terms One-to- One, One-to-All, and One-to-Some to describe whether the distribution granularity of redundant data concerns one disk (mirroring), all disks (mirroring/parity), or some disks (mirroring/parity) disks. For the One-to-One scheme, only mirroring is possible, since One-to-One for parity would mean that the size of each parity group equals 2, which consists in replicating each original block (mirroring). Hence the symbol "XXX" in the table. Mirroring Parity One-to-One Chained declustering [21, 22] XXX One-to-All Entire block replication (doubly striped) [13, 23] RAID5 with one group [18] Sub-block replication [15] One-to-Some Entire block replication RAID5 with many groups [3, 8, 7] Sub-block replication [14] Table 1: Classification of the different reliability schemes Table distinguishes seven schemes. We will give for each of these schemes an example of the data layout . Thereby, we assume that the video server contains 6 disks and stores a single video. The stored video is assumed to be divided into exactly blocks. All schemes store original blocks in the same order (round robin) starting with disk 0 (Figures 2 and 3). What remains to describe is the storage of redundant data for each of the schemes. Figure presents examples of the mirroring-based schemes. These schemes have in common that each disk is partitioned into two separate parts, the first part storing original blocks and the second part storing replicated blocks. As illustrated in Figure 2(a), the One-to-One mirroring scheme (Mirr one ) simply replicates original blocks of one disk onto another disk. If one disk fails, its load is entirely shifted to its replicated disk, which creates load-imbalances within the server (the main drawback of the One-to-One scheme). Original blocks Copies Organization MirrOne .0000001111110000001111110000001111110000001111110000001111110000001111110000001111110000001111110000001111110000001111118315105171220 21 22 23 24 26 Original blocks Copies (b) One-to-All Organization with Entire block replication Mirr all\Gammaentire .0000001111110000001111110000001111110000001111110000001111118315105171220 21 22 23 24 26 28 29 30719Original blocks Copies 7.1 7.2 7.3 7.4 7.5 13.1 13.2 13.3 13.4 13.5 19.1 19.2 19.3 19.4 19.5 25.1 25.2 25.3 25.4 25.5 1.1 1.2 1.3 1.4 1.5 (c) One-to-All Organization with Sub-blocks replication Mirr all\Gammasub .0000001111110000001111110000001111110000001111110000001111110000001111110000001111110000001111110000001111110000001111118315105171220 21 22 23 24 26 Original blocks Copies (d) One-to-Some Organization with Entire block replication Mirrsome\Gammaentire 26 28 29 30719Original blocks Copies 7.1 7.2 13.1 13.2 19.1 19.2 25.1 25.2 1.1 1.2 4.1 4.2 28.1 28.2 One-to-Some Organization with Sub-blocks replication Mirrsome\Gammasub . Figure 2: Mirroring-based schemes. In order to distribute the load of a failed disk evenly among the remaining disks of the server, the One-to-All mirroring scheme is applied as shown in Figures 2(b) and 2(c). Figure 2(b) depicts entire block replication (Mirr all\Gammaentire ) and Figure 2(c) depicts sub-block replication (Mirr all\Gammasub ). In Figure 2(c), we only show how original blocks of disk 0 are replicated over disks 1, 2, 3, 4, and 5. If we look at Figures 2(b) and 2(c), we realize that only a single disk failure is allowed. When two disk failures occur, the server cannot ensure the delivery of all video data. The One-to-Some mirroring scheme trades-off load-imbalances of the One-to-One mirroring scheme and the low reliability of the One-to-All mirroring scheme. In fact, as shown in Figures 2(d) (entire block replication, Mir some\Gammaentire ) and 2(e) (sub-block replication, Mir some\Gammasub ), the server is divided into multiple (2) independent groups. Each group locally employs the One-to-All mirroring scheme. Thus, original blocks of one disk are replicated on the remaining disks of the group and therefore the load of a failed disk is distributed over all remaining disks of the group. Further, since each group tolerates a single disk failure, the server may survive multiple disk failures. Figure 3 presents two layout examples of RAID5 that correspond to the One-to-All parity scheme, Par all (Figure 3(a)) and the One-to-Some parity scheme, Par some (Figure 3(b)). In Figure 3(a) the parity group size is 6, e.g. the 5 original blocks 16, 17, 18, 19, and 20 and the parity block P 4 build a parity group. In Figure 3(b) the parity groups size is 3, e.g. the 2 original blocks 17 and and the parity block P 8 build a parity group.000111111000111111000111111000111111000111111000011111111 26 (a) One-to-All Organization Par all .000111111000111111000111111000111111000111111 000000111000000111000000111000000111000000111000000111000000111000000111000111000000111P1 1 2 P2 3 4 26 P13 27 28 29 P14 (b) One-to-Some Organization Parsome . Figure 3: Parity-based Schemes. Looking at Figures 2 and 3, we observe that all One-to-All schemes (mirroring with entire block replication (Mirr all\Gammaentire ), mirroring with sub-block replication (Mirr all\Gammasub ), and RAID5 with one group (Par all tolerate one disk failure. All these schemes therefore have the same server reliability. The same property holds for all One-to-Some schemes (mirroring with entire block replication, mirroring with sub-block replication, and RAID5 with C groups), since they all tolerate at most a single disk failure on each group. Consequently, it is enough for our reliability study to consider the three schemes (classes): One-to-One, One-to-All, and One-to- Some. However, for our performance study we will consider in section 4 the different schemes of Table 1. 2.2 Related Work Based on RAID [1], reliability has been addressed previously in the literature either in a general context of file storage, or for video server architectures. Mechanisms to ensure fault-tolerance by adding redundant information to original content can be classified into parity-based schemes and mirroring-based schemes. An extensive amount of work has been carried out in the context of parity-based reliability, see e.g. [24, 2, 17, 10, 25, 26, 7, 27, 28]. These papers ensure a reliable real-time data delivery, even when one or some components fail. These papers differ in the way (i) they stripe data such as RAID3 (also called FGS) or RAID5 (also called CGS), and (ii) allocate parity information within the server (dedicated, shared, declustered, randomly, sequentially, SID, etc.), and (iii) the optimization goals (throughput, cost, buffer requirement, load-balancing, start-up latency for new client requests, disk bandwidth utilization, etc. Video servers using mirroring have been proposed previously, see e.g. [12, 15, 23, 14, 29, 21, 22]. However no reliability modeling has been carried out. Many mirroring schemes were compared by Merchant et al. [15], where some striping strategies for replicated disk arrays were analyzed. Depending on the striping granularity of the original and the replicated data, they distinguish between the uniform striping (CGS for original and replicated data in dedicated or in chained form) and the dual striping (original data are striped using CGS and replicated data are striped using FGS). However, their work is different from our study in many regards. First, the authors assume that both copies are used during normal and failure operation mode. Second, the comparison of different mirroring schemes is based on the mean response times and on the throughput achieved without taking into account server reliability. Finally, the authors do not analyze the impact of varying the distribution granularity of redundant data on server reliability and server performance. In a general context of Redundant Arrays of Inexpensive Disks, Trivedi et al. [30] analyzed the reliability of RAID1-5 and focused on the relationship between disk's MTTF and system reliability. Their study is based on the assumption that a RAID system is partitioned into cold and hot disks, where only hot disks are active during normal operation mode. In our case, we study reliability strategies for video servers that do not store redundant data separately on dedicated disks, but distribute original and redundant data evenly among all server disks. Gibson [31] uses continuous Markov models in his dissertation to evaluate the performance and reliability of parity-based redundant disk arrays. In the context of video servers, reliability modeling for parity-based schemes (RAID3, RAID5) has been performed in [32] and RAID3 and RAID5 were compared using Markov reward models to calculate server avail- ability. The results show that RAID5 is better than RAID3 in terms of the so-called performability (availability combined with performance). To the best of our knowledge, there is no previous work in the context of video servers that has compared several mirroring and parity schemes with various distribution granularities in terms of the server reliability and the server performance and costs. 3 Reliability Modeling 3.1 Motivation We define the server reliability at time t as the probability that the video server is able to access all videos stored on it provided that all server components are initially operational. The server survives as long as its working components deliver any video requested to the clients. As we have already mentioned, the server reliability depends on the distribution granularity of redundant data and is independent of whether mirroring or parity is used. In fact, what counts for the server reliability is the number of disks/nodes that are allowed to fail without causing the server to fail. As an example, the One-to-All mirroring scheme with entire block replication (Mirr all\Gammaentire ) only tolerates a single disk failure. This is also the case for Mirr all\Gammasub and for Par all . These three schemes have therefore the same server reliability. In light of this fact, we use the term One-to-All to denote all of the three schemes for the purpose of our reliability study. Analogous to One-to-All, the term One-to- Some will represent the three schemes Mirr some\Gammaentire , Mirr some\Gammasub , and Par some and the term One-to-One denotes the One-to-One mirroring scheme Mirr one . We use Continuous Time Markov Chains (CTMC) for the server reliability modeling [33] . CTMC has discrete state spaces and continuous time and is also referred to as Markov process in [34, 35]. We assume that the mean time to failure (MTTF ) of every component is exponentially distributed. The server MTTF s and the server reliability R s (t) have the following relationship (assuming that MTTF s ! 1): MTTF The mean time to disk failure MTTF d equals 1 and the mean time to disk repair MTTR d equals 1 To build the state-space diagram [34] of the corresponding CTMC, we introduce the following parameters: s denotes the total number of states that the server can denotes a state in the Markov chain with is the probability that the server is in state i at time t. We assume that the server is fully operational at time t is the initial state. Additionally, state (s \Gamma 1) denotes the system failure state and is assumed to be an absorbing state (unlike previous work [32], where a Markov model was used to compare the performance of RAID3 and RAID5 and allowed the repair of an overall server failure). When the video server attains state (s \Gamma 1), it is assumed to stay there for an infinite time. This previous assumption allows to concentrate our reliability study on the interval between the initial start up time and the time, at which the first server failure occurs. Thus: p 0 1. The server reliability function R s (t) can then be computed as [34]: R s We present in the remainder of this section the Markov models for the three schemes One-to-All, One-to-One, and One-to-Some, assuming both, (i) independent disk failures (section 3.2) and (ii) dependent component failures (section 3.3). 3.2 Reliability Modeling for Independent Disk Failures 3.2.1 The One-to-All Scheme With the One-to-All scheme (Mirr all\Gammaentire , Mirr all\Gammasub , Par all ), data are lost if at least two disks have failed. The corresponding state-space diagram is shown in Figure 4, where states 0, 1, and F denote respectively the initial state, the one disk failure state, and the server failure state. l d l d D . (D-1) . Figure 4: State-space diagram for the One-to-All Scheme. The generator matrix Q s of this CTMC is then: \GammaD 3.2.2 The One-to-One Scheme The One-to-One scheme (Mirr one ) is only relevant for mirroring. As the One-to-One schemes stores the original data of disk i on disk ((i D), the server fails if two consecutive disks fail. Depending on the location of the failed disks, the server therefore tolerates a number of disk failures that can take values between 1 and DThus, the number of disks that are allowed to fail without making the server fail can not be known in advance, which makes the modeling of the one-to-one scheme complicated: Let the D server disks be numbered from 0 to D \Gamma 1. Assume that the server continues to operate after failures. Let P (k) be the probability that the server does not fail after the k th disk failure. P (k) is also the probability that no disks that have failed are consecutive (adjacent). We calculate in Appendix A the probability P (k) for all k 2 [2:: D 2 ]. It is obvious that Figure 5 shows the state-space diagram for Mirr one . If the server is in state i (i - 1) and failure then the probability that the server fails (state F) equals and the probability that the server continuous operating d n+1 d D/2 F l l l d d d Figure 5: State-space diagram for the One-to-One scheme. The parameters of Figure 5 have the following values: . The corresponding generator matrix Q s of the CTMC above is: (1) 3.2.3 One-to-Some Scheme The One-to-Some scheme (mirroring/parity) builds independent groups. The server fails if one of its C groups fails and the group failure distribution is assumed to be exponential. We first model the group reliability and then derive the server reliability. Figures 6(a) and 6(b) show the state-space diagrams of one group and of the server respectively. In Figure 6(b), the parameter C denotes the number of groups in the server and - c denotes the group failure rate. The generator matrix Q c for the CTMC of a single group is: l d D c . D c l d (a) State-space diagram of one group. l c C . (b) State-space diagram of the server. Figure State-space diagrams for the One-to-Some Scheme. The group reliability function R c (t) at time t is R c (t). The group mean lifetime MTTF c is then derived from R c (t). To calculate the overall server reliability function, we assume that the group failure distribution is exponential. The server failure rate is thus C MTTFc . The server generator matrix Q s of the CTMC of Figure 6(b) is therefore: \GammaC (2) 3.3 Reliability Modeling for Dependent Component Failures Dependent component failures mean that the failure of a single component of the server can affect other server components. We recall that our server consists of a set N of server nodes, where each node contains a set of disks. Disks that belong to the same node have common components such as the node's CPU, the disk controller, the network interface, etc. When any of these components fails, all disks contained in the affected node are unable to deliver video data and are therefore considered as failed. Consequently, a single disk does not deliver video data anymore if itself fails or if one of the components of the node fails to which this disk belongs. We present below the models for the different schemes for the case of dependent component failures. Similarly to a disk failure, a node failure is assumed to be repairable. The failure rate - n of the node is exponentially distributed with - MTTFn , where MTTF n is the mean life time of the node. The repair rate - n of a failed node is exponentially distributed as - MTTRn , where MTTR n is the mean repair time of the node. For mirroring and parity schemes, we apply the so called Orthogonal RAID mechanism whenever groups must be built. Orthogonal RAID was discussed in [31] and [17]. It is based on the following idea. Disks that belong to the same group must belong to different nodes. Thus, the disks of a single group do not share any (common) node hardware components. Orthogonal RAID has the property that the video server survives a complete node failure: When one node fails, all its disks are considered as failed. Since these disks belong to different groups, each group will experience at most one disk failure. Knowing that one group tolerates a single disk failure, all groups will survive and therefore the server will continue operating. Until now, Orthogonal RAID was only applied in the context of parity groups. We generalize the usage of Orthogonal RAID for both, mirroring and parity. In order to distinguish between disk and node failure when building the models of the schemes considered, we will present each state as a tuple [i; j], where i gives the number of disks failed and j gives the number of nodes failed. The failure (absorbing) state is represented with the letter F as before. 3.3.1 The One-to-All Scheme For the One-to-All schemes (Mirr all\Gammaentire , Mirr all\Gammasub , and Par all ), double disk failures are not allowed and therefore a complete node failure causes the server to fail. Figure 7 shows the state-space diagram for the One-to- All scheme for the case of dependent component failures. The states of the model denote respectively the initial state ([0; 0]), the one disk failure state ([1; 0]), and the server failure state l d l n l d l n d N . D . (D-1) . + N . Figure 7: State-space diagram for the One-to-All scheme with dependent component failures. The generator matrix Q s is then: \GammaD 3.3.2 The Grouped One-to-One Scheme Considering dependent component failures, the One-to-One scheme as presented in Figure 2(a) would achieve a very low server reliability since the server immediately fails if a single node fails. We propose in the following an organization of the One-to-One scheme that tolerates a complete node failure and even Nnode failures in the best case. We call the new organization the Grouped One-to-One scheme. The Grouped One-to-One organization keeps the initial property of the One-to-One scheme, which consists in completely replicating the original content of one disk onto another disk. Further, the Grouped One-to-One organization divides the server into independent groups, where disks belonging to the same group have their replica inside this group. The groups are built based on the Orthogonal RAID principle and thus disks of the same group belong to different nodes as Figure 8 shows. Figure 8 assumes one video containing 40 original blocks and is stored on a server that is made of four nodes each containing two disks. Inside one group, up-to Dcdisk failures can be tolerated, where D is the number of disks inside each group. The Grouped One-to-One scheme can therefore survive N failures in the best case (the server in Figure 8 continues operating even after nodes N 1 and N 2 fail). In order to distribute the load of a failed node among possibly many and not only one of the surviving nodes, the Grouped One-to-One scheme ensures that disks belonging to the same node have their replica on disks that do not belong to the same node 1 . 1 Assume that node N1 has failed, then its load is shifted to node N3 (replica of disk 0 are stored on disk 4) and to node N4 (replica of disk 1 are stored on disk 7) Original blocks Copies 6 79251329 Figure 8: Grouped One-to-One scheme for a server with 4 nodes, each with 2 disks (D In order to model the reliability of our Grouped One-to-One scheme, we first study the behavior of a single group and then derive the overall server reliability. One group fails when two consecutive disks inside the group fail. We remind that two disks are consecutive if the original data of one of these disks are replicated on the other disk (for group 1 the disks 0 and 4 are consecutive, whereas the disks 0 and 2 are not). Note that the failure of one disk inside the group can be due to (i) the failure of the disk itself or (ii) the failure of the whole node, to which the disk belongs. After the first disk failure, the group continues operating. If the second disk failure occurs inside the group, the group may fail or not depending on whether the two failed disks are consecutive. Let P (2) denotes the probability that the two failed disks of the group are not consecutive. Generally, P (k) denotes the probability that the group does not fail after the k th disk failure inside the group. P (k) is calculated in Appendix A. The state-space diagram of one group for the example in Figure 8 is presented in Figure 9. The number of disks inside the group is D disks have failed inside the group, where i disks, themselves, have failed and j nodes have failed. Obviously, all of the i disks that have failed belong to different nodes than all of the j nodes that have failed. We describe in the following how we have built the state-space diagram of Figure 9. At time t 0 , the group is in state [0; 0]. The first disk failure within the group can be due to a single disk failure (state [1; 0]) or due to a whole node failure (state [0; 1]). Assume that the group is in state [1; 0] and one more disk of the group fails. Four transitions are possible: (i) the group goes to state [2; 0] after the second disk of the group has failed itself and the two failed disks are not consecutive, (ii) the group goes to state after the node has failed, on which the second failed disk of the group is contained and the two failed disks are not consecutive, (iii) the group goes to state F after the second disk of the group has failed (disk failure or node failure) and the two failed disks are consecutive, and finally (iv) the group goes to state [0; 1] after the node has failed, to which the first failed disk of the group belongs and thus the number of failed disks in the group does not increase (only one disk failed). The remaining of the state-space diagram is to derive in an analogous way. The parameters of Figure 9 are the following: The generator matrix Q c for a group is: l n l n l n F F F l 5 l 7 l 7 l 7 l 7 l 3 l 1 l 2 l 6 l 4 Figure 9: State-space diagram of one group for the Grouped One-to-One scheme with dependent component failures (D From the matrix Q c we get the group mean life time MTTF c , which is used to calculate the overall server reliability. The state-space diagram for the server is the one of Figure 6(b), where the parameter - takes the value denotes the failure rate of one group with - MTTFc . The server reliability is then calculated analogously to Eq. 2. Note that the example described in Figure 9 considers a small group size (D c = 4). Increasing D c increases the number of states contained in the state-space diagram of the group. In general, the number of states for a given We present in Appendix B a general method for building the state-space diagram of one group containing D c disks. 3.3.3 The One-to-Some Scheme We use Orthogonal RAID for all One-to-Some schemes (Mirr some\Gammaentire , Mirr some\Gammasub , and Par some ). If we consider again the data layouts of Figures 2(d), 2(e), and 3(b), Orthogonal RAID is then ensured if the following holds: Node 1 contains disks 0 and 3; node 2 contains disks 1 and 4; and node 3 contains disks 2 and 5. For the reliability modeling of the One-to-Some scheme, we first build the state-space diagram for a single group Figure and then compute the overall server reliability (Figure 10(b)). The states in Figure 10(a) denote the following: the initial state ([0; 0]), the state where one disk fails ([1; 0]), the state where one node fails resulting in a single disk failure within the group ([0; 1]), the state where one disk and one node have failed and the failed disk belongs to the failed node ([0; 1"]), and the one group failure state ). The parameter values used in Figure are: l 4 l 4 l 4 l 1 l 2 l 3 (a) State-space diagram for one group. l c C . (b) State-space diagram for the server. Figure 10: State-space diagrams for the One-to-Some Scheme for the case of dependent component failures. The generator matrix Q c for a group is: 3.4 Reliability Results We resolve our continuous time Markov chains using the SHARPE (Symbolic Hierarchical Automated Reliability and Performance Evaluator) [33] tool for specifying and evaluating dependability and performance models. Sharpe takes as input the generator matrix and computes the server reliability at a certain time t. The results for the server reliability are shown in Figures 11 and 12. The total number of server disks considered is and the number of server nodes is 10, each node containing 10 disks. We examine server reliability for two failure rates, 1 hours and 1 100000 hours , which are pessimistic values. Figure 11 plots the server reliability for the One-to-One, One-to-All, and One-to-Some schemes for the case of independent disk failures. As expected, the server reliability for the One-to-One scheme is the highest. The One-to-Some scheme exhibits higher server reliability than the One-to-All scheme. Figures 11(a) and 11(b) also show how much the server reliability is improved when mean time to disk failure increases (- d decreases). For example, for the One-to-One scheme and after 10 4 days of operation, the server reliability is about 0:3 for hours and is about 0:66 for - 100000 hours Figure 12 depicts the server reliability for the Grouped One-to-One, the One-to-All, and the One-to-Some schemes for the case of dependent component failures. We observe that the Grouped One-to-One scheme provides a better reliability than the One-to-Some scheme. The One-to-All scheme has the lowest server reliability, 0.40.8Time [days] Reliability Function Server Reliability (Independent Disk Failures) One-2-One One-2-Some One-2-All (a) Server reliability for hours and 72 hours Time [days] Reliability Function Server Reliability (Independent Disk Failures) One-2-One One-2-Some One-2-All (b) Server reliability for 100000 hours and 72 hours Figure 11: Server reliability for the three schemes assuming independent disk failures with 10. e.g. for - 100000 hours and after three years, the server reliability is 0 for the One-to-All scheme, 0:51 for the One-to-Some scheme, and 0:85 for the Grouped One-to-One scheme. Figures 12(a) and 12(b) show that the server reliability increases when - d (- n ) decreases. Time [days] Reliability Function Server Reliability (Dependent Component Failures) Grouped One-2-One One-2-Some One-2-All (a) Server reliability for hours and 72 hours Time [days] Reliability Function Server Reliability (Dependent Component Failures) Grouped One-2-One One-2-Some One-2-All (b) Server reliability for 100000 hours and 72 hours Figure 12: Server reliability for the three schemes assuming dependent component failures with Comparing Figures 11 and 12, we see, as expected, that the server reliability is higher for the independent disk failure case than for the dependent component failure case. We restrict our further discussion to the case of dependent component failures since it is more realistic than the case of independent disk failures. 4 Server Performance An important performance metric for the designer and the operator of a video server is the maximum number of streams Q s that the server can simultaneously admit, which is referred to as the server throughput. Adding fault-tolerance within a server requires additional resources in terms of storage volume, main memory and I/O bandwidth capacity. As we will see, the reliability schemes discussed differ not only in the throughput they achieve, but also in the amount of additional resources they need to guarantee uninterrupted service during disk failure mode. Throughput is therefore not enough to compare server performance of these schemes. Instead, we use the cost per stream. We calculate in section 4.1 the server throughput for each of the schemes. Section 4.2 focuses on buffer requirements. Section 4.3 compares then the different schemes with respect to the cost per stream. 4.1 Server Throughput The admission control policy decides, based on the remaining available resources, whether a new incoming stream is admitted. The CGS striping algorithm serves a list of streams from a single disk during one service round. During the next service round, this list of streams is shifted to the next disk. If Q d denotes the maximum number of streams that a single disk can serve simultaneously (disk throughput) in a non fault-tolerant server, then the overall server throughput Q s is simply Q . Accordingly, we will restrict our discussion to disk throughput. Disk throughput Q d is given by Eq. 3 [5], where meaning and value of the different parameters are listed in Table 2. The disk parameter values are those of Seagate and HP for the SCSI II disk drives [36]. r d r d Parameter Meaning of Parameter Value r p Video playback rate 1:5 Mbps r d Inner track transfer rate 40 Mbps t stl Settle time 1:5 ms ms t rot Worst case rotational latency 9:33 ms b Block size 1 Mbit - Service round duration b dr rp sec Table 2: Performance Parameters To allow for fault-tolerance, each disk reserves a portion of its available I/O bandwidth to be used during disk failure mode. Since the amount of reserved disk I/O bandwidth is not the same for all schemes, the disk throughput will also be different. Let us start with the Grouped One-to-One scheme Mirr one . Since the original content of a single disk is entirely replicated onto another disk, half of each disk's I/O bandwidth must be kept unused during normal operation mode to be available during disk failure mode. Consequently the disk throughput Q mirr One is simply the half of Q For the One-to-All mirroring scheme Mirr all\Gammaentire with entire block replication, the original blocks of one disk are spread among the other server disks. However, it may happen that the original blocks that would have been required from a failed disk during a particular service round are all replicated on the same disk (worst case situation). In order to guarantee deterministic service for this worst case, half of the disk I/O bandwidth must be reserved to disk failure mode. Therefore, the corresponding disk throughput Q mirr All\GammaEntire is: Q mirr worst case retrieval pattern for the One-to-Some mirroring scheme Mirr some\Gammaentire with entire block replication is the same as for the previous scheme and we get: Q mirr Since the three schemes Mirr one , Mirr all\Gammaentire , and Mirr some\Gammaentire achieve the same throughput, we will use the term MirrEntire to denote all of them and Q mirr Entire 2 to denote their disk throughput: For the One-to-All mirroring scheme Mirr all\Gammasub with sub-block replication, the situation changes. In fact, during disk failure mode, each disk retrieves at most Q mirr All\GammaSub original blocks and Q mirr All\GammaSub replicated sub-blocks during one service round. Let us assume that sub-blocks have the same size b all sub , i.e. sub The admission control formula becomes: All\GammaSub \Delta r d b all sub r d b+b all sub r d Similarly, the disk throughput Q mirr Some\GammaSub for One-to-Some mirroring with sub-block replication Mirr some\Gammasub is: b+b some sub r d where b some sub denotes the size of a sub-block as c is the number of disks contained on each group. We now consider the disk throughput for the parity schemes. Recall that we study the buffering strategy and not the second read strategy for lost block reconstruction. For the One-to-All parity scheme Par all , one parity block is needed for every (D \Gamma 1) original blocks. The additional load of each disk consisting in retrieving parity blocks when needed can be seen from Figure 3(a). In fact, for one stream in the worst case all requirements for parity blocks concern the same disk, which means that at most one parity block is retrieved from each disk every service rounds. Consequently, each disk must reserve 1 D of its I/O bandwidth for disk failure mode. The disk All is then calculated as: Analogous to the One-to-All parity scheme, the One-to-Some parity scheme Par some has the following disk These three schemes share the common property that each original block is entirely replicated into one block. In Figure 13(a), we take the throughput value Q mirr Entire of MirrEntire as base line for comparison and plot the ratios of the server throughput as a function of the total number of disks in the server. 8000.51.5Number of Disks D in the Server Throughput Server Throughput Ratio Par all Par some Mirr all-sub Mirr some-sub Mirr Entire (a) Throughput Ratios. Number of Disks in the Server Number of disks for the same throughput Mirr Entire Mirr some-sub Mirr all-sub Par some Par all (b) Number of disks required for the same Server throughput. Figure 13: Throughput results for the reliability schemes with D Mirroring schemes that use entire block replication (MirrEntire ) provide lowest throughput. The two mirroring schemes Mirr all\Gammasub and Mirr some\Gammasub that use sub-block replication have throughput ratios of about 1:5. The performance for Mirr all\Gammasub is slightly higher than the one for Mirr some\Gammasub since the sub-block size b all sub . Parity schemes achieve higher throughput ratios than mirroring schemes and the One-to-All parity scheme Par all results in the highest throughput. The throughput for the One-to-Some parity scheme Par some is slightly smaller than the throughput for Par all . In fact, the parity group size of (D \Gamma 1) for Par all is larger than D c . As a consequence, the amount of disk I/O bandwidth that must be reserved for disk failure is smaller for Par all than for Par some . In order to get a quantitative view regarding the I/O bandwidth requirements, we reverse the axes of Figure 13(a) to obtain in Figure 13(b) for each scheme the number of disks needed to achieve a given server throughput. 4.2 Buffer Requirement Another resource that affects the cost of the video server and therefore the cost per stream is main memory. Due to the speed mismatch between data retrieval from disk (transfer rate) and data consumption (playback rate), main memory is needed at the server to temporarily store the blocks retrieved. For the SCAN retrieval algorithm, the worst case buffer requirement for one served stream is twice the block size b. Assuming the normal operation mode (no component failures), the buffer requirement B of the server is therefore denotes the server throughput. Mirroring-based schemes replicate original blocks that belong to a single disk over one, all, or a set of disks. During disk failure mode, blocks that would have been retrieved from the failed disk are retrieved from the disks that store the replica. Thus, mirroring requires the same amount of buffer during normal operation mode and during component failure mode independently of the distribution granularity of replicated data. Therefore, for all mirroring schemes considered (Grouped One-to-One Mirr one , One-to-All with entire block replication Mirr all\Gammaentire , One-to-All with sub-block replication Mirr all\Gammasub , One-to-Some with entire block replication Mirr some\Gammaentire , and One-to-Some with sub-block replication Mirr some\Gammasub ) the buffer requirement during component failure is B Unlike mirroring-based schemes, parity-based schemes need to perform a X-OR operation over a set of blocks to reconstruct a lost block. In fact, during normal operation mode the buffer is immediately liberated after consumption. When a disk fails, original blocks as well as the parity block that belong to the same parity group are sequentially retrieved (during consecutive service rounds) from consecutive disks and must be temporarily stored in the buffer for as many service rounds that elapse until the lost original block will be reconstructed. Since buffer overflow must be avoided, the buffer requirement is calculated for the worst case situation where the whole parity group must be contained in the buffer before the lost block gets reconstructed. An additional buffer size of one block must be also reserved to store the first block of the next parity group. Consequently, during component failure, the buffer requirement B par all for Par all is B par B, and the buffer requirement B par some for Par some is B par Note that the buffer requirement for Par all depends on D and therefore increases linearly with the number of disks in the server. For Par some , however, the group size D c can be kept constant while the total number of disks D varies. As a result, the buffer requirement some for Par some remains unchanged when D increases. 4.3 Cost Comparison The performance metric we use is the per stream cost. We first compute the total server cost $ server and then derive the cost per stream $ stream as: Qs . We define the server cost as the cost of the hard disks and the main memory dimensioned for the component failure mode: Pmem is the price of 1 Mbyte of main memory, B the buffer requirement in Mbyte, P d is the price of 1 Mbyte of hard disk, V disk is the storage volume of a single disk in MByte, and finally D is the total number of disks in the server. Current price figures - as of 1998 - are Pmem = $13 and P these prices change frequently, we will consider the relative costs by introducing the cost ratio ff between Pmem and Thus, the server cost function becomes: Pmem ff and the per stream cost is: Pmem \Delta ff To evaluate the cost of the five different schemes, we compute for each scheme and for a given value of D the achieved and the amount of buffer B required to support this throughput. Note that we take for the schemes Mirr some\Gammaentire , Mirr some\Gammasub , and Par some . Figure 14 plots the per stream cost for the schemes Par all , Par some , MirrEntire , Mirr some\Gammasub , and Mirr all\Gammasub for different values of the cost ratio ff. We recall that the notation MirrEntire includes the three mirroring schemes Mirr all\Gammaentire , Mirr some\Gammaentire , and the Grouped One-to-One Mirr one that experience the same throughput and require the same amount of resources. In Figure 14(a), we consider 26 that presents the current memory/hard disk cost ratio. Increasing the value of means that the price for the disk storage drops faster than the price for main memory: In Figure 14(b), we multiply the current cost ratio by five to get On the other hand, decreasing the value of ff means that the price for main memory drops faster than the price for hard disk: In Figure 14(c) we divide the current cost ratio by five to get 3 To illustrate the faster decrease of the price for hard disk as compared to the one for main memory, we consider the current price for main memory (Pmem = $13) and calculate the new reduced price for hard disk 4 Analogously, to illustrate the faster decrease of the price for memory as compared to the price for hard disk, we take the current price for hard disk calculate the new reduced price for memory (Pmem Number of Disks D in the Server Cost Per Stream Cost Par all Mirr Entire Par some Mirr some-sub Mirr all-sub (a) 0:5 Number of Disks D in the Server Cost Per Stream Cost Par all Par some Mirr Entire Mirr some-sub Mirr all-sub 0:1 Number of Disks D in the Server Cost Per Stream Cost Par all Mirr Entire Mirr some-sub Mirr all-sub Par some (c) 0:5 5:2. Figure 14: Per stream cost for different values of the cost ratio ff with D The results of Figure 14 indicate the following: ffl The increase or the decrease in the value of ff as defined above means a decrease in either the price for hard disk or the price for main memory respectively. Hence the overall decrease in the per stream cost in Figures 14(b) and 14(c) as compared to Figure 14(a). Figure shows that the One-to-All parity scheme (Par all ) results in the highest per stream cost that increases when D grows. In fact, the buffer requirement for Par all is highest and also increases linearly with the number of disks D and thus resulting in the highest per stream cost. Mirroring schemes with entire block replication (MirrEntire ) have the second worst per stream cost. The per stream cost for the remaining three schemes (Mirr all\Gammasub , Mirr some\Gammasub , and Par some ) is roughly equal and is low- est. The best scheme is the One-to-All mirroring scheme with sub-block replication (Mirr all\Gammasub ). It has a slightly smaller per stream cost than the One-to-Some mirroring scheme with sub-block replication due to the difference in size between b all sub and b some sub (see the explanation in section 4.1). ffl The increase in the cost ratio ff by a factor of five Figure 14(b)) slightly decreases the per stream cost of Par all and results in a dramatic decrease in the per stream cost of all three mirroring schemes and also the parity scheme Par some . For instance the per stream cost for Par some decreases from down-to $28:72 and the per stream cost of Mirr some\Gammasub decreases from $79:79 down-to $18:55. All three mirroring schemes become more cost efficient than the two parity schemes. ffl The decrease in the cost ratio ff by a factor of five Figure 14(c)) affects the cost of the three mirroring schemes very little. As an example, the per stream cost for Mirr some\Gammasub is $79:79 in Figure 14(a) and is $77:19 in Figure 14(c). On the other hand, decreasing ff, i.e. the price for main memory decreases faster than the price for hard disk, clearly affects the cost of the two parity schemes. In fact, Par some becomes the most cost efficient scheme with a cost per stream of $65:64. Although the per stream cost of Par all decreases significantly with still remains the most expensive for high values of D. Since Par all has the highest per stream cost that linearly increases with D, we will not consider this scheme in further cost discussion. 5 Server Reliability and Performance 5.1 Server Reliability vs. Per Stream Cost Figure 15 and Table 3 depict the server reliability and the per stream cost for the different reliability schemes discussed herein. The server reliability is computed after 1 year (Figure 15(a)) and after 3 years (Figure 15(b)) of server operation. The results in Figure 15 are obtained as follows. For a given server throughput, we calculate for each reliability scheme the number of disks required to achieve that throughput. We then compute the server reliability for each scheme and its respective number of disks required. Table 3 shows the normalized per stream cost for different values of ff. We take the per stream cost of Mirr one as base line for comparison and divide the cost values for the other schemes by the cost for Mirr one . We recall again that the three schemes Mirr one , Mirr all\Gammaentire and Mirr some\Gammaentire have the same per stream cost since they achieve the same throughput given the same amount of resources (see section 4). Server Throughput Reliability Function Mirr one Par some Mirr some-sub Mirr some-entire Par all Mirr all-sub Mirr all-entire (a) Server reliability after 1 year of server operation with 100000 hours Server Throughput Reliability Function (b) Server reliability after 3 years of server operation with 100000 hours Figure 15: Server reliability for the same server throughput with D Mirr one Mirr all\Gammaentire Par some 0:688 1:129 0:588 Mirr some\Gammasub 0:698 0:729 0:691 Mirr all\Gammasub 0:661 0:696 0:653 Table 3: Normalized stream cost (by Mirr one ) for different values of ff with D The three One-to-All schemes Par all , Mirr all\Gammasub , and Mirr all\Gammaentire have poor server reliability even for a low values of server throughput, since they only survive a single disk failure. The difference in reliability between these schemes is due to the fact that Par all requires, for the same throughput, fewer disks than Mirr all\Gammasub that in turn needs fewer disks than Mirr all\Gammaentire (see Figure 13(b)). The server reliability of these three schemes decreases dramatically after three years of server operation as illustrated in Figure 15(b)). Accord- ingly, these schemes are not attractive to ensure fault tolerance in video servers and hence we are not going to discuss them more in the remainder of this paper. We further discuss the three One-to-Some schemes Par some , Mirr some\Gammasub , and Mirr some\Gammaentire and the Grouped One-to-One scheme Mirr one . Based on Figures 15(a) and 15(b), Mirr one has a higher server reliability than the three One-to-Some schemes Par some , Mirr some\Gammasub , and Mirr some\Gammaentire . From Table 3, we see that Mirr one , which has the same per stream cost as Mirr some\Gammaentire , has a per stream cost about 1:5 higher than Mirr some\Gammasub . Par some has the highest per stream cost for a high value of ff and is the most cost effective for a small value of ff In summary, we see that the best scheme among the One-to-Some schemes is Par some since it has a low per stream cost and requires fewer disks than Mirr some\Gammasub and thus provides a higher server reliability than both, Mirr some\Gammasub and Mirr some\Gammaentire . Since Mirr some\Gammaentire achieves much lower server reliability than Mirr one for the same per stream cost, we conclude that Mirr some\Gammaentire is not a good scheme for achieving fault tolerance in a video server. Based on the results of Figure 15 and Table 3, we conclude that the three schemes: Mirr one , Par some , and Mirr some\Gammasub are the good candidates to ensure fault tolerance in a video server. Note that we have assumed in Figure 15 for all these three schemes the same value D Mirr one has the highest server reliability but a higher per stream cost as compared to the per stream cost of Par some and Mirr some\Gammasub . For the value D the two schemes Par some and Mirr some\Gammasub have a lower per stream cost but also a lower server reliability than Mirr one . This difference in server reliability becomes more pronounced as the number of disks in the video server increases. We will see in the next section how to determine the parameter D c for the schemes Par some and Mirr some\Gammasub in order to improve the trade-off between the server reliability and the cost per stream. 5.2 Determining the Group Size D c This section evaluates the impact of the group size D c on the server reliability and the per stream cost. We limit our discussion to the three schemes: our Mirr one , Par some , and Mirr some\Gammasub . Remember that we use the Orthogonal RAID principle to build the independent groups (see section 3.3). Accordingly, disks that belong to the same group are attached to different nodes. Until now, we have assumed that the group size D c and the number of nodes N are constant (D In other terms, increasing D leads to an increase in the number of disks D n per node. However, the maximum number of disks D n is limited by the node's I/O capacity. Assume a video server with disks and D 5. We plot in Figure 16 two different ways to configure the video server. In Figure 16(a) the server contains five nodes 5), where each node consists of D disks. One group contains D disks, each belonging to a different node. On the other hand, Figure 16(b) configures a video server with nodes, each containing only D disks. The group size is again single group does not stretch across all nodes. Note that the number of groups C is the same for both configurations 20). When the video server grows, the second alternative suggests to add new nodes (containing new disks) to the server, whereas the first alternative suggests to add new disks to the existing nodes. Since D n must be kept under a certain limit given by the node's I/O capacity, we believe that the second alternative is more appropriate to configure a video server. We consider two values the group size: D for the remaining three schemes Mirr one , Par some , and Mirr some\Gammasub . Figures 17(a) and 17(b) depict the server reliability for Mirr one , Par some , and Mirr some\Gammasub after one year and after three years of server operation, respectively. Table 4 shows for these schemes the normalized per stream cost with different values of ff and with D the per stream cost of Mirr one as base line for comparison and divide the cost values for the other schemes by group 20 (a) group 20 group 19 Figure Configuring a video server with 5. the cost for Mirr one . Server Throughput Reliability Function Mirr one Par someMirr some-sub 5 Par someMirr some-sub 20 (a) Server reliability after 1 year of server operation with 100000 hours and Server Throughput Reliability Function (b) Server reliability after 3 years of server operation with hours and Figure 17: Server reliability with D The results of Figure 17 and Table 4 are summarized as follows: ffl The server reliability of Mirr one is higher than for the other two schemes. As expected, the server re- Mirr one Par some 20 0:798 1:739 0:584 Par some 5 0:695 0:879 0:653 Mirr some\Gammasub 20 0:678 0:717 0:671 Mirr some\Gammasub 5 0:745 0:771 0:739 Table 4: Normalized stream cost (by Mirr one ) for different values of ff and D c . liability increases for both, Par some and Mirr some\Gammasub with decreasing D c . In fact, as D c decreases, the number of groups grows and thus the number of disk failures (one disk failure per group) that can be tolerated increases as well. ffl Depending on the value of ff, the impact of varying the group size D c on the per stream cost differs for Par some and Mirr some\Gammasub . For Mirr some\Gammasub , the cost per stream decreases as the group size D c grows for all three values of ff considered. Indeed, the sub-block size to be read during disk failure is inversely proportional to the value of D c . Consequently, the server throughput becomes smaller for decreasing D c and the per stream cost increases. For Par some with 26 and ff = 130, the per stream cost decreases as D c decreases. However, this result is reversed with where the per stream cost of Par some is higher with D 20. The following explains the last observation: 1. A small value of ff (e.g. signifies that the price for main memory decreases faster than the one for hard disk and therefore main memory does not significantly affect the per stream cost for Par some independently of the group size D c . 2. As the group size D c decreases, the amount of I/O bandwidth that must be reserved on each disk for the disk failure mode increases. Consequently, the throughput is smaller with D As a result, the per stream cost for Par some increases when the group size D c decreases. 3. Since the memory cost affects only little the per stream cost of Par some for a small value of ff, the weight of the amount of I/O bandwidth to be reserved on the per stream cost becomes more visible and therefore the per stream cost of Par some increases as D c decreases. Note that the per stream cost of Par some (D lowest for it is highest for Par some has always a higher server reliability than Mirr some\Gammasub . Further, for the values 26 and Par some has a higher cost per stream than Mirr some\Gammasub given the same value of D c . However, for the value becomes more cost effective than Mirr some\Gammasub . ffl Based on the reliability results and for the high values of ff, e.g. we observe that a small group size (D c = 5) considerably increases the server reliability and decreases the per stream cost for Par some . For Mirr some\Gammasub , the server reliability increases as D c decreases, but also the per stream cost slightly increases whatever the value of ff is. In summary, we have shown that the three schemes Mirr one , Par some , and Mirr some\Gammasub are good candidates to ensure fault-tolerance in a video server. The Grouped One-to-One scheme Mirr one achieves a higher reliability than the other two schemes at the expense of the per stream cost that is about 1:5 times as high. For Par some , the value of D c must be small to achieve a high server reliability and a low per stream cost. For Mirr some\Gammasub , the value of D c must be small to achieve a high server reliability at the detriment of a slight increase in the per stream cost. 6 Conclusions In the first part we have presented an overview of several reliability schemes for distributed video servers. The schemes differ by the type of redundancy used (mirroring or parity) and by the distribution granularity of the redundant data. We have identified seven reliability schemes and compared them in terms of the server reliability and the cost per stream. We have modeled server reliability using Continuous Time Markov Chains that were evaluated using the SHARPE software package. We have considered both cases: independent disk failures and dependent component failures. The performance study of the different reliability schemes led us to introduce a novel reliability scheme, called the Grouped One-to-One mirroring scheme, which is derived by the classical One-to-One mirroring scheme. The Grouped One-to-One mirroring scheme Mirr one outperforms all other schemes in terms of server reliability. Out of the seven reliability schemes discussed the Mirr one , Par some , and Mirr some\Gammasub schemes achieve both, high server reliability and low per stream cost. We have compared these three schemes in terms of server reliability and per stream cost for several memory and hard disk prices and various group sizes. We found that the smaller the group size, the better the trade-off between high server reliability and low per stream cost. A Calculation of P (k) for the One-to-One Scheme We calculate in the following the probability P (k) that the video server that uses the One-to-One mirroring scheme does not fail after k disk/node failures. Let us consider the suite of n units. Note that the term unit can denote a disk (used for the independent disk failure case) or a node (used for the dependent component failure mode). Since we want to calculate the probability that the server does not fail having k units down, we want those units not to be adjacent. Therefore we are looking for the sub-suites Let us call S the set of these suites. We introduce a bijection of this set of suites to the set Introducing the second suite j allows to suppress condition (10) on the suite i l , since the suite j is strictly growing, whose number of elements are thus easy to count: it is the number of strictly growing functions from [1::k] in 1)], that is This result though doesn't take into account the fact that the units number 1 and n are adjacent. In fact, two scenarii are possible ffl The first scenario is when unit 1 has already failed. In this case, units 2 and the n are not allowed to fail, otherwise the server will fail. We have then a set of n \Gamma 3 units among which we are allowed to pick non-adjacent units. Referring to the case that we just solved, we obtain: C(n ffl The second scenario is when the first unit still works. In this case, k units are chosen among the remaining ones. This leads us to the value The number of possibilities N k that we are looking for is given by : Consequently, for a given number k of failed units (disks/nodes), the probability P (k) that the server does not fail after k unit failures is calculated as: Building the group state-space diagram for the Grouped One-to-One scheme We show in the following how to build the state-space diagram of one group containing D c disks for the Grouped One-to-One scheme. We focus on the transitions from state [i; j] to higher states and back. We know that state represents the total number (i + j) of disks that have failed inside the group. These failures are due to i disk failures and j node failures. We also know that the number of states in the state-space diagram is 2 Dc . The parameters i and j must respect the Dc, since at most Dcdisk failures are tolerated inside one group. We distinguish between the two cases: Figure shows the possible transitions and the corresponding rates for the case (i), whereas Figure 19 shows the transition for the case (ii). l n F l 1 l 2 Figure 18: Transitions from state [i; j] to higher states and back, for (i l Figure 19: Transition to the failure state for (i . The parameters in the two figures have the following values: --R "A Case for Redundant Arrays of Inexpensive Disks (RAID)," "Streaming raid(tm) - a disk array management system for video files," "Fault tolerant design of multimedia servers," "Striping in multi-disk video servers," "Data striping and reliablity aspects in distributed video servers," "Disk striping in video server environments," "Fault-tolerant architectures for continuous media servers," "Design and performance tradeoffs in clustered video servers," "Predictive call admission control for a disk array based video server," "Architectures and algorithms for on-line failure recovery in redundant disk arrays," "A survey of approaches to fault tolerant design of vod servers: Techniques, analysis, and comparison," "Disk shadowing," "Doubly-striped disk mirroring: Reliable storage for video servers," "The tiger video fileserver," "Analytic modeling and comparisons of striping strategies for replicated disk arrays," Performance Modeling and Analysis of Disk Arrays. "Raid: High-performance, reliable secondary storage," "Raid: High-performance, reliable secondary storage," "Architectures and algorithms for on-line failure recovery in redundant disk arrays," "Performance and cost comparison of mirroring- and parity-based reliability schemes for video servers," "Chained declustering: A new availability strategy for multiprocessor database ma- chines.," "Chained declustering: Load balancing and robustness to skew and failures," "Issues in the design of a storage server for video-on-demand," "Raid-ii: A scalabale storage architecture for high-bandwidth network file service," "Striping in multi-disk video servers," "Segmented information dispersal (SID) for efficient reconstruction in fault-tolerant video servers," "High availability for clustered multimedia servers," "Random raids with selective exploitationof redundancy for high performance video servers," "Using rotational mirrored declustering for replica placement in a disk-array-based video server," "Reliability analysis of redundant arrays of inexpensive disks," "Performability of disk-array-based video servers," Performance and Reliability Analysis of Computer Systems: An Example-Based Approach Using the SHARPE Software Package System Reliability Theory: Models and Statistical Methods Placement of Continuous Media in Multi-Zone Disks --TR --CTR Yifeng Zhu , Hong Jiang , Xiao Qin , Dan Feng , David R. Swanson, Design, implementation and performance evaluation of a cost-effective, fault-tolerant parallel virtual file system, Proceedings of the international workshop on Storage network architecture and parallel I/Os, p.53-64, September 28-28, 2003, New Orleans, Louisiana Xiaobo Zhou , Cheng-Zhong Xu, Efficient algorithms of video replication and placement on a cluster of streaming servers, Journal of Network and Computer Applications, v.30 n.2, p.515-540, April, 2007 Seon Ho Kim , Hong Zhu , Roger Zimmermann, Zoned-RAID, ACM Transactions on Storage (TOS), v.3 n.1, p.1-es, March 2007 Stergios V. Anastasiadis , Kenneth C. Sevcik , Michael Stumm, Maximizing Throughput in Replicated Disk Striping of Variable Bit-Rate Streams, Proceedings of the General Track: 2002 USENIX Annual Technical Conference, p.191-204, June 10-15, 2002 Xiaobo Zhou , Cheng-Zhong Xu, Harmonic Proportional Bandwidth Allocation and Scheduling for Service Differentiation on Streaming Servers, IEEE Transactions on Parallel and Distributed Systems, v.15 n.9, p.835-848, September 2004 Eitan Bachmat , Tao Kai Lam, On the effect of a configuration choice on the performance of a mirrored storage system, Journal of Parallel and Distributed Computing, v.65 n.3, p.382-395, March 2005 Yifeng Zhu , Hong Jiang, CEFT: a cost-effective, fault-tolerant parallel virtual file system, Journal of Parallel and Distributed Computing, v.66 n.2, p.291-306, February 2006 Stergios V. Anastasiadis , Kenneth C. Sevcik , Michael Stumm, Scalable and fault-tolerant support for variable bit-rate data in the exedra streaming server, ACM Transactions on Storage (TOS), v.1 n.4, p.419-456, November 2005

reliability modeling;performance and cost analysis;distributed video servers;SHARPE;markov chains

351186

A PTAS for Minimizing the Total Weighted Completion Time on Identical Parallel Machines.

We consider the problem of scheduling a set ofn jobs onm identical parallel machines so as to minimize the weighted sum of job completion times. This problem is NP-hard in the strong sense. The best approximation result known so far was a 1/2 (1 2)-approximation algorithm that has been derived by Kawaguchi and Kyan back in 1986. The contribution of this paper is a polynomial time approximation scheme for this setting, which settles a problem that was open for a long time. Moreover, our result constitutes the =rst known approximation scheme for a strongly NP-hard scheduling problem with minsum objective.

Introduction The problem. We consider the following machine scheduling model. We are given a set J of n independent jobs that have to be scheduled on m identical parallel machines or processors. Each job is specified by its positive processing requirement p j and by its positive weight w j . In a feasible schedule for J , every job j 2 J is processed for p j time units on one of the m machines in an uninterrupted fashion. Every machine can process at most one job at a time, and every job can be processed on at most one machine at a time. The completion time of job j in some schedule is denoted by C j . The goal is to minimize the total weighted completion time . In the standard classification scheme of Graham, Lawler, Lenstra, & Rinnooy Kan 1979, this scheduling problem is denoted by P m part of the input, and by Pm Complexity of the problem. For the special case of only one machine (i. e., the problem can be solved in polynomial time by Smith's Ratio Rule: process the jobs in order of nonincreasing . Thus, for the single machine case, the 'importance' of a job is measured by its ratio. For a constant number m - 2 of machines, the problem is NP-hard in the ordinary sense and solvable in pseudopolynomial time. For m part of the input, the problem is NP-hard in the strong sense; see problem SS13 in Garey & Johnson 1979. The special case P j j solvable in polynomial time by sorting, see Conway, Maxwell, & Miller 1967. An extended abstract will appear in the Proceedings of the 31st Annual ACM Symposium on Theory of Computing (STOC'99). y Technische Universit?t Berlin, Fachbereich Mathematik, MA 6-1, Stra-e des 17. Juni 136, D-10623 Berlin, Germany, E-mail skutella@math.tu-berlin.de. Part of this research was done while the first author was visiting C.O.R.E., Louvain- la-Neuve, Belgium. z Technische Universit?t Graz, Institut f?r Mathematik, Steyrergasse 30, A-8010 Graz, Austria, E-mail gwoegi@opt.math.tu-graz.ac.at. Supported by the START program Y43-MAT of the Austrian Ministry of Science. Approximation algorithms. In this paper we are interested in how close one can approach an optimum solution to these NP-hard scheduling problems in polynomial time. Thus, our research focuses on approximation algorithms which efficiently construct schedules whose values are within a constant factor of the optimum solution value. The number ff is called performance guarantee or performance ratio of the approximation algorithm. A family of polynomial time approximation algorithms with performance called a polynomial time approximation scheme (PTAS). If the running times of the approximation algorithms are even bounded by a polynomial in the input size and 1 " , then these algorithms build a fully polynomial time approximation scheme (FPTAS). It is known that unless P=NP, a strongly NP-hard optimization problem cannot possess an FPTAS, see Garey & Johnson 1979. Known approximation results. Sahni (1976) gives a FPTAS for the weakly NP-hard scheduling problem fixed m. Kawaguchi & Kyan (1986) analyze list scheduling in order of nonincreasing ratios w j on identical parallel machines. They prove a performance ratio 1 for the strongly NP-hard problem Till now, this was the best approximation result for this problem. Alon, Azar, Woeginger, & Yadid (1998) study scheduling problems on identical parallel machines with various objective functions that solely depend on the machine completion times. In particular, they give a polynomial time approximation scheme for the problem of minimizing denotes the finishing time of machine i. By rewriting the objective function in an appropriate way, this result implies a PTAS for the problem of minimizing the weighted sum of job completion times if all job ratios w j are equal. Generally speaking, the approximability of scheduling problems with total job completion time objective (so-called minsum scheduling problems) is not well-understood. Some minsum problems can be solved in polynomial time using straightforward algorithms (like the single machine version of the problem weakly NP-hard minsum problems allow an FPTAS based on dynamic programming formulations (see, e. g., Sahni 1976 and Woeginger 1998). Some minsum problems do not have a PTAS unless P=NP (Hoogeveen, Schuurman, & Woeginger 1998). Some of these problems cannot even be approximated in polynomial time within a constant factor (like minimizing total flow time, see Kellerer, Tautenhahn, & Woeginger 1996 and Leonardi & Raz 1997). Some minsum problems have constant factor approximation algorithms that are based on rounding and/or transforming and/or manipulating the solutions of preemptive relaxations or relaxations of integer programming formulations (see, e. g., Phillips, Stein, & Wein 1995, Hall, Schulz, Shmoys, & Wein 1997, or Skutella 1998); due to the integrality gap, these approaches can never yield a PTAS. However, there is not a single PTAS known for a strongly NP-hard minsum scheduling problem. Contribution of this paper. Our contribution is a polynomial time approximation scheme for the general problem of minimizing the total weighted completion time on identical parallel machines. This result is derived in two steps. In the first step, we derive a PTAS for the special case where the largest job ratio is only a constant factor away from the smallest job ratio. This result is derived by modifying and by generalizing a technique of Alon et al. 1998. In the second step, we derive a in its full generality. The main idea is to partition the jobs into subsets according to their ratios such that near optimal schedules can be computed for all subsets; the key observation is that these schedules can be concatenated without too much loss in the overall performance guarantee. Our result yields the first polynomial time approximation scheme for a strongly NP-hard scheduling problem with minsum objective. It confirms a conjecture of Hoogeveen, Schuurman, & Woeginger 1998, it solves an open problem posed in Alon et al. 1998, and it finally improves on the ancient result by Only very recently, several groups announced independently from each other approximation schemes for the strongly NP-hard problem 1j r j j even for more general scheduling problems, cf. Chekuri et al. 1999. Organization of the paper. Section 2 contains preliminaries which will be used throughout the paper. In Section 3 we give an approximation scheme for the special case that the ratios of jobs are within a constant range. Finally, in Section 4 we present the approximation scheme for arbitrary instances of Preliminaries By Smith's Ratio Rule, it is locally optimal to schedule the jobs that have been assigned to a machine in order of nonincreasing ratios w j ; the proof is a simple exchange argument. Throughout the paper, we will restrict to schedules meeting this property. For a given schedule that Notice that the second term on the right hand side is nonnegative and does not depend on the specific schedule. Therefore, for each " ? 0, a (1 ")-approximation algorithm for the problem to minimize the function ")-approximation algorithm for minimizing the total weighted completion time. In fact, in Section 3 it will be more convenient to consider the objective function and we will give a polynomial time approximation scheme for the problem to minimize this function there. We use the following notation: For a subset of jobs J denote the total processing time of the jobs in J 0 . Moreover, denote the average machine load caused by the jobs in J 0 by use the simpler notation L := L(J). We start with the following observation: Consider a subset of jobs J ae for all the jobs in J 0 are consecutively scheduled on a machine in an arbitrary order starting at time - , their contribution to the objective function is given by This observation together with Smith's Ratio Rule leads to the following lemma which generalizes a result of Eastman, Even, & Isaacs 1964. Lemma 2.1. Let ae 1 denote the different job ratios w j let J h := g. For a given schedule, denote the subset of J h that is being assigned to machine i by J h;i . Then, the value of the schedule is given by ae h ae ' Proof. On each machine i and for each 1 - q, the jobs in J ';i are scheduled consecutively starting at time h=1 p(J h;i ). The result thus follows from (2). Notice that the right hand side of (3) only depends on the total processing times of the sets J h;i , but not on the structure of these sets. In the analysis of the approximation scheme we will make use of the following lower bound on the value of an optimal schedule: Lemma 2.2. Using the same notation as in Lemma 2.1, the value of an arbitrary schedule is bounded from below by Proof. By (1) and Lemma 2.1 we get ae h The second inequality follows from the convexity of the function (a 1 ; a 3 The approximation scheme for a constant range of ratios In this section we consider the problem to minimize give a polynomial time approximation scheme for instances with bounded weight to length ratios of jobs, i. e., where for all job ratios w j for an arbitrary R ? 0 and a real constant that does not depend on the input. First notice that by rescaling the weights of jobs we can restrict to the case R = 1. 3.1 Structural insights be an arbitrary real constant and choose a corresponding constant \Delta 2 N with ffi \Delta+1 - ae. If we round up the weights of jobs such that the ratio of each job attains the nearest integer power of ffi, the value of an optimal schedule increases at most by a factor 1 . Since the constant can be chosen arbitrarily close to 1, we may restrict to instances of P are of the form \Deltag. We use the following notation in this section: For by J h the class of all jobs . The completion time of machine i in some schedule is denoted by M i . Lemma 3.1. Let f in an optimal schedule M i ? fM i 0 for a pair of machines machine i processes exactly one job. Proof. Let j denote the last job on machine i. Observe that j must start at or before time moving j to the end of machine i 0 would decrease \Gamma j and hence the value of the schedule. Thus, we get . By contradiction, assume that j is not the first job on machine i; denote the first job by k. Let W i and W i 0 denote the sum of the weights of jobs scheduled on machine i and i 0 , respectively. Since since the weight to length ratios of all jobs are at most 1. Thus, removing job k from machine i and inserting it at the beginning of machine i 0 changes the value of the schedule by p k (W contradicting optimality. For the following corollary notice that there is at least one machine i with M i - L in every schedule. Corollary 3.2. If a job j has size p j - fL, then j occupies a machine of its own in any optimal schedule. If no job has size greater than fL, then the completion time of each machine is at most fL in any optimal schedule. As a result of Corollary 3.2, we can iteratively reduce a given instance: as long as a job j of size remove it from the set of jobs J , assign it to a machine of its own, and decrease the number of machines by one. In the following we can thus restrict to instances of the following form: Assumption 3.3. The processing time p j of each job j as well as the completion time M i of any machine i in an optimal schedule are bounded from above by fL. 3.2 Rounding the instance We define a simplified, rounded version of the input for which we can compute an optimal schedule in polynomial time. Moreover, under some assumption that we will specify later, an optimal solution to the rounded instance will lead to a near optimal solution to the original instance. The rounding is based on the positive integral constant - which will later be chosen to be 'sufficiently large'. The rounding is done for every class J h , 0 - h - \Delta, separately. We will replace the jobs in J h by new jobs, with slightly different processing times; however, the length to weight ratio will stay at ffi h . ffl Every 'big' job j in the original instance with - , is replaced by a corresponding rounded job whose processing time rounded up to the next integer multiple of L . The weight j of the rounded job equals . Note that for the rounded processing time holds. ffl Denote by S h the total processing time of the 'small' jobs in J h whose processing times are not greater than L - . Denote by S # h the value of S h rounded up to the next integer multiple of L - . Then the rounded instance contains S # new jobs, each of length L - . The weight of every new job equals Note that the total number of jobs in the rounded instance is bounded by n. By construction we get the following lemma. Lemma 3.4. By replacing the rounded jobs with their unrounded counterparts, an arbitrary schedule for the transformed instance induces a schedule of smaller or equal value for the original instance. The rounded instance can be solved in polynomial time, e. g., by dynamic programming. However, we use a generalization of an alternative approach of Alon, Azar, Woeginger, & Yadid 1998; we formulate the problem as an integer linear program whose dimension is bounded by a constant. By Assumption 3.3 and by construction, the size p # j of each job j is bounded by Therefore, since all job sizes are integer multiples of 1 there is at most a constant number \Pi of possible Moreover, since the number of possible length to weight ratios is bounded by the constant \Delta + 1, the total number of different types of jobs is bounded by the constant For denote the number of jobs of size k L define the vector n := (n 1;0 ; n An assignment of jobs to one machine is given by a vector is the number of jobs of size k L that are assigned to the machine. The completion time of the machine is given by h=0 let c(u) denote the contribution to the objective function of a machine that is scheduled according to u. j denote the average machine load for the transformed instance; observe that by construction Thus, by Assumption 3.3 we can bound the completion time of each machine by f -+\Delta+2 L. Denote by U the set of vectors u with M(u) - f -+\Delta+2 - L. For a vector u 2 U , each entry u k;h is bounded by a number that only depends on the constants -, \Delta, f and is thus independent of the imput. Therefore the set U is of constant size. We can now formulate the problem of finding an optimal schedule as an integer linear program with a constant number of variables. For each vector u 2 U we introduce a variable xu which denotes the number of machines that are assigned jobs according to u. An optimal schedule is then given by the following program: min u2U subject to u2U u2U It has been shown by Lenstra (1983) that an integer linear program in constant dimension can be solved in polynomial time. 3.3 Proving near optimality By Lemma 3.4 it remains to show that the value of an optimal schedule for the rounded instance is at most a factor of (1 above the optimal objective value of the original instance. We will prove this under the following assumption on the original instance, and afterwards we will demonstrate how to get rid of the assumption: Assumption 3.5. There exists an optimal schedule of the following form: The completion time M i of every machine i fulfills the inequality L In order to achieve the desired precision, we now choose the integer - sufficiently large to fulfill the inequality Since \Delta, ffi, f , and " are constants that do not depend on the input, also - is constant and independent of the input. Lemma 3.6. Under Assumption 3.5, the optimal objective value of the rounded instance is at most a factor of (1 above the optimal objective value of the original instance. Proof. Take an optimal schedule for the original instance as described in Assumption 3.5. Consider some fixed machine i with finishing time M i in this optimal schedule. For h denote the subset of J h processed on machine i and let J h denote the set of all jobs processed on machine i. Note that By Lemma 2.1, the contribution of machine i to the objective function is given by Replace every big job j by its corresponding rounded big job j # . This may increase every p(J 0 a multiplicative factor of at most -+1 - . For every J 0 h , denote by s h the total size of its small jobs. Round s h up to the next integer multiple of L - and replace it by an appropriate number of rounded small jobs with length to weight ratio ffi h . This may increase p(J 0 h ) by an additive factor of at most L - . Note that by repeating this replacement procedure for all machines, one can accommodate all jobs in the rounded instance. Denote by K h the resulting set of rounded jobs that are assigned to machine i and have length to weight ratio ffi h . We have for Now compare the first term on the right hand side of (6) for J 0 h to the corresponding term for the jobs in K h :2 In the last inequality, we applied together with the second inequality in (5). In a similar way, we compare the second term on the right hand side of (6) for J 0 h to the corresponding term for the jobs in because of the first inequality in (5), we get that the right hand side of (6) fulfills:2 Putting this together with (7), (8), and (4), a short calculation shows that the objective value for the rounded instance is at most a factor of (1 above the optimal objective value for the original instance. Finally, we consider the case where Assumption 3.5 may be violated. By Assumption 3.3 we can restrict to the case that only the lower bound 1 f L can be violated for some machine completion time. As a result of Lemma 3.1 we get: Corollary 3.7. If in an optimal schedule f for some machine i, then every machine i 0 with only one job; in particular, there exists a job j of size p j - L and every such job occupies a machine of its own. In the following we assume that there exists a job j of size p j - L; otherwise, Assumption 3.5 is true by Corollary 3.7 and we are finished. Unfortunately, we do not know in advance whether or not M i ! L f for some machine i in an optimal schedule. Therefore we take both possibilities into account and compute two schedules for the given instance such that the better one is guaranteed to be a (1 ")-approximation by Corollary 3.7. On the one hand, we compute an optimal schedule for the rounded instance and turn it into a feasible schedule for the original instance as described in Lemma 3.4. On the other hand, we assign each job j with to a machine, remove those jobs from the instance, and decrease the number of machines by the number of removed jobs. For the reduced instance, we can recursively compute a (1 ")-approximation by again taking the better of two schedules. Notice that in each recursion step the number of machines is decreased by at least one; thus, after at most m \Gamma 1 steps we arrive at a trivial problem that can be solved to optimality in polynomial time. We can now state the main result of this section: Theorem 3.8. There exists a PTAS for the special case of the problem P are in a constant range [aeR; R] for an arbitrary R ? 0 and a real constant that does not depend on the input. 4 The polynomial time approximation scheme In this section we present the approximation scheme for arbitrary instances of P . The main idea for deriving this result is to partition the set of jobs into subsets according to their ratios w j . The ratios of all jobs in one subset are within a constant range such that for each subset a near optimal schedule can be computed within arbitrary precision in polynomial time, see Theorem 3.8. In a second step, these 'partial schedules' are concatenated in order of nonincreasing job ratios such that Smith's Ratio Rule is obeyed on each machine. For the sake of a more accessible analysis, we first present a randomized variant of the approximation scheme and discuss its derandomization later. Throughout this section we assume that w j all weights of jobs can be rescaled by the inverse of the maximal ratio. 4.1 The randomized approximation scheme The partitioning step. Let \Delta be a positive integer and let ffi := 1 later we will choose \Delta large such that ffi gets small. The partitioning of the set of jobs J is performed in two steps. The first step computes a fine partition which is then randomly turned into a rougher partition in the second step. 1. For h 2 N let J(h) := \Psi . 2. Draw q uniformly at random from f1; I q s J(h). Notice that for fixed q the number of nonempty subsets J q bounded by n. Of course, we only take those subsets into consideration in the algorithm. The intuition for step 2 is to compensate the undesired property of the fine partition computed in step 1 that jobs with similar ratio may lie in different subsets of the computed partition. The random choice in step 2 assures that the probability for those jobs to lie in different subsets of the rough partition is small, i. e., in O Computing partial schedules. The quotient of the biggest and the smallest ratio of jobs in a subset s is bounded by the constant ffi \Delta . Thus, by Theorem 3.8, we can compute in polynomial time for all nonempty sets of jobs J q s near optimal m-machine schedules of value Z q s , where Z q s denotes the value of an optimal schedule for J q s . The concatenation step. In the final step of the algorithm these partial schedules are concatenated. One possibility is to do it machine-wise: On machine i, all jobs that have been assigned to i in the partial schedules are processed according to Smith's Ratio Rule. However, this deterministic concatenation can lead to an undesired unbalance of load on the machines. It might for example happen that each subset of jobs J q s consists of at most one job which is always assigned to machine 1 in the corresponding partial schedule. Thus, concatenating the partial schedules as proposed above would leave all but one machine idle. Therefore, we first randomly and uniformly permute the numbering of machines in each partial schedule and then apply the machine-wise concatenation described above. In this randomly generated schedule the probability for two jobs from different subsets to be processed on the same machine is equal to 1 such that one can expect an appropriately balanced machine assignment. 4.2 The analysis of the randomized approximation scheme The analysis is based on the observation that the value of the computed schedule is composed of the sum of the values of the partial schedules plus the additional cost caused by the delay of jobs in the concatenation step. It is easy to see that the sum of the values of the near optimal partial schedules cannot substantially exceed the value of an optimal schedule. The key insight for the analysis is that the delay of jobs in one subset caused by another subset in the concatenation step can essentially be neglected. One reason is that the delayed jobs usually (i. e., with high probability) have much smaller ratio and are thus less important than the jobs which cause the delay. On the other hand, if there are too many 'unimportant' jobs to be neglected, then the total weighted completion time of the corresponding near optimal partial schedule must be large compared to the delay caused by the important jobs. The following lemma provides two lower bounds on the value of an optimal schedule Z . To simplify Lemma 4.1. For each q 2 \Deltag, the value Z of an optimal schedule is bounded by Z - Z q s and Z - mX Proof. Take an optimal schedule for the set of jobs J and denote the completion time of job j in this schedule by C j . This yields s Z q s since the completion times C also define a feasible schedule for each subset of jobs J q s . In order to prove the second lower bound, first observe that the value of an optimal schedule decreases if we round the weights of jobs j 2 J(h) to w N. The result then follows from Lemma 2.2. The next step in the analysis is to determine an expression for the expected value of the computed schedule. Lemma 4.2. The expected value of the computed schedule is given by Z q s min Proof. We first keep q fixed and analyze the conditional expectation E q \Theta P . This conditional expectation is equal to the sum of the values of the partial schedules s plus the expected cost caused by the delay of jobs in the concatenation step. The expected delay of an arbitrary job j 2 J can be determined as follows. Let t 2 N 0 and k 2 I q such that j 2 J(k) ' J q t . Then, the expected delay of job j is equal to the expected load caused by jobs from s on the machine which job j is being processed on. Since the machines are permuted uniformly at random in the concatenation step, the expected load is equal to the average load s lie in different sets of indices I q As a consequence, for fixed q, the conditional expectation of the total weighted completion time can be as: Z q Notice that for randomly chosen q the expected value of j q h;k is equal to the probability that h and k lie in different sets of indices I q By construction of the sets of indices I q r , this probability is equal to k\Gammah \Delta. The result thus follows from (11). The following theorem contains the main result of this subsection. Theorem 4.3. For a given \Upsilon . Then, the expected value of the computed schedule is bounded by times the value of an optimal solution, i. e., Proof. We compare the expected value given in Lemma 4.2 to the lower bounds on the value of an optimal schedule given in Lemma 4.1. Since, by construction of the algorithm, Z q s , for each q, and s - Z by Lemma 4.1, the first term on the right hand side of (10) can be bounded from above by Z q s In order to bound the second term on the right hand side of (10), first observe that for each k 2 N Thus, the inequality for the geometric and the arithmetic mean yields: Notice that (12) bounds the cost for the possible delay of jobs in J(k) caused by jobs in J(h) in terms of the lower bounds on optimal schedules for J(h) and J(k) in (9). In particular, if then the cost for the delay is small compared to the sum of the values of optimal schedules for J(h) and J(k). For the cases 2, however, this cost may be too large to be neglected. This is the point where we will make use of the random choice of q. To be more precise, we divide the sum over all pairs h ! k in the second term of (10) into three partial sums . The first partial sum \Sigma 1 takes all pairs with account and can thus be bounded by by (12) The second partial sum \Sigma 2 takes all pairs with account and can be bounded using the same arguments by (12) Finally, the third partial sum \Sigma 3 takes all pairs with k - h account. In this case we replace the \Psi by 1 (i. e., we do not make use of the random choice of q) and get h-k\Gamma3 h-k\Gamma3 Putting the results together, the expected value of the computed schedule is bounded by The second inequality follows from the choice of ffi and a short calculation. 4.3 The deterministic approximation scheme Up to now we have presented a randomized approximation scheme, i. e., we can efficiently compute schedules whose expected values are arbitrarily close to the optimum. However, it might be more desirable to have a deterministic approximation scheme which computes schedules with a firm performance guarantee in all cases. Therefore we discuss the derandomization of the randomized approximation scheme. Since the random variable q can only attain a constant number of different values, we can afford to derandomize the partition step of the algorithm by trying all possible assignments of values to q. In the following discussion we keep q fixed. The derandomization of the concatenation step is slightly more complicated. We use the method of conditional probabilities, i. e., we consider the random decisions one after another and always choose the most promising alternative assuming that all remaining decisions will be made randomly. Thus, starting with the partial schedule for J q 0 , we iteratively append the remaining partial schedules for J q to the current schedule. In each iteration t we use a locally optimal permutation of the machines which is given in the following way. For denote the load or completion time of machine i in the current schedule of the jobs in s . Renumber the machines such that the sum of the weights of jobs in J q t that are processed on machine i in the corresponding partial schedule. Renumber the machines in the partial schedule such that . It is an easy observation, which can be proved by a simple exchange argument, that appending the partial schedule machine-wise to the current schedule according to the given numbering of machines minimizes the cost caused by the delay of jobs in J q t . In particular, this cost is smaller than the expected cost for the randomized variant of the algorithm which is given by s Moreover, the permutation of the machines chosen in iteration s does not influence the expected delay of jobs considered in later iterations (since the expected delay is simply the average machine load). As a result of the above discussion, the value of the schedule computed by the deterministic algorithm is bounded from above by the expected value of the schedule computed by its randomized variant given in Subsection 4.1. Thus, as a consequence of Theorem 4.3 we can state the following main result of this paper. Theorem 4.4. There exists a polynomial time approximation scheme for the problem P Acknowledgements The authors would like to thank the organizers of the Dagstuhl-Seminar 98301 on 'Graph Algorithms and Applications' during which the result presented in this paper has been achieved. --R Approximation schemes for scheduling on parallel machines. Personal communication. Theory of Scheduling. Bounds for the optimal scheduling of n jobs on m processors. Computers and Intractability: A Guide to the Theory of NP- Completeness Rinnooy Kan. Scheduling to minimize average completion time: Off-line and on-line approximation algorithms Worst case bound of an LRF schedule for the mean weighted flow-time problem Approximability and nonapproximability results for minimizing total flow time on a single machine. Approximating total flow time on parallel machines. Minimizing average completion time in the presence of release dates. Algorithms for scheduling independent tasks. Various optimizers for single-stage production Semidefinite relaxations for parallel machine scheduling. When does a dynamic programming formulation guarantee the existence of an FPTAS? --TR --CTR Nicole Megow , Marc Uetz , Tjark Vredeveld, Models and Algorithms for Stochastic Online Scheduling, Mathematics of Operations Research, v.31 n.3, p.513-525, August 2006 Mark Scharbrodt , Thomas Schickinger , Angelika Steger, A new average case analysis for completion time scheduling, Journal of the ACM (JACM), v.53 n.1, p.121-146, January 2006 Martin Skutella, Convex quadratic and semidefinite programming relaxations in scheduling, Journal of the ACM (JACM), v.48 n.2, p.206-242, March 2001

approximation scheme;combinatorial optimization;worst-case ratio;scheduling theory;approximation algorithm

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Machine-adaptable dynamic binary translation.

Dynamic binary translation is the process of translating and optimizing executable code for one machine to another at runtime, while the program is executing on the target machine. Dynamic translation techniques have normally been limited to two particular machines; a competitor's machine and the hardware manufacturer's machine. This research provides for a more general framework for dynamic translations, by providing a framework based on specifications of machines that can be reused or adapted to new hardware architectures. In this way, developers of such techniques can isolate design issues from machine descriptions and reuse many components and analyses. We describe our dynamic translation framework and provide some initial results obtained by using this system.

INTRODUCTION Binary translation is a migration technique that allows software to run on other machines achieving near native code performance. Binary translation grew out of emulation techniques in the late 1980s in order to provide for a migration path from legacy CISC machines to the newer RISC machines. Such techniques were developed by hardware manufacturers interested in marketing their new RISC platforms. From mid 1990, binary translation techniques have been used to translate competitors' applications to the desired hardware platform. In the near future, we can expect to see such techniques being used to optimize programs within a family of computers, for example, by optimizing Sparc architecture binaries to UltraSparc architecture binaries. UQBT, the University of Queensland Binary Translator, has developed techniques, specification languages and a complete framework for performing static translations of code [14,19]. In static binary translation, the code is translated off-line, before the program is run, by creating a new program that uses the machine instructions of the target machine. However, static translation has its limitations. Due to the nature of the von Neumann machine, where code and data are represented in the same way, it is not always possible to discover all the code of a program statically. For example, the target(s) of indirect transfers of control such as jumps on registers are sometimes hard to analyse statically. Therefore, a fall-back mechanism is commonly used with a statically translated program, in the form of an interpreter. The interpreter processes any untranslated code at runtime and returns to translated code once a suitable path is found. The limitations of static binary translation are overcomed with dynamic translation, at the expense of performance. In a dynamic binary translator, code gets translated "on the fly", at runtime, while the user perceives ordinary execution of the program on the target machine. As oppossed to emulation, dynamic translation generates native code and performs on-demand optimizations of the code. Hot spots in the code are optimized at runtime to increase the performance of execution of such code. Further, some optimizations that are not possible statically are possible dynamically. In this paper we describe the design of a machine-adaptable dynamic binary translator based on the static UQBT framework - UQDBT. A tool is said to be machine-adaptable when it can be "configured" to handle different source and/or target machines. In this way, a machine-adaptable dynamic binary translator is capable of being configured for different source and target machines through the specification of properties of these machines and their instruction sets. In other words, the translator is not bound to two particular machines (as per existing translators) but is capable of supporting a variety of source and target machines. UQDBT differs from other dynamic translators in that it provides a clean separation of concerns, by allowing machine-dependent information to be specified, as well as performing machine-independent analyses to support machine-adaptability. In this way, UQDBT can support a variety of CISC and RISC machines at low cost. To support a new machine, the specifications for that machine need to be written and most of the UQDBT framework can be reused. New machine-specific modules may need to be added if a particular feature of a machine is not supported by the UQDBT framework and such feature is not generic across different architectures. This paper is structured in the following way. Section 2 discusses the static and dynamic frameworks for binary translation. Section 3 outlines the research problems of machine adaptable binary translation and what is addressed in UQDBT. Section 4 provides a case study of translation in the framework through an example program. Section 5 shows preliminary results in the use of the framework. Section 6 discusses effects of changing the granularity of translation and conclusions are given in Section 7. The work reported herein is work in progress. 1.1 Related work In an attempt to improve on existing emulation techniques, companies in the late 1980s began using binary translation to achieve native code performance. Perhaps the most well known binary translators are Digital's VEST and mx[1], which translate VAX and MIPS machine instructions to 64-bit Alpha instructions. Both of these translators and others, like Apple's MAE[2] and Digital's Freeport Express[3] have a runtime environment that reproduces the old machine's operating environments. The runtime environment offers a fallback interpreter for processing old machine code that was not discovered at translation time, for example, due to indirect transfers of control. In recent years, we have seen a transition to hybrid translators, which are proving to be extremely successful. The process of mixing translation with emulation and runtime profiling brought about some of the leading performers in the hybrid translation scene - Digital's FX!32[4], Executor by Ardi[5] and Sun's Wabi[6]. FX!32 emulates the program initially and statically translates it in the background, using information gathered during profiling. Embra[7], a machine simulator, is built using dynamic translation techniques that were developed in Shade; a fast instruction-set simulator for execution profiling [17]. Le[8] investigates out-of-order execution techniques in dynamic binary translators, though their results are based on an interpreter-based implementation. Many of the optimization techniques used in dynamic translators have been derived from dynamic compilers such as SELF[9] and tcc[10]. Runtime optimizations in such compilers can provide 0.9x - 2x the performance of statically compiled programs. Such techniques have also been used in Just-in-time (JIT) compilers for Java. JITs from Sun[11], and others dynamically generate native machine code at runtime. To date, none of the current binary translators can generate code for more than one source and target machine pair. The machine-dependent aspects of the translation are hard coded into the translator, making it hard to reuse the translator's code for another set of machines. Our research differs from previous research in that machine-dependent issues are separated from machine-independent translation concerns, hence providing a way of specifying different machines (source and target machines) and supporting those specifications through reusable components, which implement the machine-independent analyses. This paper shows that this process is feasible and therefore enhances the reuse of code for the creation of dynamic binary translators. However, the machine-adaptability of the translator comes at the cost of performance, which is discussed in Section 5. 2. BINARY TRANSLATION FRAMEWORKS Binary translation is a process of low-level re-engineering; that is, decoding to a higher level of abstraction, followed by encoding to a lower level of abstraction. Figure 1 gives a block- view of the UQBT static translation framework [14,19]. The re-engineering process is divided into the initial reverse engineering phase on the left-hand side and the forward engineering phase on the right-hand side. The reverse engineering steps recover the semantic meaning of the machine instructions by a three-step process of decoding the binary file, decoding the machine instructions of the code segment, and mapping such instructions to their semantic meaning in the form of register transfer lists (RTLs). The high-level analysis process lifts the level of representation of the code to a machine-independent form, performs binary translation specific optimizations on the code, and then brings down the level of abstraction to RTLs for the target machine. This is followed by the forward engineering process of optimizing the code, encoding the instructions into machine code and storing the code and data of the program in a binary file. The forward engineering process is standard optimizing compiler code generation technology. RTL is a simple, low-level register-transfer representation of the effects of machine instructions. A single instruction corresponds to a register-transfer list, which in UQBT is a sequential composition of effects. Each effect assigns an expression to a location. All side effects are explicit at the top level; expressions are evaluated without side effects, using purely function RTL operators. An RTL language is a collection of locations and operators. For a machine M, the sub-language M-RTL is defined as those RTLs that represent instructions of machine M in a single RTL. As previously mentioned, the problems with static binary translation are the inability to find all the code that belongs to a program and the limitation of optimizations to static ones, without taking advantage of dynamic optimization techniques. One of the hardest problems to solve during the decoding of the S em an tic apper O p tim izer In s truc t io n In s truc io n Enc oder Analy s is Bin ary -f ile Bin ary -f ile Enc oder As s em b ly in s tr uc tio n s in ar y in s tr uc tio n s s tr eam ary in s tr uc tio n s s tr eam As s em b ly in s tr uc tio n s F rw ar d e ng ne e r ng Reverse e ng ne e r ng ary f ile M b in ar y f ile Figure 1. Static binary translation framework machine instructions is the separation of code from data - any binary-manipulation tool faces the same problem. Unfortunately, this problem is not solvable in general as both code and data are represented in the same way in von Neumann machines. This makes static translation incomplete and hence a runtime support environment is needed in the form of an interpreter, for example. 2.1 Dynamic binary translation framework In dynamic binary translation, the actual translation process takes place on an "as needed" basis, whereas static binary translation attempts to translate the entire program at once. Figure 2 illustrates a typical framework for a dynamic translator that uses a basic block as the unit of translation (i.e. its granularity). The left-hand side is similar to that of a static translator, but the processing of code is done at a different level of granularity (typically, one basic block at a time). The right-hand side is a little different to that of static translation. The first time a basic block is translated, assembly code for the target machine is emitted and encoded to binary form. This binary form is run directly on the target machine's memory as well as being kept in a cache. A mapping of the source and target addresses of the entire program for that basic block is stored in a map. If a basic block is executed several times, when the number of executions reaches a threshold, optimizations on the code are performed dynamically to generate better code for that hot spot. Different levels of optimization are possible depending on the number of times the code is executed. Optimized code then replaces the cached version of that basic block's code. The processing of basic blocks is driven by a switch manager. The switch manager determines whether a new translation needs to be performed by determining whether there is an entry corresponding to a source machine address in the map. If an entry exists, the corresponding target machine address is retrieved and its translation is fetched from the cache. If a match is not found, the switch manager directs the decoding of another basic block at the required source address. 2.2 Machine-adaptable dynamic binary translation framework Figure 3 extends Figure 2 to enable a dynamic translator to easily adapt to different source and target machines. This effort is achieved by a clean separation of concerns between machine-dependent information and machine-independent analyses. Through the use of specifications, a developer is able to concentrate on writing descriptions of properties of machines instead of having to (re)write the tool itself. The use of specifications to support machine-dependent information can also generate parts of the system automatically and provide a skeleton for the user to work on. As seen in Figure 3, the decoding of the binary file to source Bas ic b loc k Sem an tic Mapper imp le Op tim izations Bas ic b loc k Ins tr uc tion Dec oder Bas ic b loc k Ins tr uc tion Enc oder Ms tr ans lato r Bas ic b loc k Binary-f ile Dec oder Op tim ized inary ins truc tions s tream Ms b inary ins truc tions s tream Ms b inary file Ms rans lation cac he Add r es s Mapp ing a r Ms assemb ly ins truc tions As semb ly Enc oder ly ins truc tions a r Sw it ch Manager Figure 3. Machine-adaptable dynamic binary translation framework Bas ic b lo ck Sem an tic Mapper imple izations Bas ic b lo ck Ins truc tion Dec oder Bas ic b lo ck Ins truc tion Encoder Bas ic b lo ck ans lato Bas ic b lo ck Binary - file Dec oder As s emb ly in s truc tio ns Ef fic ient M t ass em bly in s truc tio ns ary in s truc tio ns s tream Ms binary in s truc tio ns s tream Ms assemb ly in s truc tio ns Ms binary file Ms-RTLs ans lation cache Add r es s Mapping Manager Figure 2. Dynamic binary translation framework machine RTLs (Ms-RTLs) requires the description of the binary- file format of the program, and the syntax and semantics of the machine instructions for a particular processor. We have experimented with three different languages, reusing the SLED language and developing our own BFF and SSL languages: . BFF: the binary-file format language supports the description of a binary-file's structure [15]. Current formats supported are DOS EXE, Solaris ELF and to a certain extent, Windows PE. SRL, a simple resourceable loader, supports the automatic generation of code to decode files specified using the BFF language. . SLED: the specification language for encoding and decoding supports the description of the syntax of machine instructions; ie. its binary to assembly mnemonic representation [18]. SLED is supported by the New Jersey machine-code toolkit [13]. The toolkit provides partial support for automatically generating an instruction decoder for a particular SLED specification. Current machines specified in this form include the Pentium, SPARC, MIPS and Alpha. . SSL: the semantic specification language allows for the description of the semantics of machine instructions. SSL is supported by SRD [16]. SRD is the semantic mapper component, which supports the parsing of SSL files and the storing of such information in the form of a dictionary, which can be instantiated dynamically. The output of this stage is Ms-RTLs. Ms-RTLs are converted to machine-independent RTLs (I-RTLs) through analyses, which remove machine dependent concepts of the source machine. This process identifies source machine's control transfers and maps it to the more general forms in I-RTL. For example: the following SPARC Ms-RTL for a call instruction is associated as a high-level call instruction in I-RTL. Other forms of transfers of control that exist in I-RTL are jumps, returns, conditional and unconditional branches. I-RTL supports register transfers, stack pushes and pops, high-level control transfers, and condition code functions. Some of these higher-level instructions allow for abstraction from the underlying machine. I-RTLs are converted into Mt-assembly instructions by mapping the functionality of such register transfers to the instructions available on the target machine, assisted by the SSL specifications. The instruction encoding process is supported by the SLED specification language, which maps assembly instructions into binary form. This code is then stored in the translator's cache for later reference. As an example, Figure 4 shows the various instruction transformations during the translation of a Pentium machine instruction to a SPARC machine instruction. The reverse-engineering stage decodes the Pentium binary code (0000 0010 1101 1000) to produce Pentium assembly code, which is then lifted to Pentium-RTLs and finally abstracted to I-RTL by replacing machine-dependent registers with virtual registers. The forward-engineering phase encodes the I-RTL to SPARC- RTL, SPARC assembly instructions and finally SPARC binary code. 2.3 Specification requirements in dynamic binary translation Dynamic translation cannot afford time-consuming analyses to lift the level of representation to a stage that resembles a high-level language, as per UQBT [14]. In UQBT, static analyses recover procedure call signatures including parameters and return values, thereby allowing the generated code to use native calling and parameter conventions on the target machine. If such analyses were used in dynamic translation, high performance degradation would be experienced during the translation. The alternative to costly analyses to remove properties of the underlying source machine is to go halfway to a high-level representation. We support inexpensive analyses to recover a basic form of high-level instructions (such as conditional branches and calls without parameters) and we emulate (rather than abstract away from) conventions used by the hardware and operating system in the source machine (i.e. without using the native conventions on the target machine). Both these steps are possible through the specification of features of the underlying hardware. For example, we emulate the SPARC architecture register windowing mechanism on a Pentium machine by specifying how this mechanism works. On a SPARC machine, we emulate the Pentium stack parameter passing convention. However, we do not emulate the SPARC processor delayed transfers of control as we support higher level branching instructions. Clearly, this compromise has a performance impact on the translated code, but it provides a fast way of translating code, which can then be optimized at runtime if it becomes a hotspot in the program. In order to support the translation of Ms-RTLs to I-RTLs, the SSL language has been extended from machine instruction level semantics to include hardware semantics as well. For example, for the SPARC architecture, the effects of changing register windows and how each register in the current window is accessed were specified, and for the Pentium architecture, properties about the stack movement were specified. This information is currently not used by UQBT; only by UQDBT. UQBT relies on costly analysis to abstract higher-level information, without depending on very low-level details of the underlying hardware. addl %ebx, %eax add %o1, %o2, %o1 Figure 4. Pentium to SPARC example 3. RESEARCH PROBLEMS Unlike other dynamic binary translators that are written with a fixed set of source and destination machines in mind, UQDBT is designed to handle a wide range of CISC and RISC machine architectures. While some translators can directly map source machine-specific idioms to the target machine, such translators are bound to work only under that source/target pair. To extend those translators to support different machines, extensive rewriting of the code is needed, as the direct idiom mapping between machines is different. The goal of UQDBT is to provide a framework that can be modified and extended with ease to support additional source and target machines without the need to rewrite a new translator from scratch. The process of finding a generalization of all existing (and future) machines is non-trivial and cannot be fully predicted. UQDBT uses UQBT's approach of specifying properties of machine instruction sets that are widely available in today's machines and allowing the user to extend the specification language to support new features of (future) machines to reuse the rest of the translation framework. As with UQBT, we use multi-platform operating systems to concentrate on the more fundamental issues of instruction translations. UQDBT's goal was to address the following types of research problems in dynamic machine-adaptable binary translation: 1. What is the best way of supporting the machine-dependent to machine-independent RTL translation? The main criteria in the translation is efficiency, hence expensive analyses are not an option. Further, the translation needs to be supported by the underlying specification language, in order to generate Ms-RTLs that contain enough information about the underlying Ms machine. 2. How much state of the source machine is needed for dynamic translation and what effects does this have on specification of such properties of the machine? 3. What is the best way of automating the transformation of I- RTLs down to Mt-assembly code? Can a code selector be automatically generated from a target machine specification? 4. Is it possible to efficiently use specifications that contain information about operating system conventions, such as calling and parameter conventions used by the OS to communicate with the program? For example, in order to use Pentium's stack parameter convention in code that was translated from a SPARC architecture binary (which passes parameters on registers), analysis to determine the parameters needs first to be performed. 3.1 Implementation of UQDBT We have been experimenting with the right level of description required in order to support dynamic translation based on specifications. In our experience, too low-level or high-level a description of the underlying machine is unsuitable. We view UQBT's semantic specifications of machines as a high-level description, as they only describe the machine instruction semantics but do not specify the underlying hardware that supports the control transfer instructions (e.g. the register windows and delayed instructions on the SPARC architecture). In UQBT, such detailed level of information is not needed because of the specification and use of calling conventions and control transfer instruction. Further, other semantic description languages are used to describe all the low-level details of the underlying machine; such languages are suitable for emulation purposes but contain too much information for dynamic translation. The first problem above has been addressed by specifying how the hardware works in relation to control transfer instructions; this provides for a fast translation of Ms-assembly instructions into information-rich Ms-RTLs (the extended ones), avoiding the need to recover information at runtime. The following are the types of information that have been described for SPARC and Pentium processors: . The effects of the SPARC register windowing mechanism . Stack properties . Memory alignment . Parameters and return locations. The SPARC machine allocates a new set of working registers each time a SAVE instruction is called. In other words, it effectively provides an infinite number of registers for program use. The effect of the SPARC register windows is captured by extending SSL to specifying how each of the registers are accessed and how the register windows change during each save and restore instruction, hence providing a different set of working registers. This provides accurate simulation on target machines that only have a limited amount of usable registers. The effects of the stack pointer are different on different types of machines. On Pentium machines, the stack pointer can change indefinitely within a given procedure. On RISC machines, the stack pointer is normally constrained to a pre-allocated stack frame's fixed size that includes enough space for all register spills of that procedure. Specifying how the stack changes in the original machine suggests ways for the code generator to generate stack manipulation instruction on the target machine. For example, simulating stack pushing and popping on a SPARC machine. Memory alignment places constraints on how the machine state at a particular point in the program should be. In SPARC, the frame pointer and stack pointer need to be double word aligned. Thus, the code generator needs to enforce such conditions before entry to or exit from a call. Differences in machine calling conventions, namely how the parameters are passed and where return values are stored, play a crucial part on how the code generator constructs the right setup when calling native library functions. SPARC generally passes parameters in registers while Pentium pushes them on the stack. This information is needed for both source and target machines to identify the transformation of parameters and return values. Differences in endianness between source and target machines requires byte swapping to be performed when loading and storing data. Byte swapping is an expensive process. Although the Pentium can do byte swapping quite easily, it takes about SPARC V8 instructions for a 32-bit swap. This is an expensive process and should be avoided if possible. In particular, when running a Pentium binary on a SPARC, every push and pop instruction (which appears quite often in Pentium programs) will require byte swapping. Heuristics are used in UQDBT to avoid byte swapping on pushing and popping to the stack. The second problem above is related to the first one. The amount of source machine state that is carried across depends on the effectiveness of translation in the first problem. For areas that are not easily specified or unspecified, they are carried across and are apparent within the machine-independent RTL. In UQDBT, control transfer instructions will contain a tag indicating how to process its delayed slot instruction (in architectures that support delayed slots). The third problem above is current work in progress. The goal is not only to automatically construct the code generator, but also determine the best performance heuristics for selecting target machine instruction when encountering similar patterns. Some patterns may never be matched or may be nearly impossible to match. For example, trying to pattern match a SPARC save instruction with some Pentium-RTLs. Our experiences with the fourth problem above suggest that performance can be gain by using native OS conventions. UQDBT currently simulates the calling convention for Pentium programs on a SPARC machine, i.e. parameters are passed on the stack instead of in registers. To remove this simulated effect and convert it to use native conventions, one needs know: . How much improvement does it offer over direct machine simulation? . At what level should this conversion occur? . Is it worth while doing such analysis in a dynamic binary translation environment? 4. CASE STUDY In this section we show an example of a small Pentium program converted by UQDBTps (the "ps" postfix indicates translations from Pentium to SPARC architectures) to run on a SPARC machine. Both programs are for the Solaris operating system. The main differences of the two test machines are: 1. SPARC is a RISC architecture, whereas Pentium is CISC. 2. SPARC is big-endian, while Pentium is little-endian. 3. SPARC passes parameters in registers (and sometimes on the stack as well), while Pentium normally passes them on the stack. 4.1 Basic block translations and address mappings Figure 5 is the disassembly of a "Hello World" binary program compiled for the Pentium machine running Solaris. The first column is the source address seen by the Pentium processor. The second and third columns are the actual Pentium binaries and its corresponding assembly representation. There are 3 basic blocks 0x43676c8: save %sp, -132, %sp 0x43676cc: add %sp, -4, %sp 0x43676d0: add %i7, 8, %l0 0x43676d4: st %l0, [ 0x43676d8: mov %sp, %l0 0x43676dc: subcc %l0, 4, %l0 0x43676e0: mov %l0, %sp 0x43676e4: mov %fp, %l0 0x43676e8: st %l0, [ 0x43676f0: mov %l0, %fp 0x43676f4: mov %sp, %l0 0x43676f8: subcc %l0, 4, %l0 0x43676fc: mov %l0, %sp 0x4367704: add %l0, 0x3f8, %l0 ! 0x80493f8 0x4367708: st %l0, [ ld 0x4367714: mov %sp, %l1 0x4367718: sethi %hi(0xfffffc00), %l2 0x4367720: and %sp, %l2, %sp 0x4367724: and %fp, %l2, %fp 0x4367728: sethi %hi(0xef663400), %g6 0x4367730: nop 0x4367734: mov %l1, %sp 0x4367738: mov %l0, %fp 0x4367740: add %g5, 0x125, %g5 0x4367744: sethi %hi(0x41bfc00), %g6 0x4367748: call %g6 0x436774c: nop Figure 7. Generated Sparc assembly for the 1 st BB of Figure 8048919: 8b ec movl %esp,%ebp 8048920: e8 9b fe ff ff call 0xfffffe9b 8048925: 804892a: eb 00 jmp 0x0 804892c: c9 leave 804892d: c3 ret Figure 5. "Hello World" x86 disassembly 8048918: PUSH r[29] 804891b: PUSH 134517752 8048925: *32* r[tmp1] := r[28] 804892a: JUMP 0x804892c Figure (BBs) in this program: . first BB - 4 instructions (0x8048918 to 0x8048920) . second BB - 3 instructions (0x8048925 to 0x804892a) . third BB - 2 instructions (0x804892c to 0x804892d) Figure 6 shows the intermediate representation (I-RTLs) for these BBs. Note that the translation is done incrementally, i.e. each BB is decoded separately at runtime. UQDBTps works in the source machine's address space and translates a basic block at a time. Both the data and text from the source Pentium program are mapped to the actual machine's source address space even though it is actually running on a SPARC machine. For example, a Pentium program with data and text sections located at 0x8040000 and 0x8048000 will be mapped exactly at these addresses even though a typical SPARC program expects the text and data at addresses 0x10000 and 0x20000. UQDBTps also simulates the Pentium machine's environment in the SPARC generated code, i.e. the pushing and popping of temporaries and parameters from the Pentium machine is preserved in the generated code. UQDBTps tries to generate code as quickly as it possibly can with little or no optimization. 4.2 Pentium stack simulation Figure 7 is the SPARC code generated for the first BB of Figure 6. The first four instructions simulate the Pentium main prologue by setting up the stack and storing the return address (obtained by the value of %i7 bytes of stack space are reserved in the initial save %sp, -132, %sp. This space is used by the SPARC processor to store parameters, return structures, local variables and register spills. Hence the actual simulated Pentium stack pointer %esp starts at %sp+0x84 (see Figure 8), while Pentium's %ebp is mapped to the SPARC register %fp. Pushing is handled by subtracting the size of the value pushed from %sp and storing the result in [%sp+0x84]. Popping removes from [%sp+0x84] and increments %sp by the appropriate size. 4.3 Function calls and stack alignments In Pentium, actual parameters to function calls are passed on the stack while SPARC parameters are passed in registers. The printf format string to "Hello World" is at address $0x80493f8 and is pushed by the instruction 804891b (see Figure 5). To successfully call the native SPARC printf function, this address must be stored in register %o0 (instruction 0x436770c in Figure 7). The equivalent printf in SPARC is at 0xef6635b8 and a call to this function is made (instruction 0x436772c). Calls to library functions such as printf are assumed by UQDBTps to exist on the source as well as the target machine. This assumption is not restrictive as long as there is a mapping from the source library function to an equivalent function on the target machine; i.e. libraries can be reproduced on the target machine by translation or rewriting to produce such a mapping. Translators such as FX!32 make the same assumption. SPARC machines expect %sp and %fp to be aligned to double word (64 bits) boundaries. Therefore, before calling a native library function, %sp and %fp need to be 8-byte aligned. The current values are restored after the function call returns (instructions 0x4367734 and 0x4367738). At the end of a basic block, control is passed back to UQDBTps' switch manager, with the indication of the next basic block address to be processed in %g5 (0x8048925 in the above case). It is the role of the switch manager to decide whether to start the translation indicated in %g5 or fetch an already translated BB from the translation cache. In the above case (where the next BB starts at 0x8048925), the address is not in the translation map, hence the translation starts at that new address. Figure 9 shows the generated SPARC assembly for the next BB for the Pentium program. a r a m e e r s a r a m e e r s S a c k p Figure 8. Sparc stack frame 0x435ff58: sethi %hi(0x4485c00), %l0 0x435ff64: st %l1, [ %l0 0x435ff68: mov %sp, %l0 0x435ff6c: addcc %l0, 4, %l0 0x435ff70: mov %l0, %sp 0x435ff74: sethi %hi(0x42b7000), %l0 0x435ff78: add %l0, 0x3d4, %l0 ! 0x42b73d4 0x435ff7c: rd %ccr, %l1 0x435ff80: st %l1, [ %l0 0x435ff84: sethi %hi(0x4486000), %l0 0x435ff88: add %l0, 0x18, %l0 ! 0x4486018 ld [ %l1 ], %l1 0x435ff98: sethi %hi(0x4486000), %l2 ld [ %l2 ], %l2 0x435ffa4: xorcc %l1, %l2, %l1 0x435ffa8: st %l1, [ %l0 0x435ffb4: rd %ccr, %l1 0x435ffb8: st %l1, [ %l0 0x435ffc4: sethi %hi(0x41bfc00), %g6 0x435ffc8: call %g6 0x435ffcc: nop Figure 9. Generated Sparc assembly for the 2 nd BB of figure 6 4.4 Register mapping and condition codes During the translation, all Pentium registers are mapped to virtual registers (i.e. memory locations). To access a virtual register on a SPARC machine, a sethi and an add instruction are used. For example, instructions 0x435ff84 and 0x435ff88 in Figure 9 are used to access the virtual register representing the x86 register %eax. While most instructions on SPARC do not affect condition codes (flags) unless explicitly indicated by the instruction, almost all Pentium instructions affect the flags. Each Pentium instruction that affects the status of the flags is simulated using the equivalent condition code version of the same instruction on the SPARC machine (instructions 0x435ff6c and 0x435ffa4). The condition codes are read (instruction 0x435ff7c) and saved to the virtual flag register (instruction 0x435ff80) after these instructions, to preserve its current value, which can be retrieved later if required. A closer look at the above example shows that the generated code is not very efficient. Simple optimizations such as forward substitutions and dead code elimination can greatly reduce the size of the generated code. Such optimizations can yield better code but will take longer to generate - a trade off between code quality and speed. The code generator back-end of UQDBTps is very fast despite the poor quality of the generated code. We are currently implementing on-demand optimizations, to the hotspots in the program, as well as performing register allocation. 5. PRELIMINARY RESULTS UQDBT is based on the UQBT framework, as such, its front-end is re-used from UQBT, changing its granularity of decoding from the procedure level to the basic block level. The front-end uses the extended SSL specifications and generates Ms-RTLs. The machine instruction encoding routines of the back-end are automatically generated from SLED specifications using the toolkit. This section shows some preliminary results obtained by two dynamic translators instantiated from the UQDBT framework; UQDBTps (Pentium to SPARC) and UQDBTss (SPARC to SPARC). It then looks at the types of optimizations that need to be introduced in order to improve the performance of frequently executed code, and it gives the reader an idea of effort gone into the development of the framework and the amount of reuse expected. 5.1 Performance Micro-benchmark results were obtained using a Pentium MMX machine and an UltraSparc II 250 MHz machine, both running the Solaris operating system. The results reported herein are those of the translation overhead and do not currently make use of dynamic optimizations, only register caching within basic blocks. Clearly, the performance of generated binaries from UQDBT without optimization is inferior to direct native compilation. A typical 1:10 ratio (i.e. 1 source machine instruction to 10 target machine instructions) is expected in a typical emulator/interpreter without caching. UQDBTps gives figures close to this ratio. For example, for the 9 Pentium instructions of Figure 5, 95 SPARC instructions were generated. While this ratio is similar to that of emulation, the speed up gained from UQDBT comes from reusing already translated BBs from the translation cache when the same piece of code is executed again. In its present form, UQDBT is still in its early development and hence we provide preliminary results for UQDBTps, a Pentium to SPARC translator, and UQDBTss, a SPARC to SPARC translator. It is undoubtedly true that there is little practical use for SPARC to SPARC translation unless runtime optimizations can significantly speed up translated programs. The inclusion of this translation is to show the effect of machine-adaptability in UQDBT. Further, the translation from SPARC binaries to I-RTL removes any machine dependencies and thus, during I-RTL to SPARC code generation, the UQDBTss is unaware of the fact that the source machine is SPARC. This is also true for UQDBTps. Since little analysis is done to the decoded instructions and processing is concentrated on decoding and code generation requested by the switch manager, it better reflects the performance impact on the use of on-demand techniques prior to introducing optimizations. The test programs showed in the tables are: . Sieve 3000 (prints the first 3,000 prime numbers), . Fibonacci of 40, and . Mbanner (prints the banner for the "ELF" string 500,000 times). Sieve mainly contains register to register manipulation, while Fibonacci has a lot of recursive calls and Mbanner has a lot of stack operations and accesses to an array of data. Tables show the times of translation and execution of programs using UQDBTps and UQDBTss, compared to natively gcc O0 compiled programs. The source programs were also O0 compiled. Column 2 shows the preprocessing time that is needed before the actual translation takes place. Note that UQDBTps takes longer to start than UQDBTss. This is because Pentium has a larger instruction set (hence a larger SSL specification file) which takes longer to process. It is also caused by different page alignment sizes between the Pentium and SPARC; as a result, extra steps are taken to ensure that both text and data sections are loaded correctly on the SPARC machine. Column 3 shows the total time spent decoding the source instructions, transforming them to I-RTLs and generating the final SPARC code. Column 4 shows the execution time in the generated SPARC code without using register caching, i.e. every register access was done through virtual registers. Column 5 shows the execution time of the generated SPARC code with register caching, which yields between 15 to 50 percent performance gain. Column 6 is the natively compiled gcc version of the same program on SPARC. Comparing columns 5 and 6 gives the relative performance of the translators. The figures suggest 2 to 6 times slowdown when running programs using UQDBTps and UQDBTss. The slow performance of the translated Fibonacci program under UQDBTss is caused by the effects of the register windowing mechanism in SPARC, which are carried forth to the I-RTLs. Since the I-RTLs are unaware of the fact that the source and target machines are the same, this causes the entire register windowing system to be simulated in the generated code. Given that on-demand optimizations have not been performed yet, the quality of the generated code is comparable to O0 optimization level of a traditional compiler. Tables 3 and 4 show the efficiency of the translators relative to the size of the original program. Column 2 is the size of the program's text area. Note that not all bytes necessarily represent instructions and that not all code is necessarily reachable or executed at runtime. Column 3 shows the actual bytes decoded by the translator at runtime. This number varies from Column 2 since only valid paths at runtime are translated, and sometimes re-translation is needed when a jump into the middle of a BB is made. Columns 4 and 5 show the number of bytes of code generated by the translator without/with register caching. Register caching has been done at a basic block level; cached registers are copied back to their memory locations at the end of each basic block. Comparing column 5 with column 3 gives the relative ratio of bytes generated versus bytes decoded. The above figures suggest that on average, each byte from the source machine translates to around 7 to 10 bytes of target SPARC code. The last column is the ratio of machine cycles to bytes of source code. It gives a rough indication of the performance of the translation. On an UltraSparc II 250MHZ, the translators require about 180,000 machine cycles per byte of input source, which is about 10 times more cycles used than in a traditional O0 compiler. 5.2 Optimizations - future work Most programs spend 90% of the time in a small section of the code. It is these hotspots that are worthwhile for a dynamic binary translator to spend time optimizing. UQDBT currently does not perform any optimizations. The next revision of UQDBT will contain optimizations that are triggered by counters. Counters are inserted in basic blocks to indicate the number of times a particular basic block is executed at runtime. When a certain threshold is reached (indicating that the program spends significant time in a piece of code), the optimizer will be invoked in an attempt to produce efficient code. Four levels of optimization will be provided by UQDBT progressively when certain thresholds are reached: 1. Register liveness analysis, forward substitution, constant propagation - improves the quality of the generated code and reduces the number of instructions executed. 2. Register allocation - a more rigorous process for removing access to virtual registers and replacement with allocation to hardware registers on the target machine, assisted by liveness information, rather than just caching registers at a basic block level. 3. Code movement - moving and joining frequently executed BBs closer together, thus reducing transition costs (calls and jumps). 4. Customization - create specialized versions of the BBs that are found to have a fixed range of runtime values within the BB, e.g. on repeated entry to a BB, a register or variable contains the same value 90% of the time. 5.3 Effort In order to give the reader an idea of the effort that has gone into the development of UQDBT and the effort of reusing the system, we quantify such effort as follows. UQDBT has been the effort of 1 person over a period of 1.5 years, experimenting with the amount of specification required at the semantic level for different machines. This effort was performed by a person who was already familiar with the UQBT having worked on SSL in the past. UQDBT's current implementation size is of 18,500 lines of source code in C++, 3,300 lines of partially-generated code, and lines (1,000 for SPARC and 2,500 for Pentium) of Test programs Pre- processing Translation time Execution time without reg caching Execution time simple reg caching Native gcc compiled Test programs Pre- processing Translation time Execution time without reg caching Execution time simple reg caching Native gcc compiled Fibonacci 0.52 0.07 162.23 139.35 41.18 Fibonacci 0.22 0.12 256.05 198.97 41.18 mbanner 0.50 0.34 191.00 126.28 22.85 mbanner 0.21 0.50 204.09 97.09 22.85 Test programs Original program size Source bytes decoded Target bytes generated w/o reg caching Target bytes generated w/- reg caching 1,000 cycles source byte Test programs Original program size Source bytes decoded Target bytes generated w/o reg caching Target bytes generated w/- reg caching 1,000 cycles source byte Fibonacci 102 94 1188 1104 186 Fibonacci 220 176 1764 1468 170 mbanner 483 467 4548 3816 182 mbanner 748 760 7112 5032 164 Table 4: UQDBTss - SPARC to SPARC translation Table 1: UQDBTps - Pentium to SPARC translation (second) Table 2: UQDBTss - SPARC to SPARC translation (seconds) Table 3: UQDBTps - Pentium to SPARC translation specification files. A user of the UQDBT framework would be able to reuse most of this source code and would need to write syntax and semantics specification files for new machines (or reuse existing ones). These figures are not final at this stage, as most dynamic optimizations have not been implemented yet. It nevertheless gives an indication of the amount of reuse of code in the system. 6. DISCUSSION The preliminary results of UQDBT point at the tradeoffs of machine-adaptability. In return for writing less code to support two particular machines, a performance penalty in the generated code is seen at this stage. A binary translation writer would be expected to write specifications for new machines, which are in the order of a few thousand lines of code, and reuse a good part of 18,500 lines of code, reaping the benefits of reuse and time efficiency. However, at this stage, UQDBT generates code that performs at about the same speed as emulated code, therefore a user seeing a 10x performance degradation in their translated programs. The introduction of register caching in the generated code has brought down this factor to 6. We expect that the introduction of on-demand optimizations on hotspots of the program will improve the performance of the generated code, bringing down the performance factor to 2x-3x. One of the main questions we have dealt with throughout experiments in this area has been how much should be specified and how much should be supported by hand. The level of detail in a specification can make a translator faster or slower. If the full details of a machine are specified, the specification is suitable for generating an emulator that supports 100% that machine. However, if we can provide a means for eliminating part of that emulation process, then a different type of specification is needed. This is what we have tried to achieve through our semantic specifications and the use of two intermediate languages. The RTL language describes low-level and machine specific aspects of a machine, and UQDBT finds support in the specifications to perform simple analyses to lift the level of the representation to I-RTL. The aim is to perform simple transformations of the code that are not expensive on time and that are generic enough to be suitable for our intermediate representations. This is why I-RTL is different to HRTL, the high-level intermediate representation used by the static UQBT framework. In HRTL, expensive analyses recover parameters to procedures and return values, hence allowing the code generator to use native calling conventions on the target machine. In I- RTL, the code generator makes use of the specification of a stack for example, in order to pass parameters on the stack, without ever determining which locations are parameters to procedure calls. However, some notion of parameters is needed in order to interface correctly to native library functions, and to pass parameters in the right locations. It has also been our experience that some modules may be better off written by hand, without specifying the complete semantics of features of a machine that are too unique. For example, one could consider implementing an SPARC-specific module for supporting the register windowing semantics, so that better register allocation is performed in this case. At present, we have specified the register windowing mechanism and generated code that puts all these registers in virtual (memory) locations. Through the use of register caching, some of the memory locations are mirrored to hardware registers of the target machine, improving somewhat the performance of the program. However, there is still a large overhead in the copying of the registers to virtual locations at each call and return. This can be reduced with dead code elimination, but perhaps hand written code would have achieved better code. Another aspect to take into consideration is the granularity of translation. In UQDBT, the granularity unit for processing is a basic block (BB) at a time. Just after code generation, a link is made to the switch manager at the exit of each BB and flushing of cached registers to virtual registers is performed. This is needed to keep data accurately stored and consistent across transitions from one BB to another. Transitions from one BB to the next will go via the switch manager if the next BB has not been translated yet. Using BB as the unit of translation restricts the effectiveness of register allocation. Since BBs are relatively small, it is difficult to determine register live-ness information, as data is not collected across BB boundaries. If the unit of granularity is changed, this could yield better code in some cases while worse code in others. For example, the translation unit might be changed from a BB to a procedure at a time. This would allow the code generator to reduce the amount of flushing of cached registers and hence reduce the number of instructions that need to be executed at runtime. It would improve the effectiveness of allocating registers during code generation since more live-ness information could be collected. But using a larger unit of translation such as a procedure may involve decoding paths that may not be ever taken at runtime, thus generating code that is not executed. It is not obvious what the granularity unit should be since some types of programs will benefit by using a particular granularity unit while others may suffer. Program with a lot of small procedures will benefit if the unit of translation is a procedure, but suffer if the program has a lot of conditional branches. 7. CONCLUSION UQDBT is a machine-adaptable dynamic binary translator framework that is capable of being configured for different source and target machines through specifications of properties of those machines. The UQDBT framework can be modified and extended with ease to support additional source and target machine architectures without the need to write a new translator from scratch. Our case study shows that the translation process between two different architectures is both complex and challenging using machine-adaptable dynamic translation techniques. Nevertheless, preliminary results suggest that performance of implementing on-demand processing in a dynamic system can be done efficiently. Despite that, some research problems remain in building a fully machine-adaptable dynamic translation framework. UQDBT appears to be a promising model to provide a generic dynamic binary translation framework. 8. ACKNOWLEDGMENTS The authors wish to thank Mike Van Emmerik for his helpful discussions in implementation and testing strategies and the members of the Kanban group at Sun Microsystems, Inc.; whose system motivated some of this work. This work is part of the University of Queensland Binary Translation (UQBT) project. More information can be obtained about the project by visiting the following URL: http://www.csee.uq.edu.au/csm/uqbt.html. 9. --R Binary translation. Macintosh application environment. http://www. Digital FX! How to Efficiently Run Mac Programs on PCs. Embra: Fast and Flexible Machine Simulation. SELF: The power of simplicity. Kaashoek. tcc: A System for Fast Micha&lstrok The New Jersey Machine-Code Toolkit The Design of a Resourceable and Retargetable Binary Translator. Specifying the semantics of machine instructions. A Fast Instruction-Set Simulated for Execution Profiling Specifying representation of machine instructions. Preliminary Experiences with the Use of the UQBT Binary Translation Framework. --TR Self: The power of simplicity Binary translation Shade: a fast instruction-set simulator for execution profiling Embra Specifying representations of machine instructions Fast, effective code generation in a just-in-time Java compiler An out-of-order execution technique for runtime binary translators SRL 3/4?A Simple Retargetable Loader The Design of a Resourceable and Retargetable Binary Translator Specifying the Semantics of Machine Instructions --CTR Naveen Kumar , Bruce R. Childers , Daniel Williams , Jack W. Davidson , Mary Lou Soffa, Compile-time planning for overhead reduction in software dynamic translators, International Journal of Parallel Programming, v.33 n.2, p.103-114, June 2005 Lian Li , Jingling Xue, Trace-based leakage energy optimisations at link time, Journal of Systems Architecture: the EUROMICRO Journal, v.53 n.1, p.1-20, January, 2007 David Ung , Cristina Cifuentes, Dynamic binary translation using run-time feedbacks, Science of Computer Programming, v.60 n.2, p.189-204, April 2006 Lian Li , Jingling Xue, A trace-based binary compilation framework for energy-aware computing, ACM SIGPLAN Notices, v.39 n.7, July 2004 Giuseppe Desoli , Nikolay Mateev , Evelyn Duesterwald , Paolo Faraboschi , Joseph A. Fisher, DELI: a new run-time control point, Proceedings of the 35th annual ACM/IEEE international symposium on Microarchitecture, November 18-22, 2002, Istanbul, Turkey Jason D. Hiser , Daniel Williams , Wei Hu , Jack W. Davidson , Jason Mars , Bruce R. Childers, Evaluating Indirect Branch Handling Mechanisms in Software Dynamic Translation Systems, Proceedings of the International Symposium on Code Generation and Optimization, p.61-73, March 11-14, 2007 Gregory T. Sullivan , Derek L. Bruening , Iris Baron , Timothy Garnett , Saman Amarasinghe, Dynamic native optimization of interpreters, Proceedings of the workshop on Interpreters, virtual machines and emulators, p.50-57, June 12-12, 2003, San Diego, California Bruening , Timothy Garnett , Saman Amarasinghe, An infrastructure for adaptive dynamic optimization, Proceedings of the international symposium on Code generation and optimization: feedback-directed and runtime optimization, March 23-26, 2003, San Francisco, California John Aycock, A brief history of just-in-time, ACM Computing Surveys (CSUR), v.35 n.2, p.97-113, June

dynamic execution;binary translation;interpretation;emulation;dynamic compilation

351523

On the Eigenvalues of the Volume Integral Operator of Electromagnetic Scattering.

The volume integral equation of electromagnetic scattering can be used to compute the scattering by inhomogeneous or anisotropic scatterers. In this paper we compute the spectrum of the scattering integral operator for a sphere and the eigenvalues of the coefficient matrices that arise from the discretization of the integral equation. For the case of a spherical scatterer, the eigenvalues lie mostly on a line in the complex plane, with some eigenvalues lying below the line. We show how the spectrum of the integral operator can be related to the well-posedness of a modified scattering problem. The eigenvalues lying below the line segment arise from resonances in the analytical series solution of scattering by a sphere. The eigenvalues on the line are due to the branch cut of the square root in the definition of the refractive index. We try to use this information to predict the performance of iterative methods. For a normal matrix the initial guess and the eigenvalues of the coefficient matrix determine the rate of convergence of iterative solvers. We show that when the scatterer is a small sphere, the convergence rate for the nonnormal coefficient matrices can be estimated but this estimate is no longer valid for large spheres.

Introduction . The convergence of iterative solvers for systems of linear equations is closely related to the eigenvalue distribution of the coefficient matrix. To be able to predict the convergence of iterative solvers one therefore needs to look at how the matrix and its eigenvalues arise from a discretization of a physical problem. In this article we examine the eigenvalues of the coefficient matrix arising from a discretization of a volume integral equation of electromagnetic scattering and the behavior of iterative solvers for this problem. In the early stages of this research project we noticed that in the case of a spherical scatterer, most of the eigenvalues of the matrix lie on a line in the complex plane. The line segment can also be seen for other types of scatterers. The coefficient matrix arises from a discretization of a physical problem. Thus, the eigenvalues of the coefficient matrix should resemble the spectrum of the corresponding integral operator in a function space. The spectrum of an operator is the infinite-dimensional counterpart of the eigenvalues of a matrix. In the case of a spherical scatterer, we show that it is possible to relate the spectrum of the scattering integral operator to the behavior of the analytic solution of scattering by a sphere. The main result is that the eigenvalues of the coefficient matrix are related either to resonances in the analytic solution or to the branch cut of the complex square root used in the definition of the refractive index. An analysis of the eigenvalues of the surface integral operator in electromagnetic scattering has been carried out by Hsiao and Kleinman [12], who studied the mapping properties of integral operators in an appropriate Sobolev space. Colton and Kress have also studied these weakly singular surface integral operators [3]. In our case, we look at the spectrum of a strongly singular volume integral operator. We have not studied the mapping properties of this operator. The paper is organized as follows: In x 2 we give the volume integral equation formalism and discuss its discretization. In x 3 we compute the eigenvalues of the coefficient matrix. Section 4 shows how the spectrum of the integral operator can be determined with the help of the analytic solution. Numerical Parts of this work have been carried out at the Center for Scientific Computing, Finland, and at the Helsinki University of Technology. This work was supported by the Jenny and Antti Wihuri foundation and by the Cultural Foundation of Finland. y CERFACS, 42 av G. Coriolis, 31057 Toulouse Cedex, France (Jussi.Rahola@cerfacs.fr). J. RAHOLA experiments are presented that show how well the eigenvalues of the coefficient matrix can approximate the spectrum of the integral operator. In x 5 we show how the eigenvalue distribution can be used in the prediction of the convergence of iterative solvers. In x 6 we conclude and point out some areas of future research. 2. Volume integral equation for electromagnetic scattering. The volume integral equation of electromagnetic scattering is employed especially for inhomogeneous and anisotropic scatterers. For homogeneous scatterers it also offers some advantages over the surface integral equation, namely a simple description of the scatterer with the help of cubic cells and the use of the fast Fourier transform in the computation of the matrix-vector products. On the other hand, the volume integral equation uses many more unknowns than the surface integral formulation for a given computational volume. We will study the scattering of monochromatic electromagnetic radiation of frequency f , wavelength wave number In our presentation we assume that the electric field is time-harmonic and thus its time-dependence is of the form exp(\Gammai!t). The scattering material is described by its complex refractive index defined by ~ffl, where the complex permittivity ~ ffl is given by ~ is the (real) permittivity and oe is the conductivity. We have assumed that the magnetic permeability of the material is the same as that of vacuum. The volume integral equation is typically used for dielectric or weakly conducting objects. The volume integral equation of electromagnetic scattering is given by [9, 15, 16] Z where V is the volume of space the scatterer occupies, E(r) is the electric field inside the object, E inc (r) is the incident field and G is the dyadic Green's function given by where The Green's function also has an explicit representation by ae \Gammaae 2 ae The scattering integral equation can be discretized in various ways. The simplest discretization uses cubic cells and assumes that the electric field is constant inside each cube. By requiring that the integral equation (2.1) be satisfied at the centers r i of the N cubes (point-matching or collocation technique) and by using simple one-point integration, we end up with the equation [14, 15, 22] is the volume of the computational cube, M is given by ii e ikb(3=4-) 1=3 EIGENVALUES OF INTEGRAL OPERATOR IN SCATTERING 3 b is the length of the side of the computational cube and T ij is given by ae 3 ae 3 The factor M arises from the analytical integration of the self-term, using a sphere whose volume is equal to the volume of the cube [15]. Note that all the physical dimensions of the problem are of the form kr. This means that the scattering problem depends only on the ratio of the size of the object to the wavelength. In the rest of the paper we will give the dimensions of the scattering object in the form The systems of linear equations (2.5) has a dense complex symmetric coefficient matrix. Various ways of solving the linear system with iterative solvers has been studied by Rahola [17, 23, 22] who chose the complex symmetric version of QMR [7] as the iterative solver. The matrix-vector products can be computed with the help of the fast Fourier transform if the cells sit on a regular lattice. For the FFT, the computational grid must be enlarged with ghost cells to a cube [10]. 3. Eigenvalues of the coefficient matrix. In this section we show some examples of the eigen-values of the coefficient matrices. All the eigenvalues were computed with the eigenvalue routines in the library [1]. Figure 3.1 shows the eigenvalues of a coefficient matrix for a sphere of radius refractive computational cells, respectively, are used to discretize the sphere. Note how most of the eigenvalues lie on a line in the complex plane. When the discretization is refined, the line segment becomes "denser" but there remains a small number of eigenvalues off the line. An interesting feature of these coefficient matrices is that when the discretization is refined, the number of iterations needed for iterative solvers to converge remains constant even when the linear system is not preconditioned [21, 22]. This is of course the optimal situation for iterative solvers. In practical calculations, when the number of computational points is increased, the size of computational cells is kept constant and the physical size of the object is increased. In this case the number of iterations naturally grows with the size of the problem. 4. Spectrum of the integral operator. In this section we will explain what is meant by the spectrum of a linear operator, how to compute points in the spectrum of the scattering integral operator and how they correspond to the eigenvalues of the coefficient matrix of the discretized problem. The spectrum of a linear operator T is the set of points z in the complex plane for which the operator does not have an inverse operator that is a bounded linear operator defined everywhere. Here 1 stands for the identity operator. A matrix is the prototype of a finite-dimensional linear operator. The spectrum of a matrix is exactly the set of its eigenvalues, that is the point spectrum. In the rest of this paper, we reserve the word 'spectrum' only for the infinite-dimensional integral operator and talk only about eigenvalues of matrices. For an eigenvalue -, there exists an eigenvector x such that thus the matrix singular and not invertible. For an arbitrary linear operator, there is no general way of obtaining its spectrum. Each case must be analyzed separately with different analytical tools. We will consider the case of the scattering integral equation (2.1). We will write it in the form where K is the integral operator defined by Z 4 J. RAHOLA Fig. 3.1. Eigenvalues of the coefficient matrix for a spherical scatterer of radius refractive index 0:05i. The sphere is discretized with 136 computational cells (upper) and 455 computational cells (lower). We do not know of any direct way of computing the spectrum of the scattering integral operator K. However, with the following observation, we can relate the spectrum to the well-posedness of some related scattering problems. To find the spectrum of the operator K we need to find the complex points z for which the operator does not have a well-defined inverse. For a homogeneous scatterer the integral operator (4.2) has the form Z The crucial observation is the following: the operator (z1 \Gamma K) is a scaled version of the operator z which is equivalent to the scattering integral equation (2.1) by the same homogeneous object but with a different refractive index m 0 defined by z EIGENVALUES OF INTEGRAL OPERATOR IN SCATTERING 5 Thus, to find the points z in the spectrum for a scatterer with a given shape and refractive index m, it suffices to find all possible refractive indices m 0 for which the scattering problem is not well defined. After this, the points in the spectrum are recovered from In all our figures, we actually plot the points in the spectrum of the operator correspond to the eigenvalues of the matrix. 4.1. Scattering by a sphere. In this section we recall the analytic solution of scattering by a sphere. This solution will be used to find the points in the spectrum of the integral operator. The solution uses the spherical vector wave functions M lm and N lm [2, 5] that are defined by l ('; OE)); l ('; OE) are the spherical harmonics and f l (kr) stands for the spherical Bessel or Hankel function. The functions M lm (r) and N lm (r) are both solutions to the vector Helmholtz equation r \Theta r \Theta The scattering by a sphere can be computed analytically using the so-called Mie theory [2, 25]. The incoming electric field E inc , the electric field inside the object E 1 and the scattered electric field E s are all expanded in terms of the spherical vector wave functions: l m=\Gammal l M (1) l N (1) lm (kr) l m=\Gammal l M (1) l N (1) l m=\Gammal l M (3) l N (3) lm (kr) In these formulas k is the free-space wave number and k is the wave number inside the scatterer. The notation M (1) lm refers to the spherical vector wave functions that have a radial dependency given by the spherical Bessel function j n (kr) while for M (3) lm the radial dependency is given by the spherical Hankel function h (1) (kr). The magnetic field can easily be computed from these expansions. One then requires that the tangential components of the electric and magnetic field are continuous across the boundary of the sphere at radius kr = x. Now we can integrate the expansions and the vector spherical harmonics over the full solid angle and the orthogonality of the vector wave functions shows that the single mode M (1) lm (r) in the incoming field excites the corresponding modes M (1) lm (r) and M (3) lm (r) in the internal and scattered fields, respectively. This means that the coefficients of the internal and scattered fields are related to the coefficients ff m l and fi m l of the incoming field by i m l l , l , and l . At the radius kr = x, the boundary conditions and the expansions for the electric and magnetic fields give four linear equations for the coefficients a l , b l , c l , and d l . The Mie coefficients giving the field inside the object are [2] l l (x)[xj l (x)] 0 l (mx)[xh (1) l l (x)[mxj l (mx)] 0 l l (x)[xj l (x)] 0 l l (x)[mxj l (mx)] 0 6 J. RAHOLA 4.2. Resonances in the Mie series. To find points in the spectrum of the scattering integral operator, we now need to find those values of the refractive index m for which the Mie solution is not well-behaved. One such possibility is that the denominator in Equation (4.11) or (4.12) becomes zero. The case when a denominator of a Mie coefficient becomes zero or close to zero is called the Mie resonance. It means that a single spherical harmonic mode is greatly amplified and efficiently suppresses all the other modes. The resonances in Mie scattering have been studied quite extensively, see, e.g., [4, 8, 13, 26]. In these studies both the refractive index m of the scatterer and the size of the scatterer, which is measured by the size parameter kr = 2-r=-, can obtain complex values. The imaginary part of the refractive index determines how the object absorbs electromagnetic energy. A zero imaginary part means no absorption, a positive imaginary part is used for absorbing materials, while a negative imaginary part signifies a somewhat unphysical case when the objects are generating electromagnetic energy. If the size parameter is real, the denominators of Mie coefficients can become zero only if the refractive index has a negative imaginary part. However, in some cases the resonance appears with a refractive index that has a very small negative imaginary part. In this case the proximity of the resonance is visible in the scattering calculations also if we set the imaginary part of the resonant refractive index to zero. Likewise, for a object with a refractive index with zero or positive imaginary part, the resonance can appear only if the size parameter becomes complex. Sometimes the resonant size parameter has a very small imaginary part, making the resonance visible at the corresponding real size parameter. In this study we have a fixed the size parameter and try to find all the values for the complex refractive index that satisfy the resonance conditions exactly. We must stress that this is purely a mathematical trick to find the points in the spectrum of the integral equation and thus the refractive indices have no physical meaning here. To find the resonance points, we find the approximate locations of the zeros of denominators for Mie coefficients c l and d l for all orders l by visual inspection of the absolute value of the denominators when m varies in the complex plane. After this, we feed these values as initial guesses to the root-finding routine DZANLY in the IMSL library that uses Muller's method [20]. Figure 4.1 shows some of the resonant complex refractive indices m for a sphere of radius kr = 1. Note how the resonance positions gather along the positive real axis and at around \Gammai. We did not compute the resonant refractive indices with large positive real parts, because all these will be mapped to very small z-values in Equation (4.5). If the refractive index m 0 is resonant, so is \Gammam 0 , but these will be mapped to the same z-point. The preceding analysis indicates also that the coefficient matrix should become singular when we try to compute scattering by a sphere with a resonant refractive index. For a resonant refractive index, the mapping (4.5) indicates that 1 belongs to the spectrum of the integral operator K and thus there is a zero eigenvalue in the coefficient matrix. In addition to this, the matrix should become badly conditioned in the vicinity of such refractive indices and thus the convergence of iterative solvers should slow down. This has indeed been observed in another study [11]. 4.3. Branch cut of the square root. Now we turn our attention to the line segment in the eigenvalue plots of the matrix and ask in what other way besides a resonance can the Mie solution fail. To this end we will study the definition of the refractive index in the whole complex plane. Recall that the refractive index is defined by ~ffl, where ~ ffl is the complex permittivity. The square root y of a complex number z is defined as the solution of the equation y z. But this equation has two solutions. We will define the square root as being the solution that lies in the right half-plane, i.e., with non-negative real part. If the complex number z is given by re iOE , where \Gamma- ! OE -, then the square root can be uniquely defined as re iOE=2 . This is the main branch corresponding to positive square roots for positive real numbers. There is a discontinuity associated with this definition of the square root. When z approaches the negative real axis from above, z approaches the positive imaginary axis. However, when z approaches the EIGENVALUES OF INTEGRAL OPERATOR IN SCATTERING 7 -1.4 -1.2 -0.4 Fig. 4.1. Some of the complex refractive indices (m) giving rise to a resonance in the Mie series for a sphere of radius negative real axis from below, p z approaches the negative imaginary axis. Thus there is a discontinuity in p z as z crosses the negative imaginary axis. The negative imaginary axis is called the branch cut of the complex square root. We recall that in the Mie series solution, inside the scatterer the radial dependency of the solution is given by the spherical Bessel function where the wave number inside the object is given by mk. Thus in the scattering problem, when the complex permittivity ~ ffl sits on the negative real axis, the refractive index being purely imaginary, there is an ambiguity in the Mie solution due to the branch cut. Using mapping (4.5) these refractive indices imply that the corresponding points z belong to the spectrum of the integral operator and that these points form a line segment in the complex plane. The analysis of the branch cut of the refractive index and the mapping (4.4) are valid for any homogeneous scatterer, not just the sphere. Thus it is clear that the position and size of the line segment in the spectrum only depends on the refractive index, but not on the shape or size of the object. On the other hand, the part of the spectrum corresponding to the resonances does depend on the shape and size of the scatterer. 4.4. Numerical experiments. In Figure 4.2 we show the eigenvalues of the coefficient matrix for two different discretizations of a sphere with kr = 1. We also show the points in the spectrum of the found with the help of the resonances of the Mie series and the branch cut of the square root. If m 0 is a refractive index for which the scattering problem is not well defined, the corresponding points in the spectrum of operator are recovered from Most of the eigenvalues lying outside the line can be explained by the resonance points and when the discretization is refined, the agreement gets better. The endpoint of the line segment corresponding to the branch cut, i.e., scattering with purely imaginary refractive indices, is also shown. These refractive indices are of type iy, where y is real. When y approaches infinity, the points z in the spectrum approach 1. When y goes to zero, the points in the spectrum approach the point m 2 , which is marked in the plots with a large circle. In Figure 4.3 we show the same information as in Figure 4.2 but with computational cells) and cells), respectively. These figures show that when the size of the sphere is increased, 8 J. RAHOLA Fig. 4.2. The eigenvalues of the coefficient matrix (small black dots) and some points in the spectrum of the integral operator due to the resonances (small circles). The points in the spectrum of the integral operator that correspond to scattering by objects with purely imaginary refractive indices extend from the point one to the point marked with a large circle. The eigenvalues were computed from a discretization with 280 computational cells (upper) and 1000 computational cells (lower). The radius of the sphere is given by kr = 1. the number of eigenvalues off the line also increases. Figure 4.4 shows the eigenvalues for an anisotropic spherical scatterer with and the refractive indices in the direction of the coordinate axes being m Three line segments are visible in the eigenvalue plots. The segments start from 1, and they correspond to scattering with purely imaginary refractive indices in the mapping (4.13), where the refractive index m is replaced by any of the above indices. The fact that the scattering integral operator has a continuous spectrum implies that the operator cannot be compact. However, in a related situation, there is a theorem by Colton and Kress [3, Section 9] that states that the scattering integral operator is a compact operator if the refractive index is a smooth function of r. In the case of scattering by a homogeneous sphere, the refractive index is not smooth but instead has a discontinuity at the surface of the scatterer. Corresponding to the case analyzed by Colton and Kress EIGENVALUES OF INTEGRAL OPERATOR IN SCATTERING 9 -1.4 -1.2 -0.4 -0.20.2 Real(l) Real(l) kr=5, m=1.4+0.05*i Fig. 4.3. Same as Figure 4.2 but with size kr = 3 and with 480 computational cells (upper) and with size with 1064 computational cells (lower). we also computed the eigenvalues of the coefficient matrices of the integral equation when the refractive index varies smoothly. In this case, the line segment was still visible in the eigenvalue plots, but the eigenvalues no longer resided uniformly along the line but converged towards 1. This is the behavior one would expect of the discretization of the identity operator plus a compact operator. 5. Convergence estimates for iterative solvers. In this section we study the possibility of estimating the speed of convergence of iterative solvers from the knowledge of the eigenvalues of the coefficient matrix. In general, the convergence of iterative methods is dictated by the distribution of the eigenvalues, the conditioning of the eigenvalues and the right-hand side vector. We shall now give a basic convergence results for a Krylov-subspace method, i.e., iterative methods based on only the information given by successive matrix-vector products of the matrix A. We shall denote the linear system by b, the current iterate by xn and the residual by r and r 0 being the initial guess and initial residual, respectively. Iterative Krylov-subspace methods produce iterates xn such that the corresponding residual r n is J. RAHOLA Real(l) Fig. 4.4. The eigenvalues of an anisotropic sphere with refractive indices in the direction of the coordinate axes being 0:3i. The sphere is discretized with 1064 computational cells. For each of the three indices, the endpoint of the line segments corresponding to purely imaginary refractive indices in the mapping (4.13) are shown as large circles. given by r is a polynomial of degree n with pn 1. The task of iterative methods is to construct polynomials pn such that the norm of the residual decreases rapidly when n increases. Suppose that the coefficient matrix A is diagonalizable and thus has distinct eigenvectors v i , . Denote by V the matrix whose columns are v i and by the diagonal matrix of the corresponding eigenvalues - i . The matrix A can be decomposed as . Given this decomposition we can estimate the norm of the matrix polynomial pn (A) in terms of the polynomial evaluated at the eigenvalues: where the condition number of the eigenvalue matrix is given by -(V In our experiments we will use two iterative methods: the generalized minimal residual method (GMRES) [24] without restarts and the complex symmetric version of the quasi-minimal residual method (QMR) [7]. GMRES minimizes the residual of the current iterate among all the possible iterates in a Krylov subspace, a subspace generated by successive multiplications by the initial residual with the coefficient matrix. QMR has only an approximate minimization property and thus converges slower than GMRES, but its iterates are much cheaper to compute. QMR is the iterative solver used in our production code. For the iterative method GMRES, we have the following theorem [18]: kr nk pn (z) pn (0)=1 In other words, the iterative method GMRES finds a polynomial such that the maximum value of the polynomial at the eigenvalues is minimized. In the rest of this paper, we assume that the eigenvectors are well-conditioned and thus -(V ) is close to unity. The question of the optimal convergence speed in iterative methods has been studied by, e.g., Nevanlinna [19], who studied iteration of the problem g. We study the so-called linear phase of the convergence of iterations. The fastest possible linear convergence of the residual is given by kr k EIGENVALUES OF INTEGRAL OPERATOR IN SCATTERING 11 where j is the optimal reduction factor. Nevanlinna shows by potential-theoretic arguments that this is given by is the conformal map from the outside of the spectrum of K to the outside of the unit disk. Here we have assumed that the spectrum of K is a simply connected region in the complex plane. Now we will try to estimate the convergence speed of iterative solvers assuming the spectrum is a line segment in the complex plane, thus neglecting the few eigenvalues off the line. We are working with the spectrum of the scattering integral operator K defined in (4.2), not the operator that corresponds to the coefficient matrix. The line segment in the spectrum of K starts from zero, has a negative imaginary part, makes and angle of ff with the negative real axis, and has a length of d. From the analysis in the preceeding section it follows that The conformal map that maps the outside of the line segment to the outside of the unit disk consists of three parts: rotates the line segment to the negative imaginary axis (i.e., to the segment [\Gammad; 0]), OE 2 translates this segment to the segment [\Gamma1; 1] and finally OE 3 maps the outside of this segment to the outside of the unit disk. These maps are explicitly given by d Thus the optimal reduction factor is given by where d Figure 5.1 shows the convergence behavior or QMR and GMRES together with the estimated optimal convergence speed for a sphere with refractive index 0:05i and with size of kr image) and (lower image). It can be seen that the convergence estimate and the actual behavior of iterative methods are very close in the case kr = 1. Also, the convergence of QMR is very close to GMRES. In the case of kr = 3 we notice that the observed convergence of iterative methods is much slower than the predicted rate. This is due to the increased number of eigenvalues that lie off the line, as can be seen from Figure 4.3. The convergence could be better estimated by drawing a polygon around the spectrum and using the Schwarz-Cristoffel theorem [6] to compute the values of a conformal map from this polygon to the outside of the unit disk, but this approach has not been pursued further. 6. Conclusions. We have studied the eigenvalues of the coefficient matrix arising from a discretization of the volume integral equation of electromagnetic scattering. The eigenvalues of a coefficient matrix for a spherical scatterer consist of a line segment plus some isolated points. We have studied how these eigenvalues and the spectrum of the scattering integral operator are related. To find the spectrum of the integral operator, we show that it is sufficient to find all the refractive indices for which the scattering problem is not well defined. These indices are then mapped to the points in the spectrum of the operator. We have shown that the isolated eigenvalues of the coefficient matrix correspond to exact resonances in the analytical solution of scattering by a sphere. The line segment in the eigenvalue plots corresponds to scattering by purely imaginary refractive indices, which is related to the branch cut of the square root in the definition of the complex refractive index and the wave number inside the scatterer. The knowledge of the eigenvalues of the integral operator can help us to understand the convergence of iterative solution methods for the discretized scattering problem. For example, we have observed that J. RAHOLA Iteration number Residual norm Iteration number Residual norm Fig. 5.1. Convergence given by the optimal reduction factor (solid line), convergence of GMRES (dashed line) and of QMR (dash-dotted line) for a sphere with and with refractive index when the same object is discretized with increasing resolution, the number of unpreconditioned iterations is practically constant. The convergence of iterative solvers depends on the eigenvalue distribution of the coefficient matrix. Successively finer discretizations of the scattering integral equation produce coefficient matrices which have approximately the same eigenvalue distribution and thus the same convergence properties. In contrast, when partial differential equations are discretized with increasingly finer meshes, the convergence of unpreconditioned iterative solvers typically gets worse. This situation can arise if zero belongs to the spectrum of the partial differential operator. When such a problem is discretized with finer and finer meshes, some of the eigenvalues of the coefficient matrices move closer and closer to zero, giving rise to poor convergence of iterative solvers. This problem can sometimes be remedied with the help of preconditioners. Finally, we have tried to predict the convergence speed of iterative solvers based on the knowledge of the eigenvalues. In doing so, we have only used the location and length of the line segment in the eigenvalue plots, thus neglecting the isolated eigenvalues that lie outside the line. This strategy gives EIGENVALUES OF INTEGRAL OPERATOR IN SCATTERING 13 convergence estimates that are very close to the observed convergence for small scatterers. Once the size of the sphere is increased, the number of eigenvalues lying off the line is increased and thus the convergence estimate quickly becomes useless. The analysis presented in this paper could be applied to other scattering geometries, such as spheroids, for which analytical solutions exist. It would also be interesting to compute the eigenvalues arising from more refined discretization schemes of the volume integral operator. The convergence analysis of iterative solvers presented here could be augmented to account for the eigenvalues lying off the line segment in the complex plane. However, for a given physical problem it is quite tedious to compute all the resonance locations and thus this approach will probably not give a practical convergence analysis tool. Acknowledgment . I would like to thank Francis Collino for fruitful discussions on the subject. --R Absorption and Scattering of Light by Small Particles Inverse Acoustic and Electromagnetic Scattering Theory Resonant spectra of dielectric spheres Translational addition theorems for spherical vector wave functions Algorithm 756: A Matlab toolbox for Schwarz-Christoffel mapping Conjugate gradient-type methods for linear systems with complex symmetric coefficient matrices Electromagnetic resonances of free dielectric spheres Scattering by irregular inhomogeneous particles via the digitized Green's function algorithm Application of fast-Fourier-transform techniques to the discrete- dipole approximation Accuracy of internal fields in VIEF simulations of light scattering Mathematical foundations for error estimation in numerical solutions of integral equations in electromagnetics Strong and weak forms of the method of moments and the coupled dipole method for scattering of time-harmonic electromagnetic fields On two numerical techniques for light scattering by dielectric agglomerated structures Light scattering by porous dust particles in the discrete-dipole approximation How fast are nonsymmetric matrix iterations? Convergence of Iterations for Linear Equations Numerical Recipes in Fortran - The Art of Scientific Computing Solution of dense systems of linear equations in electromagnetic scattering calculations. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems Resonances and poles of weakly absorbing spheres --TR --CTR Matthys M. Botha, Solving the volume integral equations of electromagnetic scattering, Journal of Computational Physics, v.218 n.1, p.141-158, 10 October 2006

electromagnetic scattering;spectrum of linear operators;eigenvalues of matrices;iterative methods;integral equations

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A Priori Sparsity Patterns for Parallel Sparse Approximate Inverse Preconditioners.

Parallel algorithms for computing sparse approximations to the inverse of a sparse matrix either use a prescribed sparsity pattern for the approximate inverse or attempt to generate a good pattern as part of the algorithm. This paper demonstrates that, for PDE problems, the patterns of powers of sparsified matrices (PSMs) can be used a priori as effective approximate inverse patterns, and that the additional effort of adaptive sparsity pattern calculations may not be required. PSM patterns are related to various other approximate inverse sparsity patterns through matrix graph theory and heuristics concerning the PDE's Green's function. A parallel implementation shows that PSM-patterned approximate inverses are significantly faster to construct than approximate inverses constructed adaptively, while often giving preconditioners of comparable quality.

Introduction . A sparse approximate inverse approximates the inverse of a (usually sparse) matrix A by a sparse matrix M . This can be accomplished, for example in the least-squares method, by minimizing the matrix residual norm 1 F with the constraint that M is sparse. In general, the degrees of freedom of this problem are the nonzero values in M as well as their locations. A minimization that considers all these variables simultaneously, however, is very complex, and thus a simple approach is to prescribe the set of nonzeros, or the sparsity pattern S, of before performing the minimization. The objective function (1.1) can then be decoupled as the sum of squares of the 2-norms of the n individual columns, i.e., in which e j and m j are the jth columns of the identity matrix and of the matrix M , respectively. Each least-squares matrix is small, having a number of columns equal to the number of nonzeros in its corresponding m j . If A is nonsingular, then the least-squares matrices have full rank. Thus the approximate inverse can be constructed by solving n least-squares problems in parallel. However, sparse approximate inverses are attractive for parallel pre-conditioning primarily because the preconditioning operation is a sparse matrix by vector product. The cost of constructing the approximate inverse for a large matrix is usually so high, especially with the adaptive pattern selection strategies described below, that they are only competitive if they are constructed in parallel. For diagonally dominant A, the entries in A 1 decay rapidly away from the diagonal [17], and a banded pattern for M will produce a good approximate inverse. This work was performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract W-7405-ENG-48. y Center for Applied Scientic Computing, Lawrence Livermore National Laboratory, L-560, Box 808, Livermore, CA 94551 (echow@llnl.gov). 1 The form for the right approximate inverse is used here, which is notationally slightly clearer. If the matrix is distributed over parallel processors by rows, a left approximate inverse can be computed row-wise. In a general setting, without application-specic information, it is not clear how best to choose a sparsity pattern for M . Algorithms have been developed that rst compute an approximate inverse with an initial pattern S; then S is updated and a new minimization problem is solved either exactly or inexactly. This process is repeated until a threshold on the residual norm has been satised, or a maximum number of nonzeros has been reached [12, 15, 22]. We refer to these as adaptive procedures. One such procedure is to use an iterative method, such as minimal residual, starting with a sparse initial guess [12] to approximately minimize (1.2), i.e., nd sparse approximate solutions to Suppose a sparse initial guess for m j is used. The rst few iterates will be sparse. To maintain sparsity, a strategy to drop small elements is usually used, either in the search direction or the iterates. No prescribed pattern S is necessary since the sparsity pattern emerges automatically. For this method to be e-cient, sparse-sparse operations must be used: the product of a sparse matrix by a sparse vector with p nonzeros only involves p columns of the sparse matrix. Another adaptive procedure, called SPAI [15, 22], uses a numerical test to determine which nonzero locations should be added to the current sparsity pattern. For the jth column, the numerical test for adding a nonzero in location k has the form tolerance is the residual for a given sparsity pattern of column j, and m j is the current approximation. The test (1.4) is a lower bound on the improvement in the square of the residual norm when the pattern of m j is augmented. Entry k is added if the tolerance is satised, or if the left-hand side of (1.4) is large compared to its values for other k. The cost of performing this test is a sparse dot product between r and column k of A for each location to test. Interprocessor communication is needed to test a k corresponding to a column not on a local processor. These adaptive algorithms utilize the additional degrees of freedom in minimizing aorded by the locations of the nonzeros in M and have allowed much more general problems to be solved than before. Adaptive methods, however, tend to be very expensive. Thus, this paper focuses on the problem of selecting S in a preprocessing step so that a sparse approximate inverse can be computed immediately by minimizing (1.1). Section 2 rst examines the sparsity patterns that are produced by both non-adaptive and adaptive schemes. Section 3 tests the idea of using the patterns of powers of sparsied matrices (PSM's) as a priori sparsity patterns for sparse approximate inverses. Numerical tests are presented in Section 4, with comparisons to both sequential and parallel versions of some current methods. Finally, Section 5 draws some conclusions. 2. Graph interpretations of approximate inverse sparsity patterns. 2.1. Use of the pattern of A and variants. The structure of a sparse matrix A of order n is the directed graph G(A), whose vertices are the integers whose edges (i ! j) correspond to nonzero o-diagonal entries in A. (This notation usually implies matrices with all nonzero diagonal entries.) A subset of G(A) is a directed graph with the same vertices, but with a subset of the edges in G(A). The A PRIORI PATTERNS FOR SPARSE APPROXIMATE INVERSES 3 graph G(A) is a representation of the sparsity pattern of A, and when it is clear from the context, we will not distinguish between them. The structure of a vector x of order n is the subset of ng that corresponds to the nonzero entries in x. When there is an associated matrix of order n, we often refer to the structure of x as a subset of the vertices of the associated matrix. Notice then, that the structure of column j of a matrix A is the set of vertices in G(A) that have edges pointing to vertex j plus vertex j itself. The structure of row j is vertex plus the set of vertices pointed to by vertex j. The inverse of a matrix shows how each unknown in a linear system depends on the other unknowns. The structure of the matrix A shows only the immediate dependencies. This suggests that in the structure of A 1 there is an edge (i ! whenever there is a directed path from vertex i to vertex j in G(A) [21] (if A is nonsingular, and ignoring coincidental cancellation). This structure is called the transitive closure of G(A), and is denoted G (A). For an irreducible matrix, this result says that the inverse is a full matrix, but it does suggest the possibility of truncating the transitive closure process to approximate the inverse by a sparse matrix. A heuristic that is often employed is that vertices closer to vertex j along directed paths are more important, and should be retained in an approximate inverse sparsity pattern. This idea is supported by the decay in the elements observed by Tang [30] in the discrete Green's function for many problems. These sparsity patterns were rst used by Benson and Frederickson [4] in the symmetric case, who also dened matrices with these patterns to be q-local matrices. Given a graph G(A) of a structurally symmetric matrix A with a full diagonal, the structure of the jth column of a q-local matrix consists of vertex j and its qth level nearest-neighbors in G(A). A 0-local matrix is a diagonal matrix, while a 1-local matrix has the same sparsity pattern as A. The sparsity pattern of A is the most common a priori pattern used to approximate A 1 . It gives good results for many problems, but can usually be improved, or fails for many other problems. One improvement is to use higher levels of q. Unfortu- nately, the storage for these preconditioners grows very quickly when q is increased, and even impractical in many cases [20]. Huckle [24] proposed similar patterns which may be more eective when A is nonsymmetric. These include the patterns corresponding to the graphs G((A T The density of the former, in particular, grows very quickly with increasing k. Primarily, these patterns are useful as envelope patterns from which the adaptive SPAI algorithm can select its pattern. This gives an upper bound on the interprocessor communication required by a parallel implementation [23]. Cosgrove and Daz [14] proposed augmenting the pattern of A without going to the full 2-local matrix. They suggested adding nonzeros to m j in a way that minimizes the number of new rows introduced into the jth least-squares matrix (in expression (1.2)). The augmented structure is determined only from the structure of A. Kolotilina and Yeremin [28] proposed similar heuristics for augmenting the sparsity pattern for factorized sparse approximate inverses. Sparsication. Instead of augmenting the pattern of A, it is also possible to diminish the pattern of A when A is relatively full. This can be accomplished by spar- sication (dropping small elements in the matrix A) and using the resulting pattern. This was introduced by Kolotilina [26] for computing sparse approximate inverses for dense matrices (see also [10, 32]), and Kaporin [25] for sparse matrices and factorized approximate inverses. Sparsication can be combined with the use of higher level neighbors. Tang [30] showed that sparsifying a matrix prior to applying the adaptive SPAI algorithm is eective for anisotropic problems. The observation is that the storage and therefore operation count required for preconditioners produced this way are much smaller. This technique can generate patterns that are those of powers of sparsied matrices. The idea of explicitly combining sparsication with the use of higher level neighbors was used by Alleon et al. [1], who attributes the technique to Cosnuau [16]. For approximating the inverse of dense matrices in electromagnetics, however, their tests showed that higher levels were not warranted. Tang and Wan [31] also used a sparsi- cation before applying a q-local matrix pattern, for q > 1, for approximate inverses used as multigrid smoothers. They showed that the sparsication does not cause a deterioration in convergence rate for their problems. Both the work by Alleon et al. and Tang and Wan represent the rst uses of PSM patterns. Instead of applying the sparsication to A, it is also appropriate in some cases to apply the sparsication to the sparse approximate inverse after it has been computed [27, 31]. This is useful to reduce the cost of using the approximate inverse when it is relatively full. 2.2. Insights from adaptive schemes. Adaptive schemes can generate patterns that are very dierent from the pattern of A, for example, the generated patterns can be much sparser than A. Nevertheless, the patterns produced by adaptive schemes can be interpreted using the graph of A. Consider rst the approximate minimization method described in Section 1. The following algorithm nds a sparse approximate solution to minimal residual iterations. A dropping strategy for elements in the search direction r is encapsulated by the function \drop" in step 4, which may depend on the current pattern of m. Algorithm 2.1. Sparse approximate solution to 1. m := sparse initial guess 2. r := e j Am 3. Loop until krk < tol or reached max. iterations 4. d := drop(r) 5. q := Ad 6. := (r;q) 7. m := m+ d 8. r := r q 9. EndLoop The elements in r not already in m are candidates for new elements in m. The vector r is generated essentially by the product Am and thus the structure of r is the set of vertices that have edges pointing to vertices in the structure of m. If the initial guess for m consists of a single nonzero element at location j, then the structure of grows outward from vertex j in G(A) with each iteration of the above algorithm. If the search direction in the iterative method is A T r instead of r, then the kth entry in the search direction is r T Ae k . A dropping strategy based on the size of the these entries is similar to one based on the test (1.4) which attempts to minimize the updated residual norm. In this method, the candidates for new elements are the rst and second level neighbors of the vertices in m (for nonsymmetric A, the directions of the edges are important). A PRIORI PATTERNS FOR SPARSE APPROXIMATE INVERSES 5 This is exactly the graph interpretation for the SPAI algorithm (see Huckle [24]). When computing column j of M , vertices far from vertex j will not enter into the pattern, at least initially. Algebraically, this means that the nonzero locations of r and Ae k do not intersect, and the value of the test is zero. An e-cient implementation of SPAI uses these graph ideas to narrow down the indices k that need to be checked. An early parallel implementation of SPAI [18] tested only the rst level neighbors of a vertex, rather than both the rst and second levels. This is a good approximation in many cases. This implementation also assumed A is structurally symmetric, so that one-sided interprocessor communication is not necessary. A more recent parallel implementation of SPAI [2, 3] implements the algorithm exactly. This code implements one-sided communication with the Message Passing Interface (MPI), and uses dynamic load balancing in case some processors nish computing their rows earlier than others. 3. Patterns of powers of sparsied matrices. 3.1. Graph interpretation. In Section 2, we observed that prescribed sparsity patterns or patterns generated by the adaptive methods are generally subsets of the pattern of low powers of A (given that A has a full diagonal) and typically increase in accuracy with higher powers. Clearly all these methods are related to the Neumann series or characteristic polynomial for A [24]. The structure of column j of the approximate inverse of A is a subset of the vertices in the level sets (with directed edges) about vertex j in G(A). Good vertices to choose are those in the level sets in the neighborhood of vertex j, but the algorithms dier in how these vertices are selected. For convection-dominated and anisotropic problems, upstream vertices or vertices in the preferred directions will have a greater in uence than others on column j of the inverse. Figure 3.1 shows the discrete Green's function for a point on a PDE with convection. The nonsymmetry of the function shows that upstream nonzeros in a row or column of the exact inverse are greater in magnitude than others. Without additional physical information such as the direction of ow, however, it is often possible to use sparsication to identify the preferred or upstream directions.5152501020300.10.30.50.7 Fig. 3.1. Green's function for a point on a PDE with convection. We examined the sparsity patterns produced by the adaptive algorithms and tried to determine if they could have been generated by simpler graph algorithms. For some simple examples, it turned out that the structures produced are exactly or very close to the transitive closures of a subset of G(A), i.e., of the structure of a sparsied 6 EDMOND CHOW matrix. In Figure 3.2 we show the structures of several matrices: (a) ORSIRR2 from the Harwell-Boeing collection, (b) A 0 , a sparsication of the original matrix, (c) the transitive closure G (d) the structure produced by the SPAI algorithm. This latter gure was selected from [2] which shows it as an example of an eective sparse approximate inverse pattern for this problem. (There are, however, some bothersome features of this example: the approximate inverse is four independent diagonal blocks.) Note that we can approximate the adaptively generated pattern (d) very well by the pattern (c) generated using the transitive closure. (a) (b) Sparsied ORSIRR2 (c) Transitive closure of (b) (d) Pattern from SPAI Fig. 3.2. The adaptively generated pattern (d) can be approximated by the transitive closure (c). 3.2. How to sparsify. The simplest method to sparsify a matrix is to retain only those entries in a matrix greater than a global threshold, thresh. In the example of Figure was used. It was important, however, to make sure that the diagonal elements were retained, otherwise a structurally singular matrix would have resulted in this case. In general, the diagonal should always be retained. A PRIORI PATTERNS FOR SPARSE APPROXIMATE INVERSES 7 One strategy for choosing a threshold is to choose one that retains, for example, one-third of the original nonzeros in a matrix. Fewer nonzeros should be retained if powers of this sparsied matrix have numbers of nonzeros that grow too quickly. This is how thresholds were chosen for the small problems tested in Section 4. The number of levels used may be increased until a preconditioner reaches a target number of nonzeros. The best choices for these parameters will be problem-dependent. For special problems, this strategy may not be eective, for example, when a matrix contains only a few unique values. When a matrix is to be sparsied using a global threshold, how the matrix is scaled becomes important. It is often the case that a matrix contains many dierent types of equations and variables that are not scaled the same way. For example, consider the matrix@ which has its rst row and column scaled by a large number, Z. If a threshold Z is chosen, and if the diagonal of the matrix is retained, the third row of the sparsied matrix has become independent of the other rows. We thus apply the thresholding to a matrix that has been symmetrically scaled so that it has all ones on its diagonal, e.g., for the above matrix, the scaled matrix is@ A threshold less than or equal to 1 will guarantee that the diagonal is retained. The scaling also makes it easier to choose a threshold. This method of scaling is not foolproof, but does avoid some simple problems. In the graph of the sparsied matrix A 0 , each vertex should have some connections to other vertices. This can be accomplished by sparsifying the matrix A such that one retains at least a xed number of edges to or from each vertex, for example, the ones corresponding to the largest matrix values. Given a parameter ll, this can be implemented for column j by selecting the diagonal (jth) element plus the ll 1 largest o-diagonal elements in column j of the original matrix. This guarantees that there are ll 1 vertices with edges into vertex j. Applied row-wise, this guarantees ll 1 vertices with edges emanating from vertex j. Again, we choose explicitly to keep the diagonal of the matrix; thus each column (or row) has at least ll nonzeros. Choosing ll may be simpler and more meaningful than choosing a threshold on the matrix values. Dierent values of ll may be used for dierent vertices, depending on the vertex's initial degree (number of incident edges). denote the structure of a matrix A 0 that has been sparsied from A. denote the structure (with the same set of vertices) that has an edge whenever there is a path of distance i or less in S 0 . The structure S i 1 is a subset of S i . In matrix form, S ignoring coincidental cancellation. These are called level set expansions of a sparsied matrix or patterns of powers of a sparsied matrix (PSM patterns). Heuristic 3 tested by Alleon et al. [1] is equivalent to S i using a variable ll at each level to perform sparsication. We mention that it is also possible to perform sparsication on S i after every level set expansion. We denote this variant by S i . For this variant, values need to be computed, and we propose the following, which stresses the larger elements in A. If \drop" denotes a sparsication process, then we can dene A and S note that S 1 is not generally a subset of . More complicated strategies are possible; the thresholds can be dierent for each level i. Note that determining S i is much more di-cult than determining S i since values need to be computed. 3.3. Factorized forms of the approximate inverse. Sparse approximate inverses for the Cholesky or LU factors of A are often used. The analogue of the least-squares method (minimization of (1.1)) here is the factorized sparse approximate inverse (FSAI) technique of Kolotilina and Yeremin [28], implemented in parallel by Field [19]. If the normal equations method is used to solve the least-squares systems, the Cholesky or LU factors are not required to compute the approximate inverse. This means, however, that the adaptive pattern selection schemes cannot be used, since the matrix whose inverse is being approximated is not available. A priori sparsity patterns must be used instead. Given sparse approximate inverse approximates U 1 and sparse matrices G and H , respectively, so that The patterns for G and H should be chosen such that the pattern of GH is close in some sense to good patterns for approximating A 1 . Supposing that S is a good pattern for A 1 , then the upper and lower triangular parts of S are good patterns for G and H , since the pattern of GH includes the pattern S. These patterns will be tested in Section 4. It may also be possible to use the patterns of the powers of the exact or approximate L and U if they are known. These L and U factors are not discretizations of PDE's, but their inverses are often banded with elements decaying rapidly away from the main diagonal. This technique may be appropriate if approximate L and U are available, for example from a very sparse incomplete LU factorization. As opposed to the inverses of irreducible matrices, the inverses of Cholesky or LU factors are often sparse. An ordering should thus be applied to A that gives factors whose inverses can be well approximated by sparse matrices. Experimentally, fewer nonzeros in the exact inverse factors translates into lower construction cost and better performance for factorized approximate inverses computed by an incomplete biconjugation process [7, 8]. The transitive closure can be used to compute the number of nonzeros in the exact inverse of a Cholesky factor, based on the height of all the nodes in the elimination tree. This has lead to reordering strategies that approximately minimize the height of the elimination tree and thus the number of nonzeros in the inverse factors, and allows some prediction of how well these approximate inverses might perform on a given problem [7, 8]. 3.4. Approximate inverse of a Schur complement. To determine a good pattern for a Schur complement matrix, we notice that is the Schur complement. Thus a good sparsity pattern for S 1 can be determined from a good sparsity pattern for A 1 ; it is simply the (2,2) A PRIORI PATTERNS FOR SPARSE APPROXIMATE INVERSES 9 block of a good sparsity pattern for A 1 . B should be of small order compared to the global matrix or else the method will be overly costly. In a code, it may be possible to compute the approximate inverse of A and extract the approximation to S 1 , or compute a partial approximate inverse, i.e., those rows or columns of the approximate inverse that correspond to S 1 [11]. Again, S should be of almost the same order as A. 3.5. Parallel computation. Computation of S i is equivalent to structural sparse matrix-matrix products of sparsied matrices. The computation can also be viewed as n level set expansions, one for each row or column, which can be performed in parallel. For vertices that are near other vertices on a dierent processor, some communication will be necessary. Communication can be reduced by partitioning the graph of the sparsied matrix among the processors such that the number of edge-cuts is reduced. Unfortunately, in general, one-sided communication is required to compute sparse approximate inverses. Processors need to request rows from other processors, and a processor cannot predict which rows it will need to send. One-sided communication may be implemented in MPI by having each processor occasionally probe for messages from other processors. The latency between probes is a critical performance factor here. In a multithreaded environment, it is possible to dedicate some threads on a local processing node to servicing requests for rows (server threads), while the remaining threads compute each row and make requests for rows when necessary (worker threads). Consider a matrix A and an approximate inverse M to be computed that are partitioned the same way by rows across several processors. Algorithm 3.1 describes one organization of the parallel computation. Each processor computes a level set expansion for all of its rows before before continuing on to the next level. At each level, the requests and replies to and from a processor are coalesced, allowing fewer and larger messages to be used. Like in [3], external rows of A are cached on a processor in case they are needed to compute other rows. There is no communication during the numerical phase when the values of M are being computed. This algorithm was implemented using occasional probing for one-sided communication. Algorithm 3.1. Parallel level set expansions for computing S i Communicate rows 1. Initialize the set of vertices V to empty 2. Sparsify all the rows on the local processor 3. Merge the structures of all the locally sparsied rows into V 4. For level 5. For nonlocal k 2 V, request and receive row k 6. Sparsify received rows 7. Merge structures of new sparsied rows into V 8. EndFor 9. For nonlocal k 2 V, request and receive row k Compute structure of each row 10. For each local row j 11. Initialize V j to a single entry in location j 12. For level 13. For new sparsied structure of row k into V j 14. EndFor 15. EndFor Compute values of each row 16. For each local row j, nd has the pattern V j We also implemented a second parallel code, which has the following features: multithreaded, to take advantage of multiple processors per shared-memory node on symmetric multiprocessor computers uses server and worker threads to more easily implement one-sided commu- nication uses a simpler algorithm than Algorithm 3.1: computes each row and performs all the associated communications before continuing on to the next row; when multiple threads are used, this avoids worker threads needing to synchronize and coordinate which rows to request from other nodes; smaller messages are used, but communication is also spread over the entire execution time of the algorithm scalable with its use of memory, but is thus also slower than the rst version which used direct-address tables (traded memory for faster computation) Timings for this second code will be reported in the next section. Some limited timings for the rst code will also be shown. We are also working on a factorized implementation for symmetric matrices, which will guarantee that the preconditioner is also symmetric. This implementation makes a simple change in step 16 of Algorithm 3.1, and does not require one-sided communication when the full matrix is stored. 4. Numerical tests. 4.1. Preconditioning quality. First we test the quality of sparsity patterns generated by powers of sparsied matrices on small problems from the Harwell-Boeing collection. In particular, we chose problems that were tested with SPAI [22] in order to make comparisons. We performed tests in exactly the same conditions: we solve the same linear systems using GMRES(20) to a relative residual tolerance of 10 8 with a zero initial guess. We report the number of GMRES steps needed for convergence, or indicate no convergence using the symbol y. Right preconditioning was used. In Tables 4.1 to 4.5, we show test results for S i for both unfactored (column 2) and factored (column 3) forms of the approximate inverse. We compare the results to the least-squares (LS) method using the pattern of the original matrix A, and FSAI, the least-squares method for the (nonsymmetric) factored form [28], again using the pattern of the original matrix A. For the unfactored form, we also display the result of the SPAI method reported in [22], using their choice of parameters. Adaptive methods for factored forms are also available [5, 6, 29] but were not tested here. Global thresholds (shown for each table) on a scaled matrix were used to perform these sparsications. In the tables, we also show the number of nonzeros nnz in the unfactored preconditioners (the entry for LS/FSAI is the number of nonzeros in A). The results show that preconditioners of almost the same quality as the adaptive SPAI can be achieved using the S i patterns. In some cases, even better preconditioners can result, sometimes with even less storage (Table 4.1). The results also show that using the pattern of A does not generally give as good a preconditioner for these problems. SHERMAN2 is a relatively hard problem for sparse approximate inverses. The result reported in [22] shows that SPAI could reduce the residual norm by 10 5 (the preconditioner with 26327 nonzeros in 7 steps. The results are similar with S i patterns, but the full residual norm reduction can be achieved with an approximate inverse that is denser. Note that in this case, the sparsication A PRIORI PATTERNS FOR SPARSE APPROXIMATE INVERSES 11 Table Iteration counts for ORSIRR2, Pattern unfactored factored nnz LS/FSAI 335 383 5970 SPAI 84 5318 Table Iteration counts for SHERMAN1, Pattern unfactored factored nnz LS/FSAI 145 456 3750 SPAI threshold was applied to the original matrix rather than to the diagonally scaled matrix, although the matrix has values over 27 orders of magnitude; the diagonal scaling is not foolproof. Factorized approximate inverses were not eective for this problem with these patterns. SAYLR4 is a relatively hard problem for GMRES. Grote and Huckle [22] report that SPAI could not solve the problem with GMRES, but could with BiCGSTAB. This is also true for S i patterns; the results in Table 4.5 are with BiCGSTAB. There are, of course, many problems that are di-cult for PSM-patterned approximate inverses. These include NNC666 and GRE1107 from the Harwell-Boeing collection, and FIDAP problems from Navier-Stokes simulations. These problems, however, can be solved using adaptive methods [12]. These problems pose di-culties for PSM patterns because the Green's function heuristic is invalid; the problems either are not PDE problems, or have been modied (e.g., the FIDAP problems used a penalty formulation). In Tables 4.6 and 4.7, we show test results for S i for the unfactored form of the approximate inverse. S 0 and the LS patterns were the same for these problems. Since is not generally a superset of S there is no guarantee that S is a better pattern than S in terms of the norm of R = I AM . To show this, we also display these matrix residual norms. The parameter ll (shown for each table) was used to sparsify these matrices after each level set expansion. Again, the results show that the a priori methods can approach the quality of the adaptive methods very closely. 4.2. Parallel timing results. The results above show that the preconditioning quality for PDE problems is not signicantly degraded by using the non-adaptive schemes based on powers of sparsied matrices. In this section we illustrate the main advantage of these preconditioners: their very low construction costs compared to the adaptive schemes. In this and Section 4.3 we show results using the multithreaded version of our code, ParaSAILS (parallel sparse approximate inverse, least squares). Like the parallel version of SPAI [3] with which we make comparisons, ParaSAILS is implemented Table Iteration counts for SHERMAN2, Pattern unfactored factored nnz LS/FSAI y y 23094 SPAI y 26327 Table Iteration counts for PORES3, Pattern unfactored factored nnz LS/FSAI y y 3474 SPAI 599 16745 as a preconditioner object in the ISIS++ solver library [13]. Both these codes generate a sparse approximate inverse partitioned across processors by rows; thus left preconditioning is used. In all the codes, the least-squares problems that arose were solved using LAPACK routines for QR decomposition. For problems with relatively full approximate inverses, solving these least-squares problems takes the majority of the computing time. Tests were run on multiple nodes of an IBM RS/6000 SP supercomputer at the Lawrence Livermore National Laboratory. Each node contains four 332 MHz PowerPC CPU's. Timings were performed using user-space mode, which is much more e-cient than internet-protocol mode. However, nonthreaded codes can only use one processor per node in user-space mode. We tested SPAI with one processor per node, and ParaSAILS with up to four processors per node. The iterative solver and matrix-vector product codes were also nonthreaded, and used only one processor per node. The rst problem we tested is a nite element model of three concentric spherical shells with dierent material properties. The matrix has order 16881 and has nonzeros. The SPAI algorithm using the default parameters (target residual norm for each row < 0:4) produced a much sparser precondi- tioner, with 171996 nonzeros, and solved the problem using GMRES(50) to a tolerance steps. For comparison purposes, we chose parameters for ParaSAILS that gave a similar number of nonzeros in the preconditioner. In particular, S 3 with an ll parameter of 3 gave a preconditioner with 179550 nonzeros, and solved the problem in 331 steps. Figure 4.1 shows the two resulting sparsity patterns. Table 4.8 reports, for various numbers of nodes (npes), the wall-clock times for the preconditioner setup phase (Precon), the iterative solve phase (Solve), and the total time (Total). The time for constructing the preconditioner in each code includes the time for determining the sparsity pattern. Due to the relatively small size of this (and the next) problem, only one worker and one server thread was used per node (i.e., two processors per node) in the ParaSAILS runs; one processor was used in the SPAI runs. For comparison, in Table 4.9 we show the results using the rst (nonthreaded, occasional MPI probes for one-sided communication) version of the code. This code A PRIORI PATTERNS FOR SPARSE APPROXIMATE INVERSES 13 Table Iteration counts for SAYLR4, Pattern unfactored factored nnz LS/FSAI y y 22316 SPAI 67 84800 Table Iteration counts for SHERMAN3, steps nnz kRkF LS y 20033 17.3620 SPAI 264 48480 is much faster because it uses direct-address arrays to quickly merge the sparsity patterns of rows; direct-address arrays have length the global size of the matrix and are not scalable. (a) ParaSAILS (b) SPAI Fig. 4.1. Structure of the sparse approximate inverses for the concentric shells problem. We tested a second, larger problem which models the work-hardening of metal by squeezing it to make it pancake-like In this particular example, the pattern of a sparsied matrix (S 0 ) with an ll parameter of 3 lead to a good preconditioner. ParaSAILS produced a preconditioner with 141848 nonzeros and solved the problem in 142 steps; SPAI produced a preconditioner with 120192 nonzeros 14 EDMOND CHOW Table Iteration counts for SHERMAN4, steps nnz kRkF LS 199 3786 6.2503 SPAI 86 9276 Table Timings for concentric shells problem. ParaSAILS SPAI npes Precon Solve Total Precon Solve Total and solved the problem in 139 steps. Results for varying numbers of processing nodes are shown in Table 4.10. The results show that the non-adaptive ParaSAILS algorithm implemented here is many times faster than the adaptive SPAI algorithm. 4.3. Implementation scalability. In this section, we experimentally investigate the implementation scalability of constructing sparse approximate inverses with ParaSAILS. Let T (n; p) be the time to construct an approximate inverse of order n on a parallel computer using p processors. We dene the scaled e-ciency to be E(n; p) T (n; 1)=T (pn; p): If E(n; then the implementation is perfectly scalable, i.e., one could double the size of the problem and the number of processors without increasing the execution time. However, as long as E(n; p) is bounded away from zero for a xed n as p is increased, we say that the implementation is scalable. We consider the 3-D constant coe-cient PDE in @ discretized using standard nite dierences on a uniform mesh, with the anisotropic parameters This problem has been used to test the scalability of multigrid solvers [9]. The problems are a constant size per compute node, from 10 3 to local problem sizes. Node topologies of 1 3 to 5 3 were used. Thus the largest problem was over a cube with (60 unknowns. Each node used all four processors (4 worker threads, 1 server thread) for this problem in the preconditioner construction phase (i.e., 500 processors in the largest conguration). A threshold for ParaSAILS was chosen so that only the nonzeros along the strongest (z) direction are retained, and the S 3 pattern was used. Although this is a symmetric problem, the preconditioner is not symmetric, and we used GMRES(50) as the iterative solver with a zero initial guess. The convergence tolerance was 10 6 . A PRIORI PATTERNS FOR SPARSE APPROXIMATE INVERSES 15 Table Concentric shells problem: Timings for preconditioner setup using direct addressing. npes Precon Table Timings for work-hardening problem. ParaSAILS SPAI npes Precon Solve Total Precon Solve Total Table 4.11 shows the results, including wall-clock times for constructing the preconditioner and the iterative solve phase, the number of iterations required for convergence, the average time for one iteration in the solve phase, and E p and E s , the scaled e-cien- cies for constructing the preconditioner and for one step in the solve phase. Figures 4.2 and 4.3 graph the scaled e-ciencies E p and E s , respectively. The implementation seems scalable for all values of p that may be encountered. For comparison, in Table 4.12, we show results for SPAI using a 40 3 local problem size. Larger problems led to excessive preconditioner construction times. Again, one processor per node was used for SPAI. 5. Conclusions. This paper demonstrates the eectiveness of patterns of powers of sparsied matrices for sparse approximate inverses for PDE problems. As opposed to many existing methods for prescribing sparsity patterns, PSM patterns use both the values and structure of the original matrix, and very sparse patterns can be produced. PSM patterns allow simpler direct methods of constructing sparse approximate inverse preconditioners to be used, with comparable preconditioning quality to adaptive methods, but with signicantly less computational cost. The numerical tests show that the additional eort of adaptive sparsity pattern calculations is not always required. Acknowledgments . The author is indebted to Wei-Pai Tang, who was one of the rst to use sparsication for computing approximate inverse sparsity patterns. John Gilbert was instrumental in directing attention to the transitive closure of a matrix, and motivating the possibility of nding good patterns a priori. Michele Benzi made helpful comments and directed the author to [24]. The author is also grateful for the ongoing support of Robert Clay, Andrew Esmond Ng, Ivan Otero, Yousef Saad, and Alan Williams, and nally for the cogent comments of the anonymous referees. Table Timings, iteration counts, and e-ciencies for ParaSAILS. local problem size npes N Precon Solve Iter Solve/Iter Ep Es local problem size npes N Precon Solve Iter Solve/Iter Ep Es local problem size npes N Precon Solve Iter Solve/Iter Ep Es local problem size npes N Precon Solve Iter Solve/Iter Ep Es 50 50 50 local problem size npes N Precon Solve Iter Solve/Iter Ep Es local problem size npes N Precon Solve Iter Solve/Iter Ep Es Table Timings, iteration counts, and e-ciencies for SPAI. local problem size npes N Precon Solve Iter Solve/Iter Ep Es A PRIORI PATTERNS FOR SPARSE APPROXIMATE INVERSES 17 1200.10.30.50.70.9Number of nodes Scaled efficiency Fig. 4.2. Implementation scalability of ParaSAILS preconditioner construction phase. 1200.10.30.50.70.9Number of nodes Scaled efficiency Fig. 4.3. Implementation scalability of one step of iterative solution. --R An MPI implementation of the SPAI preconditioner on the T3E A portable MPI implementation of the SPAI preconditioner in ISIS Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems An ordering method for a factorized approximate inverse pre- conditioner Semicoarsening multigrid on distributed memory machines On a class of preconditioning methods for dense linear systems from boundary ele- ments Approximate inverse techniques for block-partitioned matrices Etude d'un pr Decay rates for inverses of band matrices Parallel implementation of a sparse approximate inverse preconditioner Predicting structure in sparse matrix computations Parallel preconditioning with sparse approximate inverses A preconditioned conjugate gradient method for solving discrete analogs of Explicit preconditioning of systems of linear algebraic equations with dense matrices Factorized sparse approximate inverse preconditionings. Factorized sparse approximate inverse precondition- ings I Iterative Methods for Sparse Linear Systems Towards an e Sparse approximate inverse smoother for multi-grid Preconditioning for boundary integral equations --TR --CTR Kai Wang , Sang-Bae Kim , Jun Zhang , Kengo Nakajima , Hiroshi Okuda, Global and localized parallel preconditioning techniques for large scale solid Earth simulations, Future Generation Computer Systems, v.19 n.4, p.443-456, May Robert D. Falgout , Jim E. Jones , Ulrike Meier Yang, Conceptual interfaces in hypre, Future Generation Computer Systems, v.22 n.1, p.239-251, January 2006 Robert D. Falgout , Jim E. Jones , Ulrike Meier Yang, Pursuing scalability for hypre's conceptual interfaces, ACM Transactions on Mathematical Software (TOMS), v.31 n.3, p.326-350, September 2005 Kai Wang , Jun Zhang , Chi Shen, Parallel Multilevel Sparse Approximate Inverse Preconditioners in Large Sparse Matrix Computations, Proceedings of the ACM/IEEE conference on Supercomputing, p.1, November 15-21, Chi Shen , Jun Zhang, Parallel two level block ILU Preconditioning techniques for solving large sparse linear systems, Parallel Computing, v.28 n.10, p.1451-1475, October 2002 Dennis C. Smolarski, Diagonally-striped matrices and approximate inverse preconditioners, Journal of Computational and Applied Mathematics, v.186 n.2, p.416-431, 15 February 2006 Edmond Chow, Parallel Implementation and Practical Use of Sparse Approximate Inverse Preconditioners with a Priori Sparsity Patterns, International Journal of High Performance Computing Applications, v.15 n.1, p.56-74, February 2001 Oliver Brker , Marcus J. Grote, Sparse approximate inverse smoothers for geometric and algebraic multigrid, Applied Numerical Mathematics, v.41 n.1, p.61-80, April 2002 Luca Bergamaschi , Giorgio Pini , Flavio Sartoretto, Computational experience with sequential and parallel, preconditioned Jacobi--Davidson for large, sparse symmetric matrices, Journal of Computational Physics, v.188 Michele Benzi , Miroslav Tma, A parallel solver for large-scale Markov chains, Applied Numerical Mathematics, v.41 n.1, p.135-153, April 2002 Anwar Hussein , Ke Chen, Fast computational methods for locating fold points for the power flow equations, Journal of Computational and Applied Mathematics, v.164-165 n.1, p.419-430, 1 March 2004 P. K. Jimack, Domain decomposition preconditioning for parallel PDE software, Engineering computational technology, Civil-Comp press, Edinburgh, UK, 2002 E. Flrez , M. D. Garca , L. Gonzlez , G. Montero, The effect of orderings on sparse approximate inverse preconditioners for non-symmetric problems, Advances in Engineering Software, v.33 n.7-10, p.611-619, 29 November 2002 Michele Benzi, Preconditioning techniques for large linear systems: a survey, Journal of Computational Physics, v.182 n.2, p.418-477, November 2002

preconditioned iterative methods;graph theory;sparse approximate inverses;parallel computing

351534

Mesh Independence of Matrix-Free Methods for Path Following.

In this paper we consider a matrix-free path following algorithm for nonlinear parameter-dependent compact fixed point problems. We show that if these problems are discretized so that certain collective compactness and strong convergence properties hold, then this algorithm can follow smooth folds and capture simple bifurcations in a mesh-independent way.

Introduction . The purpose of this paper is to extend the results in [4], [5], and [16] on mesh-independent convergence of the GMRES, [21], iterative method for linear equations to a class of matrix-free methods for solution of parameter dependent nonlinear equations of the form In Banach space. We present an algorithm for numerical path following and detection of simple bifurcations together with conditions on a sequence of approximate problems, (with G consistency of notation) that imply that the performance of the algorithm is independent of the level h of the discretization. Hence, for such problems, methods, and discretizations, the difficulties raised in [23] and [24] will not arise. In this section we set the notation and specify the kinds of singlarities that we will consider. This discussion, and that of algorithms for path following in x 2.1 and detection of simple bifurcation and branch switching in x 3, do not depend on the discretization and we use the notation G for both G 0 and G h for h ? 0. When we describe our assumptions on the discretization in x 4 and present an example in x 5, the distinction between G 0 and G h becomes important and the notation in those sections reflects that difference. 1.1. Notation and Simple Singularities. We let denote the solution path in X \ThetaIR of pairs (u; -) that satisfy (1.1), G u denotes the Fr-echet derivative with respect to u, and G - the derivative with respect to the scalar -. Version of May 19, 1998. y Department of Applied Mathematics, National Chiao-Tung University, Hsin-Chu 30050, Taiwan. (ferng@math.nctu.edu.tw). The research of this author was supported by National Science Council, Taiwan. z North Carolina State University, Department of Mathematics and Center for Research in Scientific Computation, Box 8205, Raleigh, N. C. 27695-8205 (Tim Kelley@ncsu.edu). The research of this author was supported by National Science Foundation grant #DMS-9700569. Throughout this paper we make ASSUMPTION 1.1. G is continuously Fr- echet differentiable in (u; -). G u is a Fredholm operator of index zero, and Here COM denotes the space of compact operators. For A 2 L(X), the space of bounded operators on X , we let N (A) be the null space of A and R(A) the range of A. Assumption 1.1 implies that R(G u (u; -)) is closed in X and the dimension of N (G u (u; -)) is the co-dimension of R(G u (u; -)). Following [14], we say that a point (u; ffl is a regular point if G u (u; -) is nonsingular, ffl is a simple singularity if 0 is an eigenvalue of G u with algebraic and geometric multiplicity one, - is a simple fold if it is a simple singularity and G - 62 R(G u ). - is a simple bifurcation point if it is a simple singularity and G - 2 R(G u ). 2. Algorithms for Path Following and Branch Switching. In this section we describe the iterative methods for path following, detection of bifurcation, and branch switching that we analyze in the subsequent sections and discuss some alternative approaches. Our approach is matrix-free, which means that we use no matrix storage or matrix factorizations at all. However, there are many related methods to which our mesh-independence analysis is applicable. We refer the reader to [13] and [7] for other methods for path following and branch switching and to [20], [25], and [10] for iterative methods for detection of bifurcation. 2.1. Path Following and Arclength Continuation. For path following in the absence of singularities the standard approach is a predictor-corrector method. The methods differ in the manner in which u 0 Assume that solution point and we want to compute at a nearby - 1 . If the Jacobian G u nonsingular, then the implicit function theorem insures the existence of a unique smooth arc of solutions (u(-), through more, with the smoothness assumption on G, it follows that u 0 d- u(-) exists and differentiation of (1.1) with respect to - gives the equation for u 0 . G u The Euler predictor uses u 0 directly and approximates While the results in this paper can be applied to the solution of (2.1) by GMRES, we choose to avoid the addtional solve and approximate u 0 with a difference. The secant predictor is solution pair with - CONTINUATION METHODS 3 The nonlinear iteration is an inexact Newton iteration [6] in which where d k satisfies the inexact Newton condition kG The inexact Newton condition can be viewed as a relative residual termination criterion for an iterative linear solver applied to the equation for the Newton step G When GMRES is used as the linear iteration, as it is in this paper, the nonlinear solver is referred to as Newton-GMRES. The same procedure can be used if there are simple folds if the problem is expanded by using pseudo-arclength continuation, [12], [13]. Here we introduce a new parameter s and solve is a normalization equation. We will use the secant normalization when two points on the path are available and the norm-based normalization to begin the path following. Both normalizations are independent of the discretization. This independence plays a role in the analyis that follows. It is known, [14], that arclength continuation turns simple folds in (u; -) to regular points in 3. Singular Point Detection and Branch Switching. Although a continuation procedure incorporated with a pseudo-arclength normalization can circumvent the computational difficulties caused by turning points and usually jump over bifurcation points, it is usually desirable and important to be able to detect and locate the singularities in both cases. In the case where direct methods are used for linear algebra a sign change in the determinant of F can be used to detect simple bifurcation and a change in sign of d-=ds to detect simple folds. We use a variation of the approach in [10], which requires only the largest (in magnitude) solution of a generalized eigenvalue problem. Suppose that (x(s a ); s a ) and (x(s b ); s b ) are two regular points and the path following procedure is going from s a to s b . Let A(s a A(s a ) and A(s b ) are nonsingular. Recall that if (x(s); s) is a bifurcation point on the solution path, then the null space N dimensional and there exists a nonzero vector v such that Applying the Lagrange interpolation to A(s a where the matrix E(s) denotes the perturbation matrix in the interpolation. Then instead of solving the usually nonlinear eigenvalue problem (3.1) we make a linear approximation A(s a which leads to a generalized eigenvalue problem A(s a with Therefore, an s value which causes the Jacobian F x (x(s); s) to be singular can be approximated by Our approach differs from that in [10] in that the roles of s a and s b are interchanged. The method of [10] solves where In the approach of [10], F x (x; s) is factored at s a in order to compute dx=ds for the predictor; that factorization is used again to precondition an Arnoldi iteration for the corrector, and one last time in the formulation of the eigenvalue problem to detect singularities. Since we do not factor F x at all, the roles of s a and s b can be interchanged. We found in our experiments that our approach, using (3.3), gave a better approximation of the location of the singularity that one using (3.6). Equation (3.4) implies that a bifurcation point s closest to the interval [s a ; s b ] corresponds to the largest eigenvalue in magnitude oe of (3.3). A negative oe signals that there is a bifurcation point between s a and s b , indicates that there is a bifurcation point close behind s a , and a "large" positive oe means that a bifurcation point is approaching. When oe - 1, it should be interpreted that no bifurcation point is nearby. regular point and A(s b ) is nonsingular, the generalized eigenvalue problem (3.3) can be solved via the equivalent linear eigenvalue problem (3. CONTINUATION METHODS 5 where only the largest eigenvalue and corresponding eigenvector are needed. We will solve (3.8) with the Arnoldi method and prove that if s a and s b are not too near a singlar point then the eigenvalue problem is well conditioned in a mesh independent way. A simple fold point can be predicted in a similar way with s replaced by - and A(s) replaced by At a simple bifurcation point two branches of solutions intersect nontangentially. We next describe how the information obtained from the solution of the eigenvalue problem can be used for branch switching. Suppose a bifurcation point x on the primary branch is determined. Since the eigenvector w corresponding to the largest eigenvalue oe for the eigenvalue problem described above is an approximation for a null vector of F x On the other hand, the tangent vector - which is the solution of G is also a null vector of [G u x 0 are linearly independent it is recommended in [10], where G u is factored and dx=ds is computed using (3.9), that N ([G u approximated by spanf - wg. In the matrix-free case considered in this paper, we approximate the tangent vector by a secant approximation We approximate the tangent direction of the new branch by orthogonalizing w against ffi x To obtain a regular point on the secondary branch, we solve the following augmented nonlinear system where and ffl is a "switching factor". This switching factor is problem dependent and should be chosen large enough so that the solution of (3.11) does not fall back to the primary branch [14]. The nonlinear equation (3.11) can be solved by Newton-type method with initial iterate x After moving onto the secondary branch, the continuation procedure for path following on the secondary branch is identical to the one on the primary branch. In [10] the linear systems for computing tangent vectors (3.9) are solved with a preconditioned Arnoldi iteration and the eigenvalues of the Hessenberg matrix produced in the Arnoldi iteration are used to predict singular points. Since a factorization of A(s a ) is used as the preconditioner in [10], the prediction comes with very little cost. In our approach, Jacobians are not factored and the tangent vector is not computed, instead, a secant vector is used as an approximation. The eigenvalue problem for singularity prediction is performed separately using some iterative algorithm. This leaves us the flexibility of choosing any robust iterative solver and the associated preconditioner for the continuation procedure. Other methods for singularity detection based on solution of eigenvalue problems have been proposed in [25], [9], and [20]. 4. Approximations and Mesh Independence. This is the only section in which the properties of the discrete problems are explicitly addressed. We assume, as is standard in the integral equations literature [1], [3], that the discretization used in (1.2) has been constructed, by interpolation if necessary, so that G h has the same domain and range as G. Recall [1] that a family of operators fT ff g ff2A is collectively compact if the set is precompact in X . In (4.1) B is the unit ball in X . In the special case that the indexing set A is an interval [0; h 0 ], we will denote the index by h. We say that strongly as h ! 0 and write in the norm of X for all u 2 X . All of the results are based on the following results from [1] and some simple consequences. THEOREM 4.1. Let fT h g be a family of collectively compact operators on X that converge strongly to T 2 COM(X). Assume that I \Gamma T is nonsingular. Then there is h 0 ? 0 such that nonsingular for all h - h 0 and converges strongly to A simple compactness argument implies uniform bounds for parameter dependent families of collectively compact strongly convergent operators. COROLLARY 4.2. Let a ! b be given and assume that is a collectively compact family of operators such that T h for each fixed s. Assume that fT h (s)g is uniformly Lipschitz continuous in s and that I \Gamma T (s) is nonsingular for all s 2 [a; b]. Then there are M; h 0 ? 0 such that for all h - h 0 and s 2 [a; b]. We let We assume that ASSUMPTION 4.1. 1. G h (u; -) ! G(u; -) for all (u; -) 2 X \Theta IR as h ! 0. 2. G h is Lipschitz continuously Fr- echet differentiable and the Lipschitz constants of G h u and are independent of h. 3. For all (u; -) 2 X \Theta IR (4. CONTINUATION METHODS 7 4. There are then the families of operators are collectively compact. We augment G with a mesh-independent Lipschitz continuously differentiable arclength normalization defining 0: Examples of normalizations N that do not depend on h are (2.6) and (2.7). We let Consistently with the notation in the previous sections, we let u h (-) and x h solutions to G h (u; 4.1. The Corrector Equation and Simple Folds. At regular points when - is used as the continuation parameter the equation for the Newton step is The theory in [1] asserts that if G u (u; -) is nonsingular, so is G h sufficiently small h and strongly convergent to G u (u; -) \Gamma1 . However, this convergence is not uniform and in order to invoke the results of [1] and [5] we must take care to remain away from singular points. One can treat simple folds as regular points by means of arclength continuation. If we solve the augmented system, (2.5), and this system has only regular points, then x k is bounded on finite segments of \Gamma. If, moreover, the normalization N(u; -; s) does not depend on the discretization, as it will not if (2.6) or (2.7) are used, then our assumptions imply that the finite dimensional problems are as well conditioned as the infinite dimensional problems. THEOREM 4.3. Let Assumptions 1.1 and 4.1 hold. Let [s a ; s b ] be such that is a single smooth arc and F x is nonsingular on - \Gamma(s) then there are ffi 1 , K, and h 0 such that Proof. Assumption 1.1 and the mesh independence of N imply that we may apply Corollary 4.2 to T h x (x(s); s). Hence, there is such that for all h - h 0 and x Let L be the (h-independent) Lipschitz constant of F h x . Then if Moreover and hence if M Lffi ! 1 the Banach Lemma implies that F h is nonsingular and 1=(2ML) the proof is complete with The bound, (4.7), part 2 of Assumption 4.1, and the Kantorovich theorem [11], [15], [17], imply convergence of x h to x. COROLLARY 4.4. Let Assumptions 1.1 and 4.1 hold. Let [s a ; s b ] be such that f - [s a ; s b ]g is a single smooth arc with at most simple fold singularities. Then there is a unique solution arc for F h , and x h So, for h sufficiently small, the secant predictor converges to x (0) uniformly in (s Hence if the steps in arclength fffi n s are independent of h and the secant predictor is used, then the accuracy of the initial iterate to the corrector equation and by Theorem 4.3 the condition of the linear equation for the Newton step is independent of h. Hence the performance of the nonlinear iteration is independent of h. As for the GMRES iteration that computes the Newton step, the methods from [16], [4], and [5], may be extended to show that the GMRES iteration for the Newton step converges r- superlinearly in a manner that is independent of h, x, and s. All that one needs is an uniform clustering of the eigenvalues of F h on M(ffi) that is independent of x, s, and h. This follows directly from a theorem in [1]. In the results that follow, we count eigenvalues by multiplicity and order them by decreasing distance from 0. THEOREM 4.5. Let fT h g be a family of collectively compact operators on X that converge strongly to K 2 COM(X). Let f- h be the eigenvalues of T h and f- j g 1 be the eigenvalues of T . Let ae ? 0 and assume that . Then there is then . Moreover, if - 1 has algebraic and geometric multiplicity one, then so does . Moreover there is a sequence of eigenfunctions fw h g of T h corresponding to the eigenvalue - h 1 so that w h ! w, and eigenfunction of T correpsonding to the eigenvalue - 1 . With this in hand, the eigenvalue clustering result can be obtained in the same way that Theorem 4.3 was derived from Theorem 4.1. We begin with the analog of Corollary 4.2. COROLLARY 4.6. Let the assumptions of Theorems 4.3 and 4.5 hold. Let fmu h be the eigenvalues, counted by multiplicity, of I \Gamma F x (x; s). Let ae ? 0 be given. There are h such that if h - h 0 then CONTINUATION METHODS 9 THEOREM 4.7. Let the assumptions of Theorems 4.3 and 4.5 hold. Let fmu h be the eigenvalues, counted by multiplicity, of I \Gamma F x (x; s). Let ae ? 0 be given. There are such that if h - h 0 and (x; s) 2 N for all Before we state our r-superlinear convergence theorem we must set some more notation. We write the equation for the Newton step for the corrector equation as We let d h denote the kth GMRES iteration and r h the kth GMRES residual. As is standard in nonlinear equations, d h s). The proof of Theorem 4.8 is similar to that of the similar results in [4]. THEOREM 4.8. Let Assumptions 1.1 and 4.1 hold. Let [s a ; s b ] be such that - [s a ; s b ]g is a single smooth arc and F x is nonsingular on - \Gamma(s) Then there are a continuous function M on (0; 1) such that for all ae 2 (0; 1) and (x; s) ae N (ffi 3 ) \Theta [s a ; s b ] and Theorem 4.8 states that any desired rate of linear convergence can be obtained in a mesh independent way and hence, for (x; s) fixed, the convergence is r-superlinear in a mesh independent way. 4.2. Simple Bifurcation. Now consider a path - which has a single simple bifurcation at be the branch of solutions that intersects - \Gamma 0 at (x(s c ); s c ). We will denote solutions on - \Gamma 1 by y and the arclength parameter on - t. Hence, for We set (y(t c ); t c ) be the bifurcation point on Hence For any ffi ? 0, the results in x 4.1 hold for the paths We define 0 in a similar way. Since x h the results in the previous section, simple bifurcation from - \Gamma hcan only arise for s 2 In this section we describe how the prediction of the bifurcation point, the conditioning of the generalized eigenproblem, the solution of that eigenproblem, and the accurate tracking of the other branch depend on h. Since perturbation of G, even in finite dimension, can change the structure of - \Gamma 0 from two intersecting arcs to two disconnected arcs [18], [19], we cannot show that the bifurcation will be preserved, however if s c is far enough away from s a and s b , the new other path will be detected even if it does not correspond to a bifurcation for the finite dimensional problem. For will be nonsingular and we can consider the eigenvalue problems where the largest eigenvalue in magnitude is sought. If (4.11) is solved via the Arnoldi method, the matrix-vector products of F h with a vector will each require a product of F h e. a linear solve. If this solve is performed with GMRES, then Theorem 4.8 implies that the number of iterations required to approximate that matrix-vector product can be bounded independently of h; x; s. Our assumptions imply, if s b \Gamma s a is sufficiently small, that oe = oe 0 is a simple eigenvalue with geometric multiplicity one. Since is a collectively compact and strongly convergent sequence, we can apply Theorem 4.5 to conclude that for h sufficiently small, oe h is a simple eigenvalue with geometric multiplicity one as well. Moreover oe h will be well separated from the next largest eigenvalue in magnitude, hence the conditioning of the eigenvalue problem is independent of h. 5. Numerical Results. In the example presented in this section we use the normalization equation are two previously computed solutions and used in the experiments. We used a step of in s for path following. We apply the secant predictor (2.3) to generate the initial iterate. For the corrector, we choose a forward-difference Newton-GMRES algorithm [15]. The outer iteration that generates the sequence when the nonlinear residual norm satisfies where the absolute error tolerance - were used. To avoid oversolving on the linear equation for the Newton step z k the forcing terms j k such that CONTINUATION METHODS 11 were adjusted with a method from [8]. We use the l 2 norm and scale the norm by a factor of 1=N for differential and integral equations so that the results are independent of the computational mesh. The generalized eigenvalue problem (3.3) that characterizes the singularity prediction and branch switching need not be solved at each new point on the path is computed (i. e. s is too frequent). In the example considered in this section, we solved the eigenvalue problem to make a predicition of bifurcation after a step of had been taken. We used a simple version of the Arnoldi, [2], method based on modified Gram-Schmidt orthogonalization to solve (3.3) with reorthogonalization at each iteration. The Jacobian of the inflated system is usually neither symmetric nor positive definite, we treat the generalized eigenvalue problem as linear eigenvalue problem and solve the linear system involved explicitly with preconditioned GMRES. After j steps the Arnoldi method produces is an upper Hessenberg matrix with the h ij 's as its nonzero entries and Q j is orthogonal with the q j 's as columns. If ('; y) is an eigenpair of H j and y, then provides a computable error bound. The Ritz pair, ('; w), is used to approximate the eigenpair of We terminated the Arnoldi iteration when 5. The prediction - - is accepted when - lies in between - a and - b , or We found this strategy sufficient for the example here and were able to use (3.10) to move onto the new branch. Of course, the values of \Delta eig is problem dependent (as it is for other methods) as is the criteria for accepting a prediction. As an example we consider the following two-point boundary value problem [14]. Uniform mesh of spacing is used to define the grid points ft j g N as order finite difference discretization leads to the discrete nonlinear system of order FIG. 5.1. Solution u( 1) versus -. Solution at the boundaries. Starting from the trivial solution the algorithm traces the solution path, locates bifurcation points, switches branches, and captures the secondary solution curves. Indeed there are total two bifurcation points at - 81; \Gamma81 and eight turning points at - \Sigma11; \Sigma110; \Sigma336 in the region of interest. The primary solution branch represents the symmetric solutions and the secondary solution branch represents nonsymmetric periodic solutions bifurcating from the primary branch. Figure 5.1 plots the solution u( showing the folds and bifurcations on the curve. In Table 5.1 we report the bifurcation predictions along the solution path going toward - \Gamma81. One can see that (3.3) is a more accurate predictor than (3.6). Tables 5.2 and 5.3 illustrate the mesh independence of the linear and nonlinear iterations at various points on the primary (5.2) and secondary ((5.3) branches. The iteration statistics remain virtually unchanged as the mesh is refined. Table 5.4 lists the total number of preconditioned GMRES iterations and Arnoldi iterations corresponding to three representative prediction intervals. In Table 5.4 we show how both the GMRES iterations needed to approximate the product of F h u with G u in the case where we predict a turning point) and the overall number of Arnoldi iterations is independent of the mesh. Each Arnoldi step requires about 12-14 GMRES iterations. This is similar to the numbers listed in Table 5.2 and 5.3. The residual in the table is the Arnoldi residual when the iteration terminates. CONTINUATION METHODS 13 Prediction of bifurcation point along the primary branch going toward - \Gamma81. - a - b prediction (3.3) prediction (3.6) 6.0998e+00 1.0574e+01 1.2614e+02 -9.3188e+03 8.4208e+00 1.1361e+00 -2.2855e+01 -3.5346e+01 1.1361e+00 -7.1699e+00 -3.6599e+01 -4.7426e+01 Total number of Newton and preconditioned GMRES iterations on the primary branch. problem size - value Newton P-GMRES Total number of Newton and preconditioned GMRES iterations at switching point and on the secondary branch. problem size - value Newton P-GMRES switching 7 21 switching 7 22 switching 6 20 Total number of preconditioned GMRES and Arnoldi iterations required at different sections on the path when a prediction procedure is performed. path N - a - b P-GMRES Arnoldi residual turning 64 7.0814e+00 1.0632e+01 53 5 2.9490e-02 point 128 6.0998e+00 1.0574e+01 50 5 1.9156e-05 near-by 256 9.1379e+00 1.0883e+00 regular 64 -3.1304e+01 -4.0215e+01 point 128 -2.4500e+01 -3.3295e+01 69 5 3.3738e-05 bifurcation point 128 -6.8734e+01 -7.7621e+01 56 4 1.3263e-06 detected 256 -7.4823e+01 -8.3605e+01 62 4 1.3251e-06 Acknowledgments . The authors wish to thank Gene Allgower for making us aware of reference [22], Klaus B-ohmer for providing us with a copy, and Alastair Spence for many other pointers to the literature. We are also grateful to Dan Sorensen for his council on the solution of eigenvalue problems. --R Collectively Compact Operator Approximation Theory The principle of minimized iterations in the solution the matrix eigenvalue problem A survey of numerical methods for Fredholm integral equations of the second kind GMRES and the minimal polynomial Convergence estimates for solution of integral equations with GMRES Lecture Notes on Numerical Analysis of Bifurcation Problems. Choosing the forcing terms in an inexact Newton method WANG, Nonequivalence deflation for the solution of matrix latent value problems A new algorithm for numerical path following applied to an example from hydrodynamic flow Functional Analysis Constructive methods for bifurcation and nonlinear eigenvalue problems Lectures on Numerical Methods in Bifurcation Theory Iterative methods for linear and nonlinear equations GMRES and integral operators Iterative Solution of Nonlinear Equations in Several Variables Column buckling-an elementary example of bifurcation Algorithms for the nonlinear eigenvalue problem GMRES a generalized minimal residual algorithm for solving nonsymmetric linear systems Andwendung von Krylov-Verfahren auf Verzweigungs-und Fortsetzungsprobleme An adaption of Krylov subspace methods to path following problems Numerical path following and eigenvalue criteria for branch switching --TR

fold point;bifurcation;arnoldi method;singularity;matrix-free method;GMRES;path following;collective compactness;mesh independence

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Robustness and Scalability of Algebraic Multigrid.

Algebraic multigrid (AMG) is currently undergoing a resurgence in popularity, due in part to the dramatic increase in the need to solve physical problems posed on very large, unstructured grids. While AMG has proved its usefulness on various problem types, it is not commonly understood how wide a range of applicability the method has. In this study, we demonstrate that range of applicability, while describing some of the recent advances in AMG technology. Moreover, in light of the imperatives of modern computer environments, we also examine AMG in terms of algorithmic scalability. Finally, we show some of the situations in which standard AMG does not work well and indicate the current directions taken by AMG researchers to alleviate these difficulties.

Introduction . Algebraic multigrid (AMG) was rst introduced in the early 1980's [11, 8, 10, 12], and immediately attracted substantial interest [32, 28, 30, 29]. Research continued at a modest pace into the late 1980's and early 1990's [18, 14, 21, 25, 20, 26, 22]. Recently, however, there has been a major resurgence of interest in the eld, for \classical" AMG as dened in [29], as well as for a host of other algebraic- type multilevel methods [3, 16, 34, 6, 2, 4, 5, 15, 33, 17, 35, 36, 37]. Largely, this resurgence in AMG research is due to the need to solve increasingly larger systems, with hundreds of millions or billions of unknowns, on unstructured grids. The size of these problems dictates the use of large-scale parallel processing, which in turn demands algorithms that scale well as problem size increases. Two dierent types of scalability are important. Implementation scalability requires that a single iteration be scalable on a parallel computer. Less commonly discussed is algorithmic scalability, which requires that the computational work per iteration be a linear function of the problem size and that the convergence factor per iteration be bounded below 1 with bound independent of problem size. This type of scalability is a property of the algorithm, independent of parallelism, but is a necessary condition before a scalable implementation can be attained. Multigrid methods are well known to be scalable (both types) for elliptic problems on regular grids. However, many modern problems involve extremely complex geometries, making structured geometric grids extremely di-cult, if not impossible, to use. Application code designers are turning in increasing numbers to very large unstructured grids, and AMG is seen by many as one of the most promising methods for solving the large-scale problems that arise in this context. This study has four components. First, we examine the performance of \classical" AMG on a variety of problems having regular structure, with the intent of determining its robustness. Second, we examine the performance of AMG on the same suite of problems, but now with unstructured grids and/or irregular domains. Third, we study the algorithmic scalability of AMG by examining its performance on several of Center for Applied Scientic Computing (CASC), Lawrence Livermore National Laboratory, Livermore, CA. Email:fcleary, rfalgout, vhenson, jjonesg@llnl.gov y Department of Applied Mathematics, University of Colorado, Boulder, CO. Email: ftmanteuf, stevemg@boulder.colorado.edu z USS Florida (SSBN-728), Naval Submarine Base, Silverdale, WA, Email: JerryTrish@aol.com x Front Range Scientic, Boulder, CO. Email: jruge@sobolev.Colorado.EDU the problems using grids of increasing sizes. Finally, we introduce a new method for computing interpolation weights, and we show that in certain troublesome cases it can signicantly improve AMG performance. Our study diers from previous reports on the performance of AMG (e.g., [29, 30]) primarily by our examination of algorithmic scalability, our emphasis on unstructured grids, and the introduction of a new algorithm for computing interpolation weights. In Section 2, a description of some details of the AMG algorithm is given to provide an understanding of the results and later discussion. In Section 3, we present results of AMG applied to a range of symmetric scalar problems, using nite element discretizations on structured and unstructured 2D and 3D meshes. AMG is also tested on nonsymmetric problems, on both structured and unstructured meshes, and the results are presented in Section 4. A version of AMG designed for systems of equations is tested, with the focus on problems in elasticity. Results are discussed in Section 5. In Section 6, we introduce and report on tests of a new method for computing interpolation weights. We concluding with some remarks in Section 7. 2. The Scalar AMG Algorithm. We begin by outlining the basic principles and techniques that comprise AMG. Detailed explanations may be found in [29]. Consider a problem of the form (1) where A is an n n matrix with entries a ij . For convenience, the indices are iden- tied with grid points, so that u i denotes the value of u at point i, and the grid is denoted by ng. In any multigrid method, the central idea is that error e not eliminated by relaxation must be removed by coarse-grid correction. Applied to elliptic problems, for example, simple relaxations (Jacobi, Gauss-Seidel) reduce high frequency error components e-ciently, but are very slow at removing smooth compo- nents. However, the smooth error that remains after relaxation can be approximated accurately on a coarser grid. This is done by solving the residual equation on a coarser grid, then interpolating the error back to the ne grid and using it to correct the ne-grid approximation. The coarse-grid problem itself is solved by a recursive application of this method. One iteration of this process, proceeding through all levels, is known as a multigrid cycle. In geometric multigrid, standard uniform coarsening and linear interpolation are often used, so the main design task is to choose a relaxation scheme that reduces errors the coarsening process cannot approximate. One purpose of AMG is to free the solver from dependence on geometry, so AMG instead xes relaxation (normally Gauss-Seidel), and its main task is to determine a coarsening process that approximates error that this relaxation cannot reduce. An underlying assumption in AMG is that smooth error is characterized by small residuals, that is, Ae 0, which is the basis for choosing coarse grids and dening interpolation weights. For simplicity of discussion here, we assume that A is a symmetric positive-denite M-matrix, with a ii > 0; a ij 0 for j 6= i, and This assumption is made for convenience; AMG will frequently work well on matrices that are not M-matrices. To dene any multigrid method, several components are required. Using superscripts to indicate level number, where 1 denotes the nest level so that A and the components that AMG needs are as follows: 1. \Grids" M . 2. Grid operators A 1 ; A 3. Grid transfer operators: Interpolation I k Restriction I k+1 4. Relaxation scheme for each level. Once these components are dened, the recursively dened cycle is as follows: Algorithm: MV k Otherwise: times on A k u Perform coarse grid correction: Set \Solve" on level k+1 with MV k+1 Correct the solution by times on A k u For this cycle to work e-ciently, relaxation and coarse-grid correction must work together to eectively reduce all error components. This gives two principles that guide the choice of the components: P1: Error components not e-ciently reduced by relaxation must be well approximated by the range of interpolation. P2: The coarse-grid problem must provide a good approximation to ne-grid error in the range of interpolation. Each of these aects a dierent set of components: given a relaxation scheme, P1 determines the coarse grids and interpolation, while P2 aects restriction and the coarse grid operators. In order to satisfy P1, AMG takes an algebraic relaxation is xed, and the coarse grid and interpolation are automatically chosen so that the range of the interpolation operator accurately approximates slowly diminishing error components (which may not always appear to be \smooth" in the usual sense). P2 is satised by dening restriction and the coarse-grid operator by the Galerkin formulation: I k+1 and A (2) When A is symmetric positive denite, this ensures that the correction from the exact solution of the coarse-grid problem is the best approximation in the range of interpolation [23], where \best" is meant in the A-norm: by jjvjj A hAv; vi 1=2 . The choice of components in AMG is done in a separate preprocessing step: 1. 2. Partition into disjoint sets C k and F k . (a) (b) Dene interpolation I k . 3. Set I k+1 and A . 4. If is small enough, set set go to step 2. Step 2 is the core of the AMG setup process. Since the focus is on coarsening a particular level k, such superscripts are omitted here and c and f are substituted for necessary to avoid confusion. The goal of the setup phase is to choose the set C of coarse-grid points and, for each ne-grid point small set C i C of interpolating points. Interpolation is then of the form: I f c u c 2.1. Dening Interpolation Weights. To dene the interpolation weights recall that slow convergence is equivalent to small residuals, Ae 0. Thus, we focus on errors satisfying a ii e i Now, for any a ij that is relatively small, we could substitute e i for e j in (4) and this approximate relation would still hold. This motivates the denition of the set of dependencies of a point i, denoted by S i , which consists of the set of points j for which a ij is large in some sense. Hence, i depends on such j because, to satisfy the ith equation, the value of u i is aected more by the value of u j than by other variables. The denition used in AMG is ( a ik ) with typically set to be 0.25. We also dene the set S T that is, the set of points j that depend on point i, and we say that i is the set of in uences of point i. Note: our terminology here diers from the classical use in [29], which refers to i as being strongly connected to or strongly dependent on j if j 2 S i and which uses no specic terminology for A basic premise of AMG is that relaxation smoothes the error in the direction of in uence. Hence, we may select C as the set of interpolation points for i, and adhere to the following criterion while choosing C and F : P3: For each i 2 F , each j 2 S i is either in C or S j \ C i 6= ;. That is, if i is a ne point, then the points in uencing i must either be coarse points or must themselves depend on the coarse points used to interpolate u i . This allows approximations necessary to dene interpolation. For can be rewritten as: a ii e i a ik e k AMG interpolation is dened by making the following approximation in (6): a jk e k a jk otherwise. Substituting this into (6) and solving for e i gives the desired interpolation weights for point 2.2. Selecting the Coarse Grid. The coarse grid is chosen to satisfy the criterion above, while attempting to control its size. We employ the two-stage process described in [29], modied slightly to re ect our modied terminology. The grid is rst \colored", providing a tentative C/F choice. Essentially, a point with the largest number of in uences (\in uence count") is colored as a C point. The points depending on this C point are colored as F points. Other points in uencing these F points are more likely to be useful as C points, so their in uence count is increased. The process is repeated until all points are either C or F points. Details of the initial C=F choice are as follows: Repeat until k . Set (the number of points depending on the point i). Pick an i 2 U with maximal i . Set T fig and fig. For (points depending on fig) do: S fjg and For all k 2 S j T U set (Increment the for points that in uence the new F -points). For T U set Next, a second pass is made, in which some F points may be recolored as C points to ensure that P3 is satised. In this pass, each F -point i is examined. The coarse interpolatory set C S C is dened. Then, if i depends on another F -point, j, the points in uencing j are scanned, to see if any of them are in C i . If this is not the case then j is tentatively converted into a C-point and added to C i . The dependencies of i are then examined anew. If all F -points depending on i now depend on a point in C i then j is permanently made a C-point and the algorithm proceeds to the next F -point and repeats. If, however, the algorithm nds another F -point dependent on i that is not dependent on a point in C i then i itself is made into a C-point and j returned to the pool of F -points. This procedure is followed to minimize the number of F -points that are converted into C-points. We make a brief comment about the computational and storage costs of the setup phase. Unlike geometric multigrid, these costs cannot be predicted precisely. Instead, computational cost must be estimated based on the average \stencil size" over all grids, the average number of interpolation points per F -point, the ratio of the total number of gridpoints on all grids to the number of points on the ne grid (grid complexity), and the ratio of the number of nonzero entries in all matrices to that of the ne-grid matrix (operator complexity). While a detailed analysis is beyond the scope of this work [29], a good rule of thumb is that the computational eort for the setup phase is typically equivalent to between four and ten V -cycles. 3. Results for Symmetric Problems. In this section, results for AMG applied to symmetric scalar problems are presented. Initially, constant-coe-cient diusion problems in 2D are tested as a baseline for comparison as we begin to introduce complications, including unstructured meshes, irregular domains, and anisotropic and discontinuous coe-cients. Results for 3D problems follow. All problems are run using the same AMG solver with xed parameters. On many problems, it is possible to improve our results by tuning some of the input parameters (there are many), but the purpose here is to show AMG's basic behavior and robustness over a range of problems. The primary indicator of the speed of the algorithm is the asymptotic convergence factor per cycle. This is determined by applying 20 cycles to the homogeneous prob- lem, starting with a random initial guess, then measuring the reduction in the norm of the residual from one cycle to the next (we use the homogeneous problem to avoid contamination by machine representation). Generally, this ratio starts out very small for the rst few cycles, then increases to some asymptotic value after 5-10 cycles, when the most slowly converging components become dominant. This asymptotic value is also a good indicator of the actual error reduction from one cycle to the next. We use the 2-norm of the residual, although it is easy to show that the asymptotic convergence factor is just the spectral radius of the AMG V -cycle iteration operator, and hence is independent of the choice of norm. The times given are for the setup and a single (1,1) V -cycle. Setup time is what it takes to choose the coarser grids, dene interpolation, and compute the coarse grid matrices. Cycle time is for one cycle, not the full solution time. Three machines are used in this study. The majority of the smaller tests are performed on a Pentium 166MHz PC, although some are performed on a Sun Sparc Ultra 1. For the larger problems that demonstrate scalability, we use a DEC Alpha. For this reason, timings should be compared only within individual problems. Additionally, timings for the smallest problems can have a high relative error, so the larger tests should give a better picture of performance. Grid complexity is dened as the number of grid points on level k. This gives an idea of how quickly the grids are reduced in size. For comparison, in standard multigrid, the number of points is reduced by a factor of 4 in 2D and 8 in 3D, yielding grid complexities of 4/3 and 8/7, respectively. AMG tends to coarsen more slowly. Operator complexity, which is a better indicator of the work per cycle, is dened as is the average number of non-zero entries per row (or \stencil size") on level k. Thus, the operator complexity is the ratio of the total number of nonzero matrix entries on all levels to those on the nest level. Since relaxation work is proportional to the number of matrix entries, this gives a good idea of the total amount of work in relaxation relative to relaxation work on the nest grid, and also of the total storage needed relative to that required for the ne grid matrix. In geometric multigrid, the grid and operator complexities are equal, but in AMG, operator complexity is usually higher since average stencil sizes tend to grow somewhat on coarser levels. Note that the convergence factors and complexities are entirely independent of the specic machine on which a test is performed. In the tests reported here, the focus is on nite element discretizations of r d 21 (x; y) d 22 (x; y) Several dierent meshes and diusion coe-cients D are used. 3.1. Regular domains, structured and unstructured grids. The rst ve problems are 2D Poisson equations, with d Dierent domains and meshes are used to demonstrate the behavior of AMG with simple equations. We begin with the simplest 2D model problem. The success of AMG on the regular-grid Poisson problem is well-documented [30, 28, 29], so our purpose here is more to assess its scalability. Problem 1 This is a simple 5-point Laplacian operator with homogeneous Dirichlet boundary conditions on the unit square. The experiment is run for uniform meshes Convergence factor vs. log of problem size Cycle Times vs. log of problem size vs. log of problem size Fig. 1. Top Left: Convergence factors, as a function of number of mesh points, for Problem 1 the uniform-mesh 5-point Laplacian. Top Right: Log-log plots of setup times (circles) and cycle times (triangles) for the uniform-mesh 5-point Laplacian. The dotted line, for reference, shows perfectly linear scaling. Bottom: Operator (circles) and grid (triangle) complexities for the uniform-mesh 5-point Laplacian. with n n interior grid points, yielding mesh sizes 90000, 250000, and 490000. Results for Problem 1 are displayed in Figure 1. The convergence factor (per cy- cle) is very stable at approximately 0.04 for all problem sizes. Both the setup and cycle time are very nearly linear in N (compare with the dotted line depicting a perfectly linear hypothetical data set). Here, setup time averages roughly the time of 6 cycles. As noted before, the operator complexities are higher than the corresponding grid complexities, but both appear to be unaected by problem size. These data indicate that AMG (applied to the uniform-mesh Laplacian) is algorithmically scalable: the computational work is O(N) per cycle and the convergence factor is O(1) per cycle. An important component of our study is to determine to what extent this algorithmic scalability is retained as we increase problem complexity. Problem 2 This is the same equation as Problem 1 ( discretized on an unstructured triangular mesh. These meshes are obtained from uniform triangulations by randomly choosing 15-20% of the nodes and \collapsing" them to neighboring nodes, then smoothing the resulting mesh. The resulting operators might be represented by M-matrices in some cases, but this is not generally the case. We use meshes with typical example is shown at top left in Figure 2. Results of the experiments are displayed in Figure 2. On the unstructured meshes, convergence factors tend to show some dependence on mesh size, growing to around 0.35 on the nest grid. It should be noted, however, that these grids tend to be less structured than many found in practice, and no care was taken to ensure a \good" mesh; the meshes may have diering characteristics (such as aspect ratios), as there is a large degree of randomness in their construction. Complexities are also higher with the unstructured meshes, and the setup time increases correspondingly. The main point here is that AMG can deal eectively with unstructured meshes without too much degradation in convergence over the uniform case. 3.2. Irregular domains. We continue to use the Laplacian, but now with irregular domains. Since our emphasis here is the eects of this irregularity, we restrict our tests to two representative mesh sizes that give just a snapshot of algorithm scalability. Problem 3 The computational domain is an unstructured triangular discretization of the torus 0:05 dierent mesh sizes were used, resulting in grids with conditions around the hole are imposed, with Neumann conditions on the outer boundary. Problem 4 The domain for this problem is shown in Figure 3. The boundary conditions are Neumann except that a Dirichlet condition is imposed around the small hole on the right. The meshes are uniform, with resulting in meshes with respectively. The domain does not easily admit much coarser meshes. Problem 5 The domain for this problem is shown on the bottom in Figure 3. Dirichlet conditions are imposed on the exterior boundary, and Neumann conditions are on the interior boundaries. A triangular unstructured mesh is used. Results for Problems 3{5 are given in Table 1. Among these problems, Problem 3 has the simplest domain, but the least structured mesh and the slowest convergence. This indicates that domain conguration generally has little eect on AMG behavior, while the structure (and perhaps the quality) of the mesh is more important. Table 1. Results for Problems 3{5. Poisson problem on unstructured meshes, irregular domains Convergence Problem N factor/cycle (sec) (sec) complexity complexity Convergence Factor vs. log of problem size Cycle Times vs. log of problem size vs. log of problem size Fig. 2. Top Left: A typical unstructured grid for Problem 2, obtained by randomly deleting 15% of the nodes in a regular grid and smoothing the result. Top Right: Convergence factors, as a function of number of mesh points, for the unstructured-mesh 5-point Laplacian. Bottom Left: Log-log plots of setup times (circles) and cycle times (triangles) for the unstructured-grid 5-point Laplacian. The dotted line, for reference, shows perfectly linear scaling. Bottom Right: Operator (circles) and grid (triangle) complexities for the unstructured-grid 5-point Laplacian. Fig. 3. Domain (Top Left) and typical grid (Top Right) for Problem 4. Note that the mesh size necessary to display the triangulation is too coarse to observe the Dirichlet hole. Finer meshes are used for the calculations. Bottom: Typical grid for Problem 5. 3.3. Isotropic diusion. The next problem set deals with isotropic diusion: Discontinuous d(x; y) can cause problems for many solution methods, including standard multigrid methods, although it is possible to get good results either by aligning the discontinuities along coarse grid lines, or by using operator-dependent interpolation [1]. In AMG, nothing special is required, since it is based on operator-dependent interpolation. The problems are categorized according to the diusion coe-cient used. The unit square is discretized on four meshes: two structured meshes with and two unstructured meshes, with 54518. The diusion coe-cients are dened in terms of a parameter, c, allowed to be either 10 or 1000, as follows: Problem 6 d(x; Problem 7 d(x; Problem 8 d(x; 1:0 0:125 max (jx 0:5j; jy 0:5j) 0:25; c otherwise: Problem 9 d(x; 1:0 0:125 c otherwise: Results for these problems are presented in Table 2, which contains observed convergence factors and operator complexities for the various combinations of grid size and type, diusion coe-cient function, and discontinuity jump size. The overall results are fairly predictable. Convergence factors are fairly uniform. On the structured meshes, they tend to grow slightly with increasing grid size. They are noticeably larger for unstructured grids, and they appear to grow somewhat with increasing grid size. (As noted before, comparison among unstructured grids of various sizes must take into account that their generation involves some randomness, so they may dier in important ways.) The convergence factor does not seem to depend signicantly on the size of the jump in the diusion coe-cient. In many cases, results were better with Indeed, AMG has been applied successfully to problems with much larger jumps [28]; see also Problem 17. Note that there are only minor variations in operator complexity for the dierent problems and dierent grid sizes. The only signicant eect on operator complexity appears to be whether the grid is structured or unstructured, with the latter showing complexity increases of about 30{40%. It should be noted, however, that even in these cases, the entire operator hierarchy can be stored in just over three times the storage required for the ne-grid alone. Table 2. Results for Problems 6{9. Poisson problem, variable and discontinuous coe-cients Uniform mesh size Unstructured mesh size Problem 16642 66049 13755 54518 # c conv. cmplxty conv. cmplxty conv. cmplxty conv. cmplxty 6 1000 0.097 2.21 0.180 2.2 0.264 3.32 0.369 3.36 9 1000 0.171 2.35 0.168 2.30 0.234 3.31 0.298 3.40 Problems 10{13 are designed to examine the case in which the diusion coe-cient is discontinuous and to determine whether the \scale" of the discontinuous regions aects performance. Accordingly, Problems 10{12 use a \checkerboard" pattern: even and i where Specically, Problem 2. Problem Problem For the last problem of this group, we have Problem 13 d(x; Results for Problems 10{13 are displayed in Table 3. The overall trend is similar to the results for Problems 6{9, showing convergence factors that grow slightly with problem size and that are noticeably larger for unstructured grids. Table 3. Results for Problems 10{13. Poisson problem, variable and discontinuous coe-cients Uniform mesh size Unstructured mesh size Problem 16642 66049 13755 54518 # c conv. cmplxty conv. cmplxty conv. cmplxty conv. cmplxty The best results for the isotropic diusion problems are obtained for Problem 6, where the coe-cient is continuous. The worst convergence factor is obtained for the 50 50 \checkerboard" pattern of Problem 12. Note that AMG performs well on Problem 8, where \smooth" functions are approximately constant in the center of the region, zero in the high-diusion zone near the boundary, and smoothly varying in be- tween. Good interpolation in the low-diusion band is essential to good convergence. AMG appears to work quite well with discontinuous diusion coe-cients, even when they vary randomly by a large factor from point to point, as in Problem 13. 3.4. Anisotropic diusion. The next series of problems deals with anisotropic diusion, which can arise in several ways. Anisotropy can be introduced by the mesh being rened dierently in each directions, perhaps to resolve a boundary layer or some other local phenomenon. Another case is a tensor product grid used in order to rene some area [x with the mesh size small for x 2 [x large elsewhere. This maintains a logically rectangular mesh, but causes anisotropic discretizations in dierent parts of the domain. This is relatively easy to deal with in geometric multigrid, where line relaxation and/or semi-coarsening can be used [9]. Non-aligned anisotropy, which is more di-cult to handle with standard multigrid, arises from the operator itself, such as the case of the full potential operator in transonic ows. The performance of AMG on grid-induced (aligned) anisotropy has been reported previously [29], so we instead focus here on non-aligned anisotropy. Both types of anisotropy can be written in terms of the diusion equation using the coe-cient matrix: cos sin sin 2 When is constant, this gives the operator @ +@ , where is in the direction . On a rectangular grid with mesh sizes h x and h y , the usual Poisson equation corresponds to the diusion equation with x . Problem 14 This problem features a non-aligned anisotropic operator on the unit square, with Dirichlet boundary conditions at conditions on the other two sides. The cases were both examined with and =4. Each such combination is discretized on a uniform square mesh (with bilinear elements) and both uniform and unstructured triangular meshes (with linear elements). The uniform meshes have the unstructured meshes have Convergence factors for Problem 14, as shown in Figure 4, generally degrade with increasing . This is to be expected, as it indicates lessened alignment of anisotropy with the grid directions. The strong anisotropy case yields convergence factors as high as 0.745. As noted above, the non-aligned case is very di-cult, even for standard multigrid, and is the subject of ongoing study. One encouraging result is that the unstructured grid formulations are relatively insensitive to grid anisotropy, with convergence factors that hover between 0.3 and 0.5 in all cases except these results indicate that AMG is rather robust for anisotropic problems, although convergence factors are somewhat higher than those typically obtained with AMG on isotropic problems. Convergence factor vs. Convergence factor vs. 0.80.20.6Fig. 4. Convergence factors, plotted as a function of anisotropy direction . Left: Moderate anisotropy, In each plot, the solid lines are the uniform meshes with square discretizations, the dotted lines are uniform meshes with triangular discretizations, and the dashed lines are the unstructured triangulations. In each case, the larger grid size is indicated by a symbol (\o", \+", or \"). Problem 15 We use the operator of Problem 4, r (Dru), but with This yields a discretization such that, on any circle centered at the origin, there are dependencies in the tangential direction, but none in the radial direction. This problem is very di-cult to solve by conventional methods. Using the same meshes as in Problem 14, AMG produced the convergence factors given in Table 4. Table 4. Results for Problem 15: Circular diusion coe-cient uniform mesh (square) uniform mesh (triangular) unstructured mesh The convergence factors illustrate the di-culty with this problem, which cannot be handled easily by geometric methods, even on regular meshes. A polar-coordinate mesh would allow block relaxation over strongly coupled points, but would suer from the di-culties of polar-coordinate grids (e.g., singularity at the origin) and would be useless for more general anisotropies. While convergence of AMG here is much slower than what we normally associate with multigrid methods, this example shows that AMG can be useful even for extremely di-cult problems. 3.5. 3D problems. Turning our attention to three dimensions, we do not expect special di-culties here, since AMG is based on the algebraic relationships between the variables. Problem This is a 3D Poisson problem on the unit cube. Discretization is by trilinear nite elements on a rectangular mesh. Dirichlet boundary conditions are imposed at conditions are imposed at the other boundaries. Mesh line spacing and the number of mesh intervals were both varied to produce several grids with dierent spacings and extents in the three coordinate directions. The various combinations of mesh sizes used, convergence factors, and operator complexities are shown in Table 5. Table 5. Results for Problem 16. 3D Poisson problem, regular rectangular mesh Convergence Operator Nx hx Ny hy Nz hz N factor/cycle Complexity Problem 17 This is a 3D unstructured mesh problem, generated by a code used at Lawrence Livermore National Laboratory, for the diusion problem g(~x). The domain is a segment of a sphere, from and to =2. The coe-cient a(~x) is a large constant for r 0:05 and a small constant for r > 0:05, with a step discontinuity of 1:010 26 . The boundaries and are Dirichlet, while the boundaries are surfaces of symmetry. Discretization is by nite elements using hexahedral elements. Figure 5 shows the locations of the nodes at the element corners for one of the prob- lems. Three problem sizes are given, with Convergence factors and operator complexities for these problems are given in Table 6. Table 6. Results for Problem 17. 3D unstructured diusion problem Convergence Operator 8000 0.166 2.84 AMG apparently works quite well for 3D problems, including those with discontinuous coe-cients. The convergence factors are good in all cases. Complexity varies signicantly, with the highest values for the uniform grids, but decreasing markedly with increasing grid anisotropy. This may be taken as further evidence that AMG automatically takes advantage of directions of in uence. performed well on this suite of symmetric scalar test problems. Many of these problems are designed to be very di-cult, often unrealistically so, especially those with the circular anisotropic diusion pattern and the random diu- sion coe-cients. Recall that the same AMG algorithm, with no parameter tuning, was used in all cases. There are a number of tools for increasing the e-ciency of AMG, especially on symmetric problems, that have proved useful in many cases. One 0.10.010.030.050.07Fig. 5. Nodes at element corners, 3D diusion problem. Top Left: View of all the node locations. Top Right: View from directly overhead, showing radial lines of nodes in the azimuthal direction. Bottom: Distribution of nodes within a plane of constant azimuth. is the so-called V ? cycle [30], in which the coarse-grid corrections are multiplied by an optimal parameter, determined by minimizing the A-norm of the corrected error. Another, which has been successful in applying AMG to Maxwell's equations [27], is to use an outer conjugate gradient iteration, with AMG cycling as a preconditioner. Often, when AMG fails to perform well, the problem lies in a small number of components that are not reduced e-ciently by relaxation or coarse-grid correction, and conjugate gradients can be very e-cient in such cases. Other methods for improving e-ciency include the F cycle [7, 31] and the full multigrid (FMG) method, whose applicability to AMG is the subject for future research. 4. AMG applied to Nonsymmetric Scalar Problems. Although much of the motivation and theory for AMG is based on symmetry of the matrix, this is not at all a requirement for good convergence behavior. Mildly nonsymmetric problems behave essentially like their symmetric counterparts. Such cases arise when a nonsymmetric discretization of a symmetric problem is used or when the original problem is predominantly elliptic. An important requirement for current versions of AMG is that point Gauss-Seidel relaxation converge, however slowly. Thus, central dierencing of rst-order terms, when they dominate, cannot be used because of severe loss of diagonal dominance. Even in these cases, successful versions of AMG can be developed using Kaczmarz relaxation [9]. Nevertheless, we restrict ourselves here to upstream dierencing so that we can retain our use of Gauss-Seidel relaxation. Problem This is a convection-diusion problem of the form sin with Dirichlet boundary conditions. Triangular meshes are used, both structured unstructured 54518). The diusion term is discretized by nite elements. The convection term is discretized using upstream dierencing, that is, the integral of the convection term is computed over the triangle and added to the equation corresponding to the node with the largest coe-cient (the node \most upstream"). Note that this can result in a matrix that has o-diagonal entries of both signs. Two choices for are employed: are conducted with Results are presented in Figure 6. The curves in the top left graph are for the structured grid results are displayed with solid lines, and the unstructured-grid results are displayed with dashed lines. For each pair of curves, the curve with the marker (\o" or \") indicates the mesh with larger N . The convection-dominated case is shown at the top right (structured grids) and on the bottom (unstructured grids). In each case, the smaller N is shown with solid lines and the larger N with dashed lines. Note that convergence is generally good and fairly uniform, particularly for the unstructured cases. Results on the smaller uniform mesh are especially good when the ow is aligned with the directions or =2. This is due to the triangulation: to obtain the uniform mesh, the domain is partitioned into squares, and then each square is split into two triangles, with the diagonal going from the lower left to the upper right; the \good" directions are aligned with the edges of the triangles. This also has an eect on the quality of the discretization, and on convergence, when the ow is in the direction 3=4 and 7=4. Here, the discretization used for the convection term causes a rather severe loss of positivity in the o-diagonals. This is more the fault of the discretization than AMG. For with the smaller uniform mesh, Convergence factor vs. Convergence factor vs. structured grid Convergence factor vs. unstructured grid Fig. 6. Convergence factors, plotted as a function of direction, , for the nonsymmetric prob- lem. Top Left: Weaker convection case, 0:1 The solid lines are the uniform meshes, and the dashed lines are the unstructured triangulations. In each case the larger grid size is indicated by the placement of a symbol (\o" or \"). Top Right: Convection-dominated case, structured grid. The solid line is the smaller N , the larger N is indicated by the dashed line. Bottom: Convection-dominated case, unstructured grid. The solid line is the smaller N , the larger N is indicated by the dashed line. AMG was unable to handle the discretization produced and failed in the setup phase. Often in multigrid applications, problems in convergence indicate problems with the discretization, as is the case here. Note that for unstructured grids where ow cannot align with (or against) the grid, convergence is generally more uniform. An interesting point is that, in many cases, a smaller diusion coe-cient reduces the convergence factor. This is particularly striking with the unstructured meshes. Finally, note that there is generally not much dierence between results for and for + , so there is no benet from accidental alignment of relaxation with the ow direction, and, conversely, there is no slowing of convergence due to upstream relaxation. This is due to the C/F ordering of relaxation. These tests show that AMG can be applied to nonsymmetric problems. While the convergence factors in this test are generally less than 0:2{0:25, which is certainly acceptable, some concern may be raised about the scalability issue, since the convergence factors for the larger unstructured mesh are noticeably greater than those for the smaller unstructured grids. It remains to be determined if the convergence factors continue to grow with increasing problem size, or if they reach an asymptotic limit. 5. AMG for Systems of Equations. The extension of AMG to \systems" problems, where more than one function is being approximated, is not straightforward. Many dierent approaches can be formulated. Consider a problem with two unknown functions of the form u The scalar algorithm could work in special circumstances (for example, if B and C are relatively small in some sense), but, generally, the scalar ideas of smoothness break down. One approach would be to iterate in a block fashion on the two equations, with two separate applications of AMG, one using A as the matrix (solving for u, holding v xed) and one using B (solving for v, holding u xed), and repeating until convergence. This is often very slow. An alternative is to use this block iteration as a preconditioner in an outer conjugate gradient solution process. Another fairly simple alternative is to couple the block iteration process on all levels, that is, to coarsen separately for each function, obtaining two interpolation operators I u and I v , then dene a full interpolation operator of the form I The Galerkin approach can then be used to construct the coarse grid operator, I T I T Once the setup process is completed, multigrid cycles are performed as usual. We will call this the function approach since it treats each function separately in determining coarsening and interpolation. When u and v are dened on the same grid, it is also possible to couple the coarse-grid choices for both, allowing for nodal relaxation, where both unknowns are updated simultaneously at a point. Following are results for the function approach applied to several problems in 2D and 3D elasticity. Problem 19 This problem is plane-stress elasticity: where u and v are displacements in the x and y directions, respectively. This can be a di-cult problem for standard multigrid methods, especially when the domain is long and thin. The problem is discretized on a rectangular grid using bilinear nite elements, and several dierent problem sizes and domain congurations are used. Problem 20 This problem is 3D elasticity: where u, v, and w are displacements in the three coordinate directions. The problem is discretized on a 3D rectangular grid using trilinear nite elements. Several dierent problem sizes and domain congurations are used. In all tests, we take 0:3. The function approach with (1,1) V -cycles is used in all tests. Results for Problem 19 are contained in Table 7, and for Problem 20 in Table 8. Note that complexities are stable in 2D, with some dependence on problem size in 3D. Convergence depends fairly heavily on the number of xed boundaries in both 2D and 3D, with convergence degrading as the number of free boundaries increases. Table 7. Results for 2D elasticity, Problem 19 # of xed Convergence Operator Table 8. Results for 3D elasticity, Problem 20 # of xed Convergence Operator 6. Iterative Interpolation Weights. Occasionally, we encounter situations where convergence of AMG is poor, yet no specic reason is apparent. Our experience leads us to believe that the fundamental problem, in many cases, stems from the limitation of the matrix entry a jk to re ect the true "smoothness" between e j and e k . Often the true in uences between variables are not clear. One case where this limitation is quite evident is where nite elements with extreme aspect ratios are used, especially in cases of extreme grid anisotropy or of thin-body elasticity. As a simple example, consider the 2D nine-point negative Laplacian based on quadrilateral elements that are stretched in the x-direction. The stencil changes as y !1: The limiting case is no longer an M-matrix. Indeed, even moderate aspect ratios (e.g., o-diagonal entries of both signs. It is not immediately clear how the neighbors to the east and west of the central point should be treated. Do they in uence the central point? Should they be in S i ? Even if they are not treated as in uences, similar questions arise about how the corner points relate to the central point. Geometric intuition indicates they are decoupled from the center, and should not be treated as in uences. Yet, for the most common choice of in (5), AMG treats them as in uences. Another di-culty arises when two F points i and j in uence each other. Then e j must be approximated in the second sum on the right side of (6) to determine the weights for i, while e i must be approximated, in (6) but with the roles of i and j reversed, to determine the weights for j. However, since both e i and e j are to be interpolated (being F -point values), it makes sense to use the interpolations to obtain these approximations, that is, the approximations for e i and e j in (6) should be and e i respectively. Note that the approximations for any points in C are unchanged in these equations. This gives an implicit system for the interpolation weights, which is solved by an iterative scheme with the initial approximation . The new interpolation weights are then calculated in a Gauss-Seidel-like manner, using the most recently computed weights to make the approximations in (12). Two sweeps are generally su-cient. An important addition to the process is that, after the rst sweep, the interpolation sets are modied by removing from C i any point for which a negative interpolation weight is computed. The second sweep is then used to compute the nal interpolation weights. We present results of two experiments illustrating the eectiveness, on certain types of problems, of using this iterative weight denition scheme. Other examples may be found in [24]. Problem 21 This operator is the \stretched quadrilateral" Laplacian mentioned above, discretized on an 10000. The stretching factor " represents the ratio of the x-dimension to the y-dimension of the quadrilateral. Values used are In each case, the convergence factor is computed for the choices 0:5. In the latter case, the corner points are not treated as in uences, and AMG selects a semi-coarsened coarse grid, which is the method geometric multigrid would take. Table 9. Results for problem 21, Convergence Factor standard weights iterative weights 100 900 0.83 0.53 0.82 0.23 100 10000 0.93 0.55 0.93 0.28 Results are displayed in Table 9. On the smallest problem, iterative weights have no eect. However, the convergence rate on that problem is quite good, even for standard weights. For moderate stretching (" < 100), the eect of iterative weighting is to correct for misidentied in uence (i.e., improvement for and to improve the results even for correctly identied in uence 0:5). For extreme stretching, only the latter eect applies. Problem 22 Here we use the unstructured 3D diusion operator from Problem 17, whose grid is displayed in Figure 5. Problem sizes are 8000. The problem includes a very large jump discontinuity, O(10 26 ), in the diusion coe-cients. Table 10. Results for problem 22 Convergence Factor standard weights iterative weights Results are displayed in Table 10. Again we see that, on the smallest problem, iterative weights have no eect, but that convergence there is fairly good anyway, even for standard weights. On the two larger problems, iterative weights produce signicant improvement for both choices of . Apparently, iterative weighting is countering the eects of both poor element aspect ratios near the boundaries and jump discontinuities in identifying in uences among variables. On these problems, and similar problems characterized by coe-cient discontinuities and/or extreme aspect ratios in the elements, iterative weight denition proves to be quite eective. However, iterative weighting is not always eective at improving slow AMG convergence, and in a few cases it can actually cause very minor degradation in performance [24]. We study a new approach in [13], called element interpolation (AMGe), which has the promise of overcoming the di-culties associated with poor aspect ratios, misidentied in uences, and thin-body elasticity, provided the individual element stiness matrices are available. 7. Conclusions. The need for fast solvers for many types of problems, especially those discretized on unstructured meshes, is a clear indication that there is a market for software with the capabilities that AMG oers. Our study here demonstrates the robustness of AMG as a solver over a wide range of problems. Our tests indicate that it can be further extended, and that robust, e-cient codes can be developed for problems that are very di-cult to solve by other techniques. AMG is also shown to have good scalability on model problems. This scalability does tend to degrade somewhat with increasing problem complexity, but the convergence factors remain tractable even in the worst of these situations. --R The multi-grid methods for the diusion equation with strongly discontinuous coe-cients Stabilization of algebraic multilevel iteration The algebraic multilevel iteration methods - theory and applications A class of hybrid algebraic multilevel preconditioning methods Towards algebraic multigrid for elliptic problems of second order Guide to multigrid development guide with applications to uid dynamics Outlines of a modular algebraic multilevel method Interpolation and related coarsening techniques for the algebraic multigrid method Additive multilevel-preconditioners based on bi-linear 4interpolation A note on the vectorization of algebraic multigrid algorithms Matrix Analysis Convergence of algebraic multigrid methods for symmetric positive de Multigrid methods for variational problems: general theory for the V-cycle Interpolation weights of algebraic multigrid On smoothing properties of SOR relaxation for algebraic multigrid method A simple parallel algebraic multigrid Multigrid methods for solving the time- harmonic Maxwell equations with variable material parameters An energy-minimizinginterpolation for robust multi-grid methods --TR --CTR Emden Henson , Ulrike Meier Yang, BoomerAMG: a parallel algebraic multigrid solver and preconditioner, Applied Numerical Mathematics, v.41 n.1, p.155-177, April 2002 Chi Shen , Jun Zhang , Kai Wang, Distributed block independent set algorithms and parallel multilevel ILU preconditioners, Journal of Parallel and Distributed Computing, v.65 n.3, p.331-346, March 2005 Randolph E. Bank, Compatible coarsening in the multigraph algorithm, Advances in Engineering Software, v.38 n.5, p.287-294, May, 2007 K. Kraus , J. Schicho, Algebraic Multigrid Based on Computational Molecules, 1: Scalar Elliptic Problems, Computing, v.77 n.1, p.57-75, February 2006 Oliver Brker , Marcus J. Grote, Sparse approximate inverse smoothers for geometric and algebraic multigrid, Applied Numerical Mathematics, v.41 n.1, p.61-80, April 2002 Touihri, Variations on algebraic recursive multilevel solvers (ARMS) for the solution of CFD problems, Applied Numerical Mathematics, v.51 n.2-3, p.305-327, November 2004 J. J. Heys , T. A. Manteuffel , S. F. McCormick , L. N. Olson, Algebraic multigrid for higher-order finite elements, Journal of Computational Physics, v.204 Michele Benzi, Preconditioning techniques for large linear systems: a survey, Journal of Computational Physics, v.182 n.2, p.418-477, November 2002

scalability;algebraic multigrid;interpolation;unstructured meshes

351544

Observability of 3D Motion.

This paper examines the inherent difficulties in observing 3D rigid motion from image sequences. It does so without considering a particular estimator. Instead, it presents a statistical analysis of all the possible computational models which can be used for estimating 3D motion from an image sequence. These computational models are classified according to the mathematical constraints that they employ and the characteristics of the imaging sensor (restricted field of view and full field of view). Regarding the mathematical constraints, there exist two principles relating a sequence of images taken by a moving camera. One is the epipolar constraint, applied to motion fields, and the other the positive depth constraint, applied to normal flow fields. 3D motion estimation amounts to optimizing these constraints over the image. A statistical modeling of these constraints leads to functions which are studied with regard to their topographic structure, specifically as regards the errors in the 3D motion parameters at the places representing the minima of the functions. For conventional video cameras possessing a restricted field of view, the analysis shows that for algorithms in both classes which estimate all motion parameters simultaneously, the obtained solution has an error such that the projections of the translational and rotational errors on the image plane are perpendicular to each other. Furthermore, the estimated projection of the translation on the image lies on a line through the origin and the projection of the real translation. The situation is different for a camera with a full (360 degree) field of view (achieved by a panoramic sensor or by a system of conventional cameras). In this case, at the locations of the minima of the above two functions, either the translational or the rotational error becomes zero, while in the case of a restricted field of view both errors are non-zero. Although some ambiguities still remain in the full field of view case, the implication is that visual navigation tasks, such as visual servoing, involving 3D motion estimation are easier to solve by employing panoramic vision. Also, the analysis makes it possible to compare properties of algorithms that first estimate the translation and on the basis of the translational result estimate the rotation, algorithms that do the opposite, and algorithms that estimate all motion parameters simultaneously, thus providing a sound framework for the observability of 3D motion. Finally, the introduced framework points to new avenues for studying the stability of image-based servoing schemes.

Introduction Visual Servoing and Motion Estimation A broad definition of visual servoing amounts to the control of motion on the basis of image analysis. Thus, a driver adjusting the turning of the wheel by monitoring some features in the scene, a system attempting to insert a peg through a hole, a controller at some control panel perceiving a number of screens and adjusting a number of levers, a navigating system trying to find its home, all perform visual servoing tasks. Such systems must respond "appropriately" to changes in their environment; thus, on a high level, they can be described as evolving or dynamical systems, which can be represented as a function from states and control signals to new states; both the states and the control variables may be functions of time. One approach to controlling such a system is to design an observer or state estimator to obtain an estimate of the state; for example, this estimator might implement a partial visual recovery process. This estimate is used by a controller or state regulator to compute a control signal to drive the dynamical system. Ideally, observation and control are separable in the sense that if we have an optimal controller and an optimal observer then the control system that results from coupling the two is guaranteed to be optimal [13]. Different agents, i.e., dynamical systems, have different capabilities and different amounts of memory, with simple reactive systems at one end of the spectrum and highly sophisticated and flexible systems, making use of scene descriptions and reasoning processes, at the other end. In the first studies on visual control, efforts concentrated on the upper echelon of this spectrum, trying to equip systems with the capability of estimating accurate 3D motion and the shape of the environment. Assuming that this information could be acquired exactly, sensory feedback robotics was concerned with the planning and execution of the robot's activities. This was characterized as the ``look and move'' approach and the servoing approaches were "position-or-scene-based." The problem with such separation of perception from action was that both computational goals turned out to be very difficult. On the one hand, the reconstruction of 3D motion and shape is hard to achieve accurately and, in addition, many difficult calibration problems had to be addressed. On the other hand, spatial planning and motion control are very sensitive to errors in the description of the spatiotemporal environment. After this realization and with the emergence of active vision [3-6], attention turned to the lower part of the spectrum, minimizing visual processing and placing emphasis on the regulator, as opposed to the estimator. One of the main lessons learned from research on the static look and move control strategy was that there ought to be easier things to do with images than using them to compute 3D motion and shape. This led to formulations of visual servoing tasks which were such that the controller had to act on the image, i.e., to move the manipulator's joints in such a way that the scene ends up looking a particular way. In technical terms, the controller had to act in such a way that specific coordinate systems (hand, eye, scene, etc.) were put in a particular relationship with each other. Furthermore, this relationship could be realized by using directly available image measurements as feedback for the control loop. This approach is known as image-based robot servoing [16, 18, 21], and in recent years it has given very interesting research results [14, 23, 24, 28]. Most of these results deal with the computation of the image Jacobian (i.e., the differential relationship between the camera frame and the scene frame), along with the camera's intrinsic and extrinsic parameters. Regardless of the philosophical approach one adopts in visual servoing (scene-based or image-based), the essential aspects of the problem amount to the recovery of the relationship between different coordinate systems (such as the ones between camera, gripper, robot, scene, object, etc. This could be the relationship itself or a representation of the change of the relationship. Since a servoing system is in general moving (or observing moving parts), a fundamental problem is the recovery of 3D motion from an image sequence. Whether this recovery is explicit or implicit, without it the relationships between different coordinate frames cannot be maintained, and servoing tasks cannot be achieved. Thus, it is important to understand how accurately 3D motion can be observed, in order to understand how successfully servoing can be accomplished. Experience has shown that in practice 3D motion is very difficult to accurately observe, involving many ambiguities and sensitivities. Are there any regularities in which the ambiguity in 3D motion expresses itself? In trying to estimate 3D motion, do the introduced errors satisfy any constraints? As 3D rigid motion consists of the sum of a translation and a rotation, are there differences in the inherent ambiguities governing the recovery of the different components of 3D motion? In addition, we need to study this question independently of specific optimization algorithms. This is the problem studied in this paper. 2 The Approach There is a veritable cornucopia of techniques for estimating 3D motion but our approach should be algorithm independent, in order for the results to be of general use. Equivalently, our approach should encompass all the possible computational models that can be used to estimate 3D motion from image sequences. To do so, we need to classify all possible approaches to 3D motion on the basis of the input used and the mathematical constraints that are employed. As a system moves in some environment, every point in the scene has a velocity with regard to the system. The projection of these velocity vectors on the system's eye constitutes the so-called motion field. An estimate of the motion field, called optical flow, starts by first estimating the spatiotemporal derivatives of the image intensity function. These derivatives comprise the so-called normal flow which is the component of the flow along the local image gradient, i.e., normal to the local edge. A system could start 3D motion estimation using normal flow as input or it could first attempt to estimate the optical flow, though that is a very difficult problem, and subsequently 3D motion. This means that an analysis of the difficulty of 3D motion estimation must consider both inputs, optic flow fields and normal flow fields, and algorithms in the literature use one or the other input. Regarding the mathematical constraints through which 3D motion is encoded in the input image motion, extensive work in this area has established that there exist only two such constraints if no information about the scene is available. The first is the "epipolar constraint" and the second, the "positive depth constraint." The epipolar constraint ensures that corresponding points in the sequence are the projection of the same point in the scene. The positive depth constraint requires that the depth of every scene point be positive, since the scene is in front of the camera. Since knowledge of normal flow does not imply knowledge of corresponding points in the sequence, the epipolar constraint cannot be used when normal flow is available. The epipolar constraint has attracted most work. It can only be used when optic flow is available. Since many measurements are present, one develops a function that represents deviation from the epipolar constraint all over the image. A variety of approaches can be found in the literature using different metrics in representing epipolar deviation and using different techniques to seek the optimization of the resulting functions. Furthermore, there exist techniques that first estimate rotation and, on the basis of the result subsequently estimate translation [9, 36, 37]; techniques that do the opposite [1, 22, 26, 31, 33, 38, 40]; and techniques that estimate all motion parameters simultaneously [7, 8, 17, 35, 43]. The positive depth constraint, which has been used for normal flow fields, is relatively new and is employed in the so-called direct algorithms [8, 20, 27]. One has to search for the 3D motion that is consistent with the input and produces the minimum amount of negative depth. Put differently, in these approaches the function representing the amount of negative depth must be minimized. Finally, one may be able to use the "positive depth constraint" when optic flow is available, but there exist no algorithms yet in the literature implementing this principle. Looking at nature we observe that there exists a great variety of eye designs in organisms with vision. An important characteristic is the field of view. There exist systems whose vision has a restricted field of view. This is achieved by the corneal eyes of land vertebrates. Human eyes and common video cameras fall in that category and they are geometrically modeled through central projection on a plane. We refer to these eyes as planar or camera-type eyes. There exist also systems whose vision has a full 360 degree field of view. This is achieved by the compound eyes of insects or by placing camera-type eyes on opposite sides of the head, as in birds and fish. Panoramic vision is adequately modeled geometrically by projecting on a sphere using the sphere's center as the center of projection. In studying the observability of 3D motion, we investigate both restricted field of view and panoramic vision. It turns out that they have different properties regarding 3D motion estimation. In our approach we employ a statistical model to represent the constraints to derive functions representing deviation from epipolar geometry or the amount of negative depth, for both a camera-type eye and a spherical eye. All possible, meaningful algorithms for 3D motion estimation can be understood from the minimization of these functions. Thus, we perform a topographic analysis of these functions and study their global and local minima. Specifically, we are interested in the relationships between the errors in 3D motion at the points representing the minima of these surfaces. The idea behind this is that in practical situations any estimation procedure is hampered by errors and usually local minima of the functions to be minimized are found as solutions. 3 Problem Statement 3.1 Prerequisites We use the standard conventions for expressing image motion measurements for a monocular observer moving rigidly in a static environment. We describe the motion of the observer by its translational velocity and its rotational velocity with respect to a coordinate system OXY Z fixed to the nodal point of the camera. Each scene point P with coordinates has a velocity - R relative to the camera. Projecting - R onto a retina of a given shape gives the image motion field. If the image is formed on a plane (Figure 1a) orthogonal to the Z axis at distance f (focal length) from the nodal point, then an image point f) and its corresponding scene point R are related by R, where z 0 is a unit vector in the direction of the Z axis. The motion field becomes (R f z 0 \Theta (r \Theta (! \Theta Z R r Y Z x y f O z 0 r R f r O r (a) (b) Figure 1: Image formation on the plane (a) and on the sphere (b). The system moves with a rigid motion with translational velocity t and rotational velocity !. Scene points R project onto image points r and the 3D velocity R of a scene point is observed in the image as image velocity - r. with representing the depth. If the image is formed on a sphere of radius f (Figure 1b) having the center of projection as its origin, the image r of any point R is jRj , with R being the norm of R (the range), and the image motion is R u tr (t) The motion field is the sum of two components, one, u tr , due to translation and the other, u rot , due to rotation. The depth Z or range R of a scene point is inversely proportional to the translational flow, while the rotational flow is independent of the scene in view. As can be seen from (1) and (2), the effects of translation and scene depth cannot be separated, so only the direction of translation, t=jtj, can be computed. We can thus choose the length of t; throughout the following analysis f is set to 1, and the length of t is assumed to be 1 on the sphere and the Z-component of t to be 1 on the plane. The problem of 3D motion estimation then amounts to finding the scaled vector t and the vector ! from a representation of the motion field. In the following, to make the analysis easier, for the camera-type eye we will employ a non-vector notation. Then, the first two coordinates of r denote the image point in the Cartesian system oxy, with oxkOX, oykOY and o the intersection of OZ with the image plane. Denote by the image point representing the direction of the translation vector t (referred to as the Focus of Expansion (FOE) or Focus of Contraction (FOC) depending on whether W is positive or negative). Equations (1) become then the well known equations expressing the flow measurement fly where v tr (x;y) are the translational and rotational flow components, respectively. Regarding the value of the normal flow, if n is a unit vector at an image point denoting the orientation of the gradient at that point, the normal flow v n satisfies r Finally, the following convention is employed throughout the paper. We use letters with hat signs to represent estimated quantities, unmarked letters to represent the actual quantities and the subscript "ffl" to denote errors, where the error quantity is defined as the actual quantity minus the estimated one. For example, u rot (!) represents actual rotational flow, u rot ( - estimated rotational flow, t ffl the translational error vector, x ff, etc. 3.2 The model The classic approach to 3D motion estimation is to minimize the deviation from the epipolar constraint. This constraint is obtained by eliminating depth (or range) from equation (1) (or (2)). For both planar and spherical eyes it is r +! \Theta r) = 0: (5) Equating image motion with optic flow, this constraint allows for the derivation of 3D rigid motion on the basis of optic flow measurements. One is interested in the estimates of translation - t and rotation - which best satisfy the epipolar constraint at every point r according to some criterion of deviation. The Euclidean norm is usually used, leading to the minimization [25, 34] of the function 1 Z Z image \Theta r)dr: (6) On the other hand, if normal flow is given, the vector equations (1) and (2) cannot be used directly. The only constraint is scalar equation (4), along with the inequality Z ? 0 which states that since the surface in view is in front of the eye its depth must be positive. Substituting (1) or (2) into (4) and solving for the estimated depth Z or range - R, we obtain for a given estimate - t; - ! at each point r: Z(or - If the numerator and denominator of (7) have opposite signs, negative depth is computed. Thus, to utilize the positivity constraint one must search for the motion - t; - that produces a minimum number of negative depth estimates. Formally, if r is an image point, define the indicator function I nd ae Then estimation of 3D motion from normal flow amounts to minimizing [19, 20, 27] the function Z Z image I nd (r)dr: (8) Expressing - r in terms of the real motion from (1) and (2), functions (6) and (8) can be expressed in terms of the actual and estimated motion parameters t, !, - t and - (or, equivalently, the actual motion parameters t; ! and the errors t !) and the depth Z (or range R) of the viewed scene. To conduct any analysis, a model for the scene is needed. We are interested in the statistically expected values of the motion estimates resulting from all possible scenes. Thus, as our probabilistic model we assume that the depth values of the scene are uniformly distributed between two arbitrary values Z min (or R min ) and Zmax (or Rmax For the minimization of negative depth values, we further assume that the directions in which flow measurements Because t \Theta r introduces the sine of the angle between t and r, the minimization prefers vectors t close to the center of gravity of the points r. This bias has been recognized [40] and alternatives have been proposed that reduce this bias, but without eliminating the confusion between rotation and translation. are made are uniformly distributed in every direction for every depth. Parameterizing n by /, the angle between n and the x axis, we thus obtain the following two functions: Z=Zmin Z Z=Zmin nd dZ d/; (10) measuring deviation from the epipolar constraint and the amount of negative depth, respectively. Functions and (10) are five-dimensional surfaces in t ffl the errors in the motion parameters. Finally, since for the scene in view we employ a probabilistic model, the results are of a statistical nature, that is, the geometric constraints between ffl at the minima of (9) and (10) that we shall uncover should be interpreted as being likely to occur. 3.3 Negative depth and depth distortion This section contains a few technical prerequisites needed for the study of negative depth minimization and a geometric observation that show the relationship of epipolar minimization to minimization of negative depth. Equation (7) shows the estimated depth - Z (or range - R) given normal flow - r \Delta n and estimates - t; - ! of the motion. It can be further written as: Z(or - with D in the case of noiseless motion measurements (that is, - of the form Equation (11) shows how wrong depth estimates are produced due to inaccurate 3D motion values. The distortion D, multiplies the real depth value to produce the estimate. Equation (12), for a fixed value of D and describes a surface in (r; Z) (or (r; R)) space which is called an iso-distortion surface. Any such surface is to be understood as the locus of points in space which are distorted in depth by the same multiplicative factor if the image measurements are in direction n. If we fix n and vary D, the iso-distortion surfaces of the resulting family change continuously as D varies. Thus all scene points giving rise to negative depth estimates lie between the 0 and \Gamma1 distortion surfaces. The integral over all points (for all directions) giving rise to negative depth estimates we call the negative depth volume. In Section 5 we will make use of the iso-distortion surfaces and the negative depth volume to study in a geometric way function (10) resulting from minimization of the negative depth values. Let us now examine the two different minimizations from a geometric perspective. When deriving the deviation from the epipolar constraint we consider integration over all points and depth values of the expression or, equivalently, denotes the vector perpendicular to u tr ( - t) in the plane of the image coordinates. When deriving the number of negative depth values we consider integration over all points and depth values of the angle ff between the two vectors An illustration is given in Figure 2. The two measures considered in the two different minimizations are clearly related to each other. In the case of negative depth we consider the angle ff, whereas in the case of the epipolar constraint we consider the squared distance a 2 , which amounts to (sin ffku tr ( - t)k The major difference in using the two measures arises for angles ff ? 90 ffi . The measure for negative depth is monotonic in ff, but the measure for the deviation from the epipolar constraint is not, because it does not differentiate between depth estimates of positive and negative value. The fact that the computed depth has to be positive has not been considered in past approaches employing minimization of the epipolar constraint. As this fact provides an additional constraint which could be utilized in a negative depth a a Figure 2: Different measures used in minimization constraints. (a) 6 ff in the minimization of negative depth values from projections of motion fields. (b) a 2 in the epipolar constraint. the future, maybe in conjunction with the epipolar constraint, we are interested in the influence of this constraint on 3D motion estimation as well. Thus, in addition to minimization of the functions (9) and (10) discussed above, we study the minimization of a third function. This function amounts to the number of values giving rise to negative depth when full flow measurements (optical flow) are assumed and the analysis is presented in Appendix B. 3.4 A model for the noise in the flow In the analysis on the plane we will also consider noise in the image measurements, that is, we will consider the flow values of the form (u The choice of the noise model is motivated by the following considerations: First, the noise should be such that no specific directions are favored. Second, assuming that noise is both additive and multiplicative, there should be a dependence between noise and depth, because the translational flow component is proportional to the inverse depth. Therefore we define noise using two stochastic variables Z with the first and second moments being and all stochastic variables being independent of each other, independent of image position, and independent of the depth. 3.5 Overview of the paper, summary of results and related work Our approach expresses functions (9) and (10) in terms of t, !, t ffl and ! ffl and finds the conditions that t ffl ffl satisfy at local minima which represent solutions of the different estimation algorithms. Procedures for estimating 3D motion can be classified into those estimating either the translation or rotation as a first step and the remaining component (that is, the rotation or translation) as a second step, and those estimating all components simultaneously. Procedures of the former kind result when systems utilize inertial sensors which provide them with estimates of one of the components of the motion, or when two-step motion estimation algorithms are used. Thus, three cases need to be studied: the case were no prior information about 3D motion is available and the cases where an estimate of translation or rotation is available with some error. Imagine that somehow the rotation has been estimated, with an error ! ffl . Then our functions become two-dimensional in the variables t ffl and represent the space of translational error parameters corresponding to a fixed rotational error. Similarly, given a translational error t ffl , the functions become three-dimensional in the variables ! ffl and represent the space of rotational errors corresponding to a fixed translational error. To study the general case, one needs to consider the lowest valleys of the functions in 2D subspaces which pass through 0. In the image processing literature, such local are often referred to as ravine lines or courses. Each of the three cases is studied for four optimizations: epipolar minimization for the sphere and the plane (full field of view and restricted field of view vision) and minimization of negative depth for the sphere and the plane. Thus, there are twelve (four times three) cases, but since the effects of rotation on the image are independent of depth, it makes no sense to perform minimization of negative depth assuming an estimate of translation is available. Thus, we are left with ten different cases which are studied below. These ten cases represent all the possible, meaningful motion estimation procedures. The analysis shows that: 1. In the case of a camera-type eye (restricted field of view) for algorithms in both classes which estimate all motion parameters simultaneously (i.e., there is no prior information), the obtained solution will have an error in the translation and (ff ffl ; in the rotation such that x0 ffl and means that the projections of the translational and rotational errors on the image are perpendicular to each other and that the rotation around the Z axis has the least ambiguity. We refer to this constraint as the "orthogonality constraint." In addition, the estimated translation (-x lie on a line passing from the origin of the image and the real translation . We refer to this second constraint as the "line constraint." Similar results are achieved for the case where translation is estimated first and on that basis rotation is subsequently found, while the case where rotation is first estimated and subsequently translation provides different results. The work is performed both in the absence of error in the image measurements-in which case it becomes a geometric analysis of the inherent confusion between rotation and translation-and in the case where the image measurements are corrupted by noise that satisfies the model from Section 3.4. In this case, we derive the expected values of the local and global minima. As will be shown, the noise does not alter the local minima, and the global minima fall within the valleys of the function without noise. Thus, the noise does not alter the functions' overall structure. 2. In the case of panoramic vision, for algorithms in both classes which estimate all motion parameters si- multaneously, the obtained solution will have no error in the translation and the rotational error will be perpendicular to the translation. In addition, for the case of epipolar minimization from optic flow, given a translational error t ffl , the obtained solution will have no error in the rotation (! while for the case of negative depth minimization from normal flow, given a rotational error the obtained solution will have no error in the translation. In other cases ambiguities remain. In the spherical eye case the analysis is simply performed for noiseless flow. A large number of error analyses have been carried out [2, 11, 12, 15, 29, 39, 41, 42] in the past for a camera-type eye, while there is no published research of this kind for the full field of view case. None of the existing studies, however, has attempted a topographic characterization of the function to be minimized for the purpose of analyzing different motion techniques. All the studies consider optical flow or correspondence as image measurements and investigate minimizations based on the epipolar constraint. Often, restrictive assumptions about the structure of the scene or the estimator have been made, but the main results obtained are in accordance with our findings. In particular, the following two results already occur in the literature. (a) Translation along the x axis can be easily confounded with rotation around the y axis, and translation along the y axis can be easily confounded with rotation around the x axis, for small fields of view and insufficient depth variation. This fact has long been known from experimental observation, and has been proved for planar scene structures and unbiased estimators [10]. The orthogonality constraint found here confirms these findings, and imposes even more restrictive constraints. It shows, in addition, that the x-translation and y-rotation, and y-translation and x-rotation, are not decoupled. Furthermore, we have found that rotation around the Z axis can be most easily distinguished from the other motion components. (b) Maybank [32, 33] and Jepson and Heeger [29] established before the line constraint, but under much more restrictive assumptions. In particular, they showed that for a small field of view, a translation far away from the image center and an irregular surface, the function in (9) has its minima along a line in the space of translation directions which passes through the true translation and the viewing direction. Under fixation the viewing direction becomes the image center. 4 Epipolar Minimization: Camera-type Eye the field of view is small, the quadratic terms in the image coordinates are very small relative to the linear and constant terms, and are therefore ignored. All the computations are carried out with the symbolic algebraic computation software Maple, and, for abbreviation, intermediate results are not given. First the case of noise-free flow is studied. Considering a circular aperture of radius e, setting the focal length the function in Z=Zmin e Z Z Z r oe dr dOE dZ where (r; OE) are polar coordinates sin OE). Performing the integration, one obtains y 'Z min (a) Assume that the translation has been estimated with a certain error t 0). Then the relationship among the errors in 3D motion at the minima of (14) is obtained from the first-order conditions @E1 = 0, which yield It follows that ff ffl =fi (b) Assuming that rotation has been estimated with an error (ff among the errors is obtained from @E1 In this case, the relationship is very elaborate and the translational error depends on all the other parameters-that is, the rotational error, the actual translation, the image size and the depth interval. See Appendix A. (c) In the general case, we need to study the subspaces in which E 1 changes least at its absolute minimum; that is, we are interested in the direction of the smallest second derivative at 0, the point where the motion errors are zero. To find this direction, we compute the Hessian at 0, that is the matrix of the second derivatives of respect to the five motion error parameters, and compute the eigenvector corresponding to the smallest eigenvalue. The scaled components of this vector amount to As can be seen, for points defined by this direction, the translational and rotational errors are characterized by the "orthogonality constraint" ff ffl =fi and by the "line constraint" x 0 =y Next we consider noise in the flow measurements. We model the noise (N x ; N y ) as defined in Section 3.4 and derive E(E ep ), the expected value of E ep , which amounts to Z=Zmin Z e Z ae' r oe dr dOE dZ (16) and thus 'Z min (a) If we fix x and y 0 ffl and solve for we obtain the same relationship as for noiseless flow described in (15). This shows that the noise does not alter the expected behavior of techniques which in a first step minimize the translation. (b) If we fix ff ffl , fi ffl , and fl ffl , and solve for @y we obtain as before a complicated relationship between the translational error, the actual translation, and the rotational error. (c) To analyze the behavior of techniques which minimize for all 3D motion parameters, we study the global minimum of E(E). From minimization with regard to the rotational parameters, we obtained (15) (the orthogonality constraint and Substituting (15) into (16) and solving for @y we get, in addition, \GammaZ Thus the absolute minimum of E(E ep ) is to be found in the direction of smallest increase in E 1 , and is described by the constraint by the orthogonality constraint, and by the line constraint. 5 Minimization of Negative Depth Volume: Camera-type Eye In the following analysis we study the function describing the negative depth values geometrically, by means of the negative depth volumes, that is, the points corresponding to negative depth distortion as defined in Section 3.3. This allows us to incrementally derive properties of the function without considering it with respect to all its parameters at once. For simplicity, we assume that the FOE and the estimated FOE are inside the image, and we do not consider the exact effects resulting from volumes of negative depth in different directions being outside the field of view. We first concentrate on the noiseless case. If, as before, we ignore terms quadratic in the image coordinates, the 0 distortion surface (from equation (11)) becomes and the \Gamma1 distortion surface takes the form The flow directions (n x alternatively be written as (cos /; sin /), with / 2 [0; -] denoting the angle between and the x axis. To simplify the visualization of the volumes of negative depth in different directions, we perform the following coordinate transformation to align the flow direction with the x axis: for every / we rotate the coordinate system by angle /, to obtain the new coordinates [x cos / sin / sin / cos / Equations (17) and (18) thus become (a) To investigate techniques which, as a first step, estimate the rotation, we first study the case of fl then extend the analysis to the general case of fl ffl 6= 0. If the volume of negative depth values for every direction / lies between the surfaces The equation describes a plane parallel to the y 0 Z plane at distance - x 0 0 from the origin, and describes a plane parallel to the y 0 axis of slope 1 , which intersects the x 0 y 0 plane in the . Thus we obtain a wedge-shaped volume parallel to the y 0 axis. Figure 3 illustrates the volume through a slice parallel to the x 0 Z plane. The scene in view extends between the depth values Z min and Zmax . We denote by A/ the area of the cross section parallel to the x 0 Z plane through the negative depth volume in direction /. As can be seen from Figure 3, to obtain the minimum, - 0 has to lie between x then Z=Zmin which amounts to If we fix fi 0 ffl and solve for x 0, we obtain -that is, the 0 distortion surface has to intersect the \Gamma1 distortion surface in the middle of the depth interval in the plane . depends only on the depth interval, and thus is independent of the direction, /. Therefore, the negative depth volume is minimized if (20) holds for every direction. Since fi ffl sin /y 0 ffl , we obtain x0 ffl If the \Gamma1 distortion surface becomes This surface can be most easily understood by slicing it with planes parallel to the x 0 y 0 plane. At every depth value Z, we obtain a line of slope \Gamma1 which intersects the x 0 axis at x Figure 4a). For any given Z the slopes of the lines in different directions are the same. An illustration of the volume of negative depth is given in Figure 4b. Z min Z Figure 3: Slice parallel to the x 0 Z plane through the volume of negative estimated depth for a single direction. z4002000 volume volume (a) (b) Figure 4: (a) Slices parallel to the x 0 y 0 plane through the 0 distortion surface (C 0 ) and the \Gamma1 distortion surface at depth values 0: volume of negative depth values between the 0 and \Gamma1 distortion surfaces. Let us express the value found for - x 0 0 in the case of fl I . In order to derive the position of - 0 that minimizes the negative depth volume for the general case of fl ffl 6= 0, we study the change of volume as 0 changes from x Referring to Figure 5, it can be seen that for any depth value Z, a change in the position of - 0 to - assuming Z 6= 0, causes the area of negative depth values to change by A c , where A and y 0 1 and y 0 2 denote the y 0 coordinates of the intersection point of the \Gamma1 distortion contour at depth Z with the 0 distortion contours x I and x d. These coordinates are and y 0 Therefore Z The change V c in negative depth volume for any direction is given by Zmin A c dZ which amounts to Substituting for Z I = Zmin+Zmax , it can be verified that in order for V c to be negative, we must have \Gammasgn(d). We are interested in the d which minimizes V c . By solving @d we obtain Thus depends only on the depth interval, the total negative depth volume is obtained if the volume in every direction is minimized. Therefore, for any rotational error (ff independent of fl ffl , we have the orthogonality constraint: A comment on the finiteness of the image is necessary here. The values A c and V c have been derived for an infinitely large image. If fl ffl is very small or some of the depth values Z in the interval [Z min are small, the coordinates of the intersections y 0 1 and y 0 2 do not lie inside the image. The value of A c can be at most the length of the image times d. Since the slope of the \Gamma1 distortion contour for a given Z is the same for all directions, this will have very little effect on the relationship between the directions of the translational and rotational motion errors. It has an effect, however, on the relative values of the motion errors. Only if the intersections are inside the image can (22) be used to describe the value of x 0 as a function of fi 0 ffl and the interval of depth values of the scene in view. (b) Next we investigate techniques which minimize both the translation and rotation at once. First consider a certain translational error and change the value of fl ffl . An increase in fl ffl decreases the slope of the \Gamma1 distortion surfaces, and thus, as can be inferred from Figure 4a, the area of negative depth values for every direction / and every depth Z increases. Thus In addition, from above, we know that x0 ffl . The exact relationship of fi 0 ffl to x 0 is characterized by the locations of local minima of the function A/ , which in the image processing literature are often referred to as "courses." (These are not the courses in the topographical sense [30].) To be more precise, we are interested in the local minima of A/ in the direction corresponding to the largest second derivative. We compute the largest eigenvalue, - 1 , of the Hessian, H, of A/ , that is the matrix of the second derivatives of A/ with respect to x 0 and fi 0 ffl , which amounts to and obtain . The corresponding eigenvector, e 1 , is (fi 0 Solving for we obtain Last in an analysis of the noiseless case, we consider the effects due to the finiteness of the aperture. As before, we consider a circular aperture. We assume a certain amount of translational error , and we seek the direction of translational error that results in the smallest negative depth volume. Independent of the direction of translation, (23) describes the relationship of x and fi ffl for the smallest negative depth volume. Substituting (23) into (19), we obtain the cross-sections through the negative depth volume as a function of x 0 and the depth interval. The negative depth volume for every direction / amounts to A/ l / , where l / denotes the average extent of the wedge-shaped negative depth volume in direction /. The total negative depth volume is minimized if Considering a circular aperture, this minimization is achieved if the largest A/ corresponds to the smallest extent l / and the smallest A/ corresponds to the largest l / . This happens when the line constraint holds, that is, x0 y0 (see Figure 6). It remains to be shown that noise in the flow measurements does not alter the qualitative characteristics of the negative depth volume and thus the results obtained. First we analyze the orthogonality constraint. The analysis is carried out without considering the size of the aperture; as will be shown, this analysis leads to the orthogonality constraint. First, let Ignoring the image size, we are interested in E(V ), the expected value of the integral of the cross sections A/ , which amounts to Z=Zmin Z or Z=Zmin dZA We approximate the expectation of the integrand by performing a Taylor expansion at 0 up to second order, which gives x y Figure 5: A change of - x 0 0 to - causes the area of negative depth values A c to increase by A 1 and to decrease by A 2 . This change amounts to A OE A y 2 l y 1 l y 2 Figure Cross-sectional view of the wedge-shaped negative depth volumes in a circular aperture. The minimization of the negative depth volume for a given amount of translational error occurs when x0 y0 . and l / i denote the areas of the cross sections and average extents respectively, for two angles / 1 and / 2 . The two circles bounding A/ i are given by the equations ffl Z min and x ffl Zmax . We are interested in the angle - between the translational and rotational error which minimizes the negative depth volume. If we align the translational error with the x axis, that is, y 0, and if we express the rotational error as sin -), we obtain Z=Zmin \Delta0 Solving for @E(V ) ) to be perpendicular to (ff ffl ; fi ffl ), if minimizations considered. Next, we allow fl ffl to be different from zero. For the case of a fixed translational error, again, the volume increases as fl ffl increases, and thus the smallest negative depth volume occurs for For the case of a fixed rotational error we have to extend the previous analysis (studying the change of volume when changing the estimated translation) to noisy motion fields: The \Gamma1 distortion surface becomes x and the 0 distortion surfaces remain the same. Therefore, VC becomes \Gammasgn and E(VC ) takes the same form as VC in (21). We thus obtain the orthogonality constraint for Finally, we take into account the limited extent of a circular aperture for the case of global minimization. As noise does not change the structure of the iso-distortion surfaces, as shown in Figure 6, in the presence of noise, too, the smallest negative depth volume is obtained if the FOE and the estimated FOE lie on a line passing through the image center. This proves the orthogonality constraint for the full model as well as the line constraint. The global minimum of the negative depth volume is thus described by the constraint the orthogonality constraint and the line constraint. 6 Epipolar Minimization: Spherical Eye The function representing deviation from the epipolar constraint on the sphere takes the simple form Rmin Z Z sphere ae' r \Theta (r \Theta t) dAdR where A refers to a surface element. Due to the sphere's symmetry, for each point r on the sphere, there exists a point with coordinates \Gammar. Since u tr when the integrand is expanded the product terms integrated over the sphere vanish. Thus Rmin Z Z sphere t \Theta - t dAdR (a) Assuming that translation - t has been estimated, the ! ffl that minimizes E ep is since the resulting function is non-negative quadratic in ! ffl (minimum at zero). The difference between sphere and plane is already clear. In the spherical case, as shown here, if an error in the translation is made we do not need to compensate for it by making an error in the rotation (! while in the planar case we need to compensate to ensure that the orthogonality constraint is satisfied! (b) Assuming that rotation has been estimated with an error ! ffl , what is the translation - t that minimizes Since R is uniformly distributed, integrating over R does not alter the form of the error in the optimization. consists of the sum of two terms: Z Z sphere t \Theta - t dA and Z Z sphere are multiplicative factors depending only on R min and Rmax . For angles between t; - t and - t; ! ffl in the range of 0 to -=2, K and L are monotonic functions. K attains its minimum when Fix the distance between t and - t leading to a certain value K, and change the position of - t. L takes its minimum as follows from the cosine theorem. Thus E ep achieves its minimum when - t lies on the great circle passing through t and ! ffl , with the exact position depending on j! ffl j and the scene in view. (c) For the general case where no information about rotation or translation is available, we study the subspaces changes the least at its absolute minimum, i.e., we are again interested in the direction of the smallest second derivative at 0. For points defined by this direction we calculate, using Maple, t. 7 Minimizing Negative Depth Volume on the Sphere (a) Assuming that the rotation has been estimated with an error ! ffl , what is the optimal translation - t that minimizes the negative depth volume? Since the motion field along different orientations n is considered, a parameterization is needed to express all possible orientations on the sphere. This is achieved by selecting an arbitrary vector s; then, at each point r of the sphere, s\Thetar ks\Thetark defines a direction in the tangent plane. As s moves along half a circle, s\Thetar ks\Thetark takes on every possible orientation (with the exception of the points r lying on the great circle of s). Let us pick ! ffl perpendicular to s (s We are interested in the points in space with estimated negative range values - R. ks\Thetark and s the estimated range - R amounts to - . - where sgn(x) provides the sign of x. This constraint divides the surface of the sphere into four areas, I to IV, whose locations are defined by the signs of the functions ( - t \Theta s) \Delta r, (t \Theta s) \Delta r and (! ffl \Delta r)(s \Delta r), as shown in Figure 7. s I 0 IV III I II IV s IIV I II IV III area location constraint on R I sgn(t \Theta s) \Delta Figure 7: Classification of image points according to constraints on R. The four areas are marked by different colors. The textured parts (parallel lines) in areas I and III denote the image points for which negative depth values exist if the scene is bounded. The two hemispheres correspond to the front of the sphere and the back of the sphere, both as seen from the front of the sphere. For any direction n a volume of negative range values is obtained consisting of the volumes above areas I, II and III. Areas II and III cover the same amount of area between the great circles (t \Theta s) \Delta and area I covers a hemisphere minus the area between (t \Theta s) \Delta If the scene in view is unbounded, that is, R 2 [0; +1], there is for every r a range of values above areas I and III which result in negative depth estimates; in area I the volume at each point r is bounded from below by (! ffl \Deltar)(s\Deltar) , and in area III it is bounded from above by . If there exist lower and upper bounds R min and Rmax in the scene, we obtain two additional curves C min and Cmax with C and we obtain negative depth values in area I only between Cmax in area III only between C min and (! ffl \Theta r)(s \Theta r) = 0. We are given ! ffl and t, and we are interested in the - t which minimizes the negative range volume. For any s the corresponding negative range volume becomes smallest if - t is on the great circle through t and s, that is, (t \Theta s) \Delta - as will be shown next. Let us consider a - t such that (t \Theta s) \Delta - t 6= 0 and let us change - t so that (t \Theta s) \Delta - changes, the area of type II becomes an area of type IV and the area of type III becomes an area of type I. The negative depth volume is changed as follows: It is decreased by the spaces above area II and area III, and it is increased by the space above area I (which changed from type III to type I). Clearly, the decrease is larger than the increase, which implies that the smallest volume is obtained for s; t; - t lying on a great circle. Since this is true for any s, the minimum negative depth volume is attained for t. 2 (b) Next, assume that no prior knowledge about the 3D motion is available. We want to know for which configurations of - t and ! ffl the negative depth values change the least in the neighborhood of the absolute minimum, that is, at From the analysis above, it is known that for any ! ffl 6= 0, t. Next, we show that ! ffl is indeed different from zero: Take t 6= - t on the great circle of s and let ! ffl , as before, be perpendicular to s. the curves Cmax and C min can be expressed as C 6 (t;s) 0, where sin denotes the angle between vectors t and s. These curves consist of the great circle and the circle sin 6 (t;s) parallel to the great circle (s Figure 8). If sin 6 (t;s) this circle disappears. s I IV Figure 8: Configuration for t and - t on the great circle of s and ! ffl perpendicular to s. The textured part of area I denotes image points for which negative depth values exist if the scene is bounded. Consider next two flow directions defined by vectors s 1 and s 2 with and - t. For every point r 1 in area III defined by s 1 there exists a point r 2 in area I defined by s 2 such that the negative estimated ranges above r 1 and r 2 add up to Rmax \Gamma R min . Thus the volume of negative range obtained from s 1 and s 2 amounts to the area of the sphere times (Rmax \Gamma R min ) (area II of s 1 contributes a hemisphere; area III of area I of s 2 together contribute a hemisphere). The total negative range volume can be decomposed into A word of caution about the parameterization used for directions ks\Thetark is needed. It does not treat all orientations equally (as s varies along a great circle with constant speed, s \Theta r accelerates and decelerates). Thus to obtain a uniform distribution, normalization is necessary. The normalization factors, however, do not affect the previous proof, due to symmetry. three components: a component V 1 originating from the set of s between t and - t, a component V 2 originating from the set of s symmetric in t to the set in V 1 , and a component V 3 corresponding to the remaining s, which consists of range values above areas of type I only. If for all s in V 3 , sin 6 (t;s) zero. Thus for all Rmax , the negative range volume is equally large and amounts to the area on the sphere times (Rmax \Gamma R min ) times takes on values different from zero. This shows that for any t ffl 6= 0, there exist vectors ! ffl 6= 0 which give rise to the same negative depth volume as However, for any such ! ffl 6= 0 this volume is larger than the volume obtained by setting follows that t. From Figure 7, it can furthermore be deduced that for a given ! ffl the negative depth volume, which for only lies above areas of type I, decreases as t moves along a great circle away from ! ffl , as the areas between C min and Cmax and between C min and (t \Theta s) \Delta decrease. This proves that in addition to Conclusions The preceding results constitute a geometric statistical investigation of the observability of 3D motion from images. On their basis, a number of striking conclusions can be achieved. First, they clearly demonstrate the advantages of panoramic vision in the process of 3D motion estimation. Table 1 lists the eight out of ten cases which lead to clearly defined error configurations. It shows that 3D motion can be estimated more accurately with spherical eyes. Depending on the estimation procedure used-and systems might use different procedures for different tasks-either the translation or the rotation can be estimated very accurately. For planar eyes, this is not the case, as for all possible procedures there exists confusion between the translation and rotation. The error configurations also allow systems with inertial sensors to use more efficient estimation procedures. If a system utilizes a gyrosensor which provides an approximate estimate of its rotation, it can employ a simple algorithm based on the positive depth constraint for only translational motion fields to derive its translation and obtain a very accurate estimate. Such algorithms are much easier to implement than algorithms designed for completely unknown rigid motions, as they amount to searches in 2D as opposed to 5D spaces [19]. Similarly, there exist computational advantages for systems with translational inertial sensors in estimating the remaining unknown rotation. Since the positive depth constraint turns out to be very powerful and since epipolar minimization does not consider depth positivity, an interesting research question that arises for the future is how to couple the epipolar constraint with the positive depth constraint. We attempted a first investigation into this problem through a study of negative depth on the basis of optic flow for the plane in Appendix B, which gave very interesting results. Specifically, we found that estimating all motion parameters simultaneously by minimizing negative depth from optic flow provides a solution with no error in the translation. However, the rotation cannot be decoupled from the translation, which makes it clear that for cameras with restricted fields of view the problem of rotation/translation confusion cannot be escaped. Camera-type eyes are found in nature in systems that walk and perform sophisticated manipulation because such systems have a need for very accurate segmentation and shape estimation and thus high resolution in a limited field of view. Panoramic vision, either through compound eyes or a pair of camera-type eyes positioned on opposite sides of the head is usually found in flying systems which have the obvious need for a larger field of view but also rely on accurate 3D motion estimation as they always move in an unconstrained way. When we face the task of equipping robots with visual sensors, we do not have to necessarily copy nature, and we also do not have to necessarily use what is commercially available. Instead, we could construct new, powerful eyes by taking advantage of both the panoramic vision of flying systems and the high-resolution vision of primates. An eye like the one in Figure 9, assembled from a few video cameras arranged on the surface of a sphere, can easily estimate 3D motion since, while it is moving, it is sampling a spherical motion field! Such an eye not only has panoramic properties, allowing very accurate determination of the transformations relating multiple views, but it has the unexpected benefit of making it easy to estimate image motion with high accuracy. Any two cameras with overlapping fields of view also provide high-resolution stereo vision, and this collection of stereo systems makes it possible to locate a large number of depth discontinuities. Given scene discontinuities, image motion can be estimated very accurately. As a consequence, having accurate 3D motion Table 1: Summary of results I II Spherical Eye Camera-type Eye Epipolar minimization, given optic flow (a) Given a translational error ffl , the rotational error (b) Without any prior infor- mation, (a) For a fixed translational error ), the rotational error the form (b) Without any a priori information about the mo- tion, the errors satisfy Minimization of negative depth volume, given normal flow (a) Given a rotational error ffl , the translational error (b) Without any prior infor- mation, (a) Given a rotational error, the translational error is of the form \Gammax (b) Without any error infor- mation, the errors satisfy and image motion, the eye in Figure 9 is very well suited to developing accurate models of the world necessary for many robotic/servoing applications. Figure 9: A compound-like eye composed of conventional video cameras, arranged on a sphere and looking outward. --R Determining 3D motion and structure from optical flow generated by several moving objects. Inherent ambiguities in recovering 3-D motion and structure from a noisy flow field Active vision. Active perception. Principles of animate vision. Rigid body motion from depth and optical flow. A Computational Approach to Visual Motion Perception. Estimating 3-D egomotion from perspective image sequences On the Error Sensitivity in the Recovery of Object Descriptions. Analytical results on error sensitivity of motion estimation from two views. Understanding noise sensitivity in structure from motion. Planning and Control. Simultaneous robot-world and hand-eye calibration Robustness of correspondence-based structure from motion A new approach to visual servoing in robotics. Motion and structure from motion from point and line matches. Direct perception of three-dimensional motion from patterns of visual motion Qualitative egomotion. Subspace methods for recovering rigid motion I: Algorithm and implemen- tation Visual guided object grasping. Robot Vision. Relative orientation. A tutorial on visual servo control. Subspace methods for recovering rigid motion II: Theory. The interpretation of a moving retinal image. Algorithm for analysing optical flow based on the least-squares method A Theoretical Study of Optical Flow. Theory of Reconstruction from Image Motion. Estimation of three-dimensional motion of rigid objects from noisy observations Egomotion and relative depth map from optical flow. Determining instantaneous direction of motion from optical flow generated by a curvilinear moving observer. Processing differential image motion. Optimal computing of structure from motion using point correspondence. Optimal motion estimation. Understanding noise: The critical role of motion error in scene reconstruction. Statistical analysis of inherent ambiguities in recovering 3-D motion from a noisy flow field --TR Algorithm for analysing optical flow based on the least-squares method Inherent Ambiguities in Recovering 3-D Motion and Structure from a Noisy Flow Field Estimating 3D Egomotion from Perspective Image Sequence Relative orientation Estimation Three-Dimensional Motion of Rigid Objects from Noisy Observations Analytical results on error sensitivity of motion estimation from two views Planning and control Subspace methods for recovering rigid motion I Statistical Analysis of Inherent Ambiguities in Recovering 3-D Motion from a Noisy Flow Field Principles of animate vision Two-plus-one-dimensional differential geometry Qualitative egomotion Robot Vision --CTR John Oliensis, The least-squares error for structure from infinitesimal motion, International Journal of Computer Vision, v.61 n.3, p.259-299, February/March 2005 Tao Xiang , Loong-Fah Cheong, Understanding the Behavior of SFM Algorithms: A Geometric Approach, International Journal of Computer Vision, v.51 n.2, p.111-137, February Jan Neumann , Cornelia Fermller , Yiannis Aloimonos, A hierarchy of cameras for 3D photography, Computer Vision and Image Understanding, v.96 n.3, p.274-293, December 2004 Abhijit S. Ogale , Cornelia Fermuller , Yiannis Aloimonos, Motion Segmentation Using Occlusions, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.27 n.6, p.988-992, June 2005

positive depth constraint;error analysis;epipolar constraint;3D motion estimation

351545

First-Order System Least-Squares for the Helmholtz Equation.

This paper develops a multilevel least-squares approach for the numerical solution of the complex scalar exterior Helmholtz equation. This second-order equation is first recast into an equivalent first-order system by introducing several "field" variables. A combination of scaled L2 and H-1 norms is then applied to the residual of this system to create a least-squares functional. It is shown that, in an appropriate Hilbert space, the homogeneous part of this functional is equivalent to a squared graph norm, that is, a product norm on the space of individual variables. This equivalence to a norm that decouples the variables means that standard finite element discretization techniques and standard multigrid solvers can be applied to obtain optimal performance. However, this equivalence is not uniform in the wavenumber k, which can signal degrading performance of the numerical solution process as k increases. To counter this difficulty, we obtain a result that characterizes the error components causing performance degradation. We do this by defining a finite-dimensional subspace of these components on whose orthogonal complement k-uniform equivalence is proved for this functional and an analogous functional that is based only on L2 norms. This subspace equivalence motivates a nonstandard multigrid method that attempts to achieve optimal convergence uniformly in k. We report on numerical experiments that empirically confirm k-uniform optimal performance of this multigrid solver. We also report on tests of the error in our discretization that seem to confirm optimal accuracy that is free of the so-called pollution effect.

Introduction . The scalar Helmholtz equation with exterior radiation boundary conditions describes a variety of wave propagation phenomena. One such phenomenon is the electromagnetic scattering of time-harmonic waves, which, since the advent of stealth technology, has helped propel an interest in e#cient numerical solution techniques. However, such boundary value problems are challenging because they are both indefinite and non-self-adjoint. Hence, standard numerical procedures for solving them generally su#er from poor discretization accuracy and slow convergence of the algebraic solver. Accuracy can be improved by taking meshes that are refined enough for the given wavenumber, but such refinements are impractical for many problems. Also, convergence of the iterative solver can be improved by preconditioning the algebraic system, but finding an appropriate preconditioner is problematic and, in general, robust iterative solvers for indefinite and non-self-adjoint problems are di#cult to design. # Received by the editors May 29, 1998; accepted for publication (in revised form) February 16, 1999; published electronically April 28, 2000. This work was sponsored by the National Science Foundation under grant DMS-9706866 and the Department of Energy under grant DE-FG03-93ER25165. http://www.siam.org/journals/sisc/21-5/33977.html Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, P.O. Box 808, L-661, Livermore, CA 94551 (lee123@llnl.gov). Applied Math Department, Campus Box 526, University of Colorado at Boulder, Boulder, CO 80309-0526 (tmanteuf@boulder.colorado.edu, stevem@boulder.colorado.edu, jruge@boulder. colorado.edu). 1928 LEE, MANTEUFFEL, MCCORMICK, AND RUGE Standard multigrid methods are no exception. Helmholtz problems tax multi-grid methods by admitting certain highly oscillatory error components that yield relatively small residuals. Because these components are oscillatory, standard coarse grids cannot represent them well, so coarsening cannot eliminate them e#ectively. Because they yield small residuals, standard relaxation methods cannot e#ectively reduce them. Compounding these di#culties is the property that the dimension of the subspace of troublesome components increases with increasing wavenumber k. One approach to ameliorate these di#culties (cf. [9], [10], [21], and [23]) is to introduce ray-like basis functions on the coarser grids, via exponential interpolation and weighting, and to use multiple coarsening, where several coarse grids are used at a given discretization level to resolve some of these error components. Introduction of ray basis functions allows the oscillatory error components to be represented on coarser grids, and successive coarsening with an increasing but controlled number of grids allows a full range of these components to be resolved on the coarser levels. Since the magnitude of the wavenumber k dictates the oscillatory nature of the error components, then judicious introduction of exponential interpolation and weighting and of new coarse grid problems as a function of k leads to multigrid schemes whose convergence is k-uniform. Direct application of such a multigrid algorithm to the scalar Helmholtz equation would not generally achieve optimal discretization accuracy that is uniform in k. Because k-uniform optimality of the discretization and multigrid solver is the central aim of this paper, the multigrid algorithm will be applied instead to a carefully designed first-order system least-squares (FOSLS) (cf. [11], [12], [21], and [22]) formulation of the Helmholtz problem. This FOSLS methodology involves recasting the scalar equation into a first-order system by introducing "field" variables, deriving boundary conditions for these new variables, and applying a least-squares approach to the resulting first-order system boundary value problem. An important consideration in the numerical solution of Helmholtz problems is the so-called pollution e#ect. The results in [3], [5], and [19] show that, for large wavenumber k, even when kh # 1, the dispersive nature of the Galerkin finite element discretization introduces a large phase lead error term and, in fact, a sharp H 1 error estimate is contaminated by a pollution term unless k 2 h # 1. A stabilized Galerkin least-squares finite element method was developed in [3], [5], and [19] to reduce the pollution e#ect and, hence, to ameliorate the e#ects of this restrictive condition. In the work presented below, because the FOSLS methodology leads to a minimization principle, a Rayleigh-Ritz principle can be applied. Thus, standard bases can be used to achieve O(kh) discretization error bounds in the least-squares norm (that is, the norm determined by the functional itself) for the FOSLS formulation. Our numerical results will suggest that this pollution-free performance is achieved by the FOSLS formulation-not only the least-squares norm but in a scaled H 1 norm as well. Thus, the aim of this paper is two-fold: k-uniform multigrid convergence factors and pollution-free discretization accuracy. Since a fast k-uniformly convergent solver has been the most di#cult to obtain historically, this will be the primary focus of our work. However, we will also show in our final numerical experiment that the FOSLS discretization appears to be pollution free. Substantial literature is available on computational electromagnetics. Algorithms that apply to the time-dependent Maxwell equations or div-curl systems in general include those described in [13], [20], [24], [26], and [27]. FOSLS FOR HELMHOLTZ 1929 Studies based on first-order system formulations of the Helmholtz equation include [16], [18], [21], and [23]. The approach described in [16] does not include a curl term in the functional, so its discretizations are somewhat restrictive and its standard multigrid methods could not perform well with large wavenumbers, as expected. The functional in [18] incorporates a curl expression but not in a way that achieves uniformity in the discretization (or the multigrid solver, had they analyzed it). In fact, except for [21] and [23] which is the basis of the work presented in the present paper, none of the methods cited above were shown to achieve optimal discretization accuracy and multigrid convergence. For general literature on FOSLS, see [8], [11], [12], [21], and [22], which treat least-squares functionals based on L 2 and L 2 (that is, a combination of L 2 and products. The least-squares functionals described in the next section are also based on these norms. In fact, the analytical techniques used in these papers are also used to derive some of the theoretical results presented below. These techniques enable the establishment of several uniform norm equivalence results that would not be possible by way of the more restrictive Agmon-Douglis-Nirenberg theory (cf. [2]). This paper is organized as follows. Section 2 introduces the functional setting, the first-order system, and the L 2 and L 2 least-squares formulations. In section 3, a coercivity result for the L 2 defined over an appropriate Hilbert space is established using a compactness argument. Unfortunately, this coercivity estimate does not hold uniformly with respect to the wavenumber, so in section 4 we introduce an appropriate subspace on which we do prove k-uniform coercivity. These estimates provide an intuitive basis for the nonstandard multigrid method developed in section 5. In the final section, we report on numerical tests that demonstrate k-uniform pollution-free optimality of the discretization technique and multigrid solver. 2. FOSLS formulation. Let D # 2 be a bounded domain, which for technical reasons (e.g., scaled H 1 equivalence of the L 2 FOSLS functional; cf. Theorems 3.2 and 4.2) will be assumed to have a C 1,1 or convex polygonal boundary # i of positive measure. Consider the exterior Helmholtz boundary value problem D, Here, k is the wavenumber, f is in L 2 loc #r denotes the derivative in the radial direction used in the asymptotic Sommerfeld radiation condition (cf. [14]). To solve (2.1) numerically, the unbounded domain is truncated and the Sommerfeld radiation condition is approximated. Thus, let B be a ball containing D (D # B) with C 1,1 boundary # e , so that the region bounded by # i # e is annulus-like. Assuming now that f # L 2(# , consider the reduced boundary value problem in# , Here, a first-order approximation to the Sommerfeld radiation condition is used. Assume henceforth that (2.2) has a unique solution in H Introducing the "field" variable k #p, (2.2) can be recast into the first-order system in# , or where L is the first-order operator corresponding to (2.3). Note that (2.3) is essentially a scaled form of Maxwell's equations that has been reduced by assuming a transverse magnetic wave solution. Variable p gives the electric field and variable u gives the magnetic field. To analyze (2.3) and the corresponding least-squares formulations, several functional spaces are required. With s being a nonnegative integer, let H s(# denote the usual Sobolev space of order s with norm # s . With denote the usual Sobolev trace space with norm # t (cf. [1]). Let # denote the L 2 (#) inner product. The following spaces and scaled H 1 norm will be needed: 0,d The boundary condition imposed in the definition of W(# is to be taken in the trace sense. Finally, let H 0,d denote the completion of L 2(# in the respective inverse norms (, w) #w# 1,k FOSLS FOR HELMHOLTZ 1931 and 0,d (, w) #w# 1,k It is well known (cf. [17]) that [H(div; # H 1(# is a Hilbert space in the norm where and Thus, it is also a Hilbert space in the norm Using the continuity of the trace operators from H 2 (#) and H(div; (#), it is easy to see that W is also a Hilbert space in the # k norm (cf. [22]). Define the L 2 and L 2 least-squares functionals by f and f #-1,d respectively. Both functionals incorporate curl terms that, with essentially no increase in complexity for F and only moderate increase for G, allow for simplified finite element discretization and multigrid solution processes. For brevity, the equivalence results of the next two sections focus on G; we will state but not prove the analogous H 1,k equivalence results for F (cf. [21] for detailed proofs of similar results). 3. Nonuniform coercivity: W(#3 A compactness argument (cf. [7] and [8]) is used here to establish equivalence between the homogeneous part of the L 2 least-squares functional, G(u, p; 0), and the square of the product norm [#] 2 # 1,k . In what follows, C denotes a generic constant that may change meaning at each occurrence but is independent of k. The notation C k will be used (in this section when dependence on k may occur. Theorem 3.1. There exist constants C and C k such for all (u, p) # W 1932 LEE, MANTEUFFEL, MCCORMICK, AND RUGE Proof. Let (u, p) # W(# . To prove the upper bound, note that the triangle inequality yields -1,d # . Integration by parts, the triangle and Cauchy-Schwarz inequalities, the trace theorem, and the #-inequality give 0,d #w# 1,k 0,d u, #w # 1 #w# 1,k 0,d #w# 1,k 0,d #w# 1,k 0,d Also, integration by parts and the boundary condition on H 1 give #w# 1,k #w# 1,k #w# 1,k #u# . Finally, #p# -1,d #p# #p# 1,k . The upper bound then follows from (3.1)-(3.4). To prove the lower bound, let W(# be the completion of W(# in the #u# norm. To prove first that FOSLS FOR HELMHOLTZ 1933 on W(#5 note that the triangle inequality used twice and (3.4) yield # . Hence, (3.5) would follow if it could be shown that To this end, note that integration by parts producesk 2 #p# Moreover, since 1 real, then the right-hand side of (3.8) must be real, which, together with the triangle inequality, complex modulus arithmetic, and the Cauchy-Schwarz and #-inequalities, yieldsk 2 #p# so bound (3.5) holds on all of W(#1 Now, to prove that 1934 LEE, MANTEUFFEL, MCCORMICK, AND RUGE on W(# by contradiction, suppose that (3.9) does not hold. Then there exists a sequence W(# such that and then the Rellich selection theorem (cf. [7]) implies that p j # p for some p in L 2(#3 which, with (3.5) and (3.10), shows that is a Cauchy sequence in the #u#p# 1,k has a limit {(u, p)} # . Also, consider the scaled Helmholtz equation in# , with the corresponding bilinear form 0,d . Integration by parts gives Convergence of p j to p in the # 1,k norm, the Cauchy-Schwarz and #-inequalities, and (3.10) then imply that lim FOSLS FOR HELMHOLTZ 1935 or, equivalently, because of the solution uniqueness of (3.11). Finally, by (3.5), it follows that so that which is a contradiction. Thus, (3.9) must hold. The theorem now follows by restricting this bound to the subspace The analogous result for F is essentially obtained using a Helmholtz decomposition of u. Its proof is very similar to the proofs of Thorems 3.5 and 4.5 of [21], and, thus, will not be included here. Theorem 3.2. There exist constants C and C k such for all (u, p) # W(# . These theorems show that the homogeneous parts of functionals F and G are bounded (with constant C) and coercive (with constant 1 Ck ) in W(# . Unfortunately, the coercivity constant depends on the wavenumber k, which means that standard numerical solution processes can degrade in performance as k increases. In the next section, a subspace of W(# is introduced on which we prove the coercivity constant to be independent of k. 4. Uniform coercivity: Z #2 A fundamental problem with the Helmholtz equation is that approximate Fourier components with wavenumbers near k produce relatively small residuals; that is, oscillatory components are in the near nullspace of (2.1). This degradation of the usual sense of ellipticity is also present in system (2.3), which means that the homogeneous parts of respective functionals G and F are not equivalent to the square of the product norms [#] 2 # 1,k and [# 1,k uniformly in k. However, as we will see, these troublesome functions can be treated specially in the multigrid coarsening and finite element approximation processes. To guide the design of these processes, this section will establish equivalence results on a subspace of W(# that excludes these near-nullspace components. To this end, note that 1936 LEE, MANTEUFFEL, MCCORMICK, AND RUGE is a dense subspace of H 1 0,d (see [6]) that corresponds to the range of the inverse Laplace operator for in# , (that is, N(#07 The operator in (4.1) is self-adjoint, so there exists an L with lim j# in# , To isolate the near-nullspace components of H 1 0,d(#2 let # (0, 1) be given and define the spaces k or #2 Z Here we use span to mean finite linear combinations and overbar to denote closure in the H 1,k norm. The subspace Z # of W(#0 which is closed in the #| k norm, avoids the oscillatory near-nullspace error components of system (2.3) that yield small residuals. Hence, in this subspace, uniform equivalence statements can be made between the least-squares norms corresponding to G and F (that is, the square roots of their homogeneous parts) and the respective product norms [#] 2 Theorem 4.1. Let # (0, 1) be given. Then there exists a constant C, independent of # and k, such for all (u, p) # Z # . Proof. The upper bound follows from that of Theorem 3.1 and the fact that Z W(# . The proof of the lower bound rests on the observation that Z # is contained in the closure of Z 0,# in the product norm [#] 2 # 1,k . A proof analogous to that of the upper bound in Theorem 3.1 shows that G(u, p; 0) is continuous on Z (# in this norm. It is therefore su#cient to establish the lower bound for (u, p) # Z 0,#3 To this end, consider the Helmholtz decomposition (cf. [12]) FOSLS FOR HELMHOLTZ 1937 2(# , and # is divergence-free such that n # e . Substituting this decomposition into G(u, p; 0) and using the L orthogonality property of this decomposition and the divergence-free property of # , we have Also, integration by parts and the Cauchy-Schwarz inequality give 0,d #w# 1,k 0,d #w# 1,k Now, the triangle inequality and (4.4) yield which gives But [-#] is positive-definite and self-adjoint on S # and p # S # , so it is easy to verify the bound 1,k . Hence, (4.3), (4.5), and (4.6) imply that for any (u, p) # Z 0,#1 Finally, combining (4.7) with the triangle inequality and the restriction # < 1, we obtain (4. 1938 LEE, MANTEUFFEL, MCCORMICK, AND RUGE The lower bound now follows from (4.7) and (4.8). We again state the analogous result for F without proof. Theorem 4.2. Let # (0, 1) be given and k # 1. Then there exists a constant C, independent of # and k, such for all (u, p) # Z # . Remark 4.1. While the coercivity constant in Theorem 4.2 is uniform in k, the dimension of Z # grows with k (for fixed #). However, these troublesome components can still be specially treated in the multigrid coarsening process without increasing the order of complexity, as we will show in the next section. 5. Nonstandard multigrid. Some of the basic concepts of this section were taken from or inspired by the work of Achi Brandt (cf. [9] and Brandt and Livshitz [10]). The L 2 functional F , when it applies, is usually more practical than the L 2 functional G, which involves the somewhat cumbersome negative norms. For this rea- son, the remaining two sections will focus on minimizing F (u, p; f). A Rayleigh-Ritz finite element method is used for the discretization process. Let T h be a triangulation of domain# into finite elements of maximal length let W 1 be a finite-dimensional subspace of W(# having the approximation property for (v, q) # H W(# . The discrete fine grid minimization problem is the following. . Find Equivalently, defining the bilinear form for (u, p), (v, q) # W(# , the discrete fine grid variational problem is the following. . Find f for all (v 1 , A standard multilevel scheme for solving either of these discrete problems is fairly straightforward. Let be a conforming sequence of coarsenings of triangulation T h , FOSLS FOR HELMHOLTZ 1939 be a set of nested coarse grid subspaces of W 1 , the finest subspace, and be a suitable (generally local) basis for W j , referred to here as level j. Given an initial approximation level j, the level j relaxation sweep consists of the following cycle: . For each chosen to minimize (By #(b, c) is meant (#), where # f) is a quadratic function in #, this local minimization procedure is simple and very inexpensive (see [23] and [25]). In fact, this minimization process is just a block Gauss-Seidel iteration with blocks determined by the choice of (b recalling that L is the first-order operator corresponding to (2.3), the block system is expressed as where # . Now, given a fine grid approximation level 1, the level 2 coarse grid problem is to find a correction Having obtained the fine grid approximation is corrected according to Applying this process recursively yields a multilevel scheme in the usual way. Unfortunately, if the coarser level basis functions are chosen in the natural way, then this coarse grid correction process may not be very e#ective for the FOSLS formulation of the Helmholtz equation. Di#culties can arise with error components for elements in Z # . Specifically, trouble arises from error components that have the approximate form sin ## e k(x cos #+y sin #) , where # [0, 2#). The main problem is that, on a grid where of this type are highly oscillatory and, hence, poorly approximated by standard 1940 LEE, MANTEUFFEL, MCCORMICK, AND RUGE coarse grids, yet they produce small residuals and are therefore poorly reduced by relaxation. A compounding problem is that there are, in principle, infinitely many of these components since # can be virtually any angle in [0, 2#). More precisely, as k increases, so does the dimension of Z # , which consists of elements that are approximately of the form in (5.6). Fortunately, these problems can be circumvented by introducing exponential interpolation and multiple coarsening into the multilevel scheme ([10] and [23]). Exponential interpolation. Exponential interpolation is used to approximate the troublesome oscillatory error components of approximate form (5.6). But the ray e k(x cos #+y sin #) is the problematic factor of this form, so a more useful characterization of these error components is given by s(x, y)e k(x cos #+y sin #) , where s(x, y) is a smooth function satisfying appropriate boundary conditions. j be the components of the product space W j , which is assumed for concreteness to consist only of continuous piecewise bilinear functions. (These spaces generally di#er over # only at the boundary.) Also, let level j correspond to a grid that is fine enough relative to k that components (5.7) can be adequately approximated with bilinear functions, although only marginally so in the sense that this approximation just begins to deteriorate on level j + 1. Now, exact coarse level correction from level j is assumed to be e#ective at eliminating the smooth components left by relaxation on level j - 1. At level however, the ability of bilinear functions to approximate components (5.7) deteriorates enough to contaminate two-level performance between levels j. Some heuristics show that this occurs when 2 To approximate these exponential components from levels j basis functions are simply rescaled by the ray elements sin #) . To be more specific, fix # [0, 2#). For each l,# denote entry number # of element number # of the standard basis B l and, on level l, define the level j ray element by l,# , l is the element of W # that agrees with e k(x cos #+y sin #) d at the level j node values; that is, E # (d) is the level j nodal interpolant of e k(x cos #+y sin #) d. This has the e#ect of altering the usual bilinear interpolation formula that relates coarse to fine node values so that the rays are better approximated. In fact, this ensures that the level j interpolant of the given ray is in the range of interpolation on all coarser levels determined by the corresponding #. Figure 5.1 sketches the real part of a one-dimensional level 3 basis function d 3 and a typical ray basis element E # determined by an oscillatory level 1 ray (assuming for simplicity that Note that exponential interpolation is not much more costly than bilinear inter- polation: a given function d on level l > j is bilinearly interpolated up to level j, then simply rescaled by e k(x cos #+y sin #) at the level j nodes. Multiple coarsening. Consider now the case that level j is where approximation of components (5.7) just begins to deteriorate. It may be enough to introduce one ray (e.g., with in the sense that an exact correction from level might then adequately correct "smooth" level j error. However, it is perhaps more e#ective here to introduce a few coarse grids, one for each of, say, four rays (e.g., with FOSLS FOR HELMHOLTZ 1941 ray h Fig. 5.1. One-dimensional real parts of a typical standard d 3 basis function and a E # basis element. critically, as each level is coarsened, scale k2 j-1 h doubles and essentially twice as many rays become oscillatory. To be more specific, suppose that are angles corresponding to two neighboring level j rays in the sense that these two rays were used to coarsen level perhaps others, but none were needed with angles between # 1 and # 2 . Now, on any of the level will be one for each ray, that is, for each coarsening), the ray with angle #1 +#2will be poorly reduced in relaxation and poorly approximated by any of the level j grids. This means that twice as many rays must somehow be used in coarsening to level which in turn will again require ray doubling to level This can be accomplished e#ectively by introducing two separate ray coarsenings on a level that may itself have come from ray coarsening. Suppose that level was created from a level j ray with angle #, and that the task is now to coarsen level using rays of angles #. One way to accomplish this is to introduce the level ray # j+2,# . But now # oscillates on level j scale, so applying an operator to this function will generally require level j evaluations. Alternatively, these ray coarsenings can be e#ciently introduced using intermediate level Suppose the level j rays # j+1,# have already been introduced on level j + 1. Since e k[x cos(#)+y then letting E # j denote the level j interpolant of the ray perturbation e k{x[cos(#)-cos #]+y[sin(#)-sin #]} means that the perturbed rays can be computed directly from level appealing to level j. In particular, since is a level j+2,# can be constructed by exponentially interpolating the level Figure 5.2 for a sketch 1942 LEE, MANTEUFFEL, MCCORMICK, AND RUGE of this process for creating a level 3 ray basis function from levels 1 and 2.) Hence, given an operator M and assuming that has been constructed for the ray basis of level j +1, then M# j+2,# can be constructed by exponentially weighting the M# 's: M# c d # c M# which is a weighting of elements of (5.9) with exponential weights c . Recursively applying this exponential weighting process, analogous coarser level calculations can be computed without appealing to level j. having solved a coarse grid problem, its solution can be exponentially interpolated up to level j using the angle perturbations in a recursive way. For example, if j+2 is the solution of the level problem corresponding to perturbation #, then s # j+2 can be interpolated up to level j by first exponentially interpolating up to level then exponentially interpolating this result to level j using #. Since exponential interpolation consists of a standard bilinear interpolation and an exponential scaling, then this successive interpolation process is easy to implement. Moreover, since the exponential interpolation operator is the adjoint of the exponential weighting operator, then exponential weighting is also easy to implement. Angles (#). The angles cannot be arbitrary, but must be chosen such that an optimal number of exponential components are adequately approximated by the perturbed ray elements. To describe this selection procedure, assume that rays are first introduced on level j + 1. Then, to obtain adequate approximation, a simple but tedious analysis [21] shows that the perturbation angle # must be chosen so that the following approximation is within the level j discretization error: j+2,# . For example, for 2 , the level 3 angle perturbations must be #, as depicted in Figure 5.4. The angle perturbations are simply halved at each successively finer level. Nonstandard multigrid scheme and computational cost. Exponential in- terpolation/weighting and multiple coarsening are essentially all that is required to FOSLS FOR HELMHOLTZ 1943 d ray 2h "perturbed" ray hFig. 5.2. Level 3 ray element determined from a perturbation corresponding to angle # of the level 2 ray element. Level (j-1) Level j Level (j+1) Level (j+2) Fig. 5.3. Multiple coarsening. develop an e#ective nonstandard multigrid scheme for the Helmholtz equation. With these components, ray basis elements are naturally introduced, and this nonstandard multigrid scheme has the same form as the standard multigrid scheme defined by (5.2) and (5.5). To see this, again assume that rays are first introduced on level that all perturbation angles are determined before the multigrid cycling begins. Here, we choose j to be the largest integer satisfying since this guarantees essentially 8 grid points per wavelength in the piecewise bilinear approximation on level j. Now, for levels l # j, the nested spaces still consist of products of piecewise bilinear functions, but, for levels l # must introduce the product ray element spaces l := span #r 1944 LEE, MANTEUFFEL, MCCORMICK, AND RUGE Level 3 angles Level 2 angles Fig. 5.4. Angle choices (uniformly distributed) for level 2 and level 3 rays. where t l is the total number of rays introduced on level l, # r is the angle of the rth ray, and #r is the corresponding perturbed vector ray basis introduced on level l. We first introduce four rays on level double that number for each successively coarser level, so we have t . Note that l only if # s is a perturbation of angle # r (that is, only if As before, for levels l # j, relaxation is defined by (5.2) with (b l , c l ) # the standard basis for W l and, for levels l < j, the level (l problem is to find (u l+1 , p l+1 ) # W l+1 such that (v l+1 ,q l+1 )#W l+1 where l ) is the level l approximation. However, for the ray levels l > j, relaxation is again defined by (5.2); but now with (b l , c l ) # the product ray basis element #r l,# , and, for levels l # j, the two level (l problems corresponding to each r , are to find (u l+1 , (v l+1 ,q l+1 )#W #s where l is now the approximation on level l with angle # r . Note that each level l with angle # r is corrected by two level (l (actually, four when This gives the structure of the multigrid process. The scheduling that determines how the various levels are visited is best described by studying the schematic in Figure 5.5. We will continue to refer to this process as a V -cycle, although it looks very di#erent from a standard V-cycle below level j. For this nonstandard multigrid scheme (which can be interpreted as a multilevel multiplicative Schwarz method; cf. [28]), the level j coarse grid problem involves subspace corrections from the standard finite element space W j and the ray subspaces l l > j. Also, relaxation on the ray spaces is simply a block Gauss-Seidel iteration FOSLS FOR HELMHOLTZ 1945 Level (j-1) Level Level (j+1) Level (j+2) Fig. 5.5. V-cycle for the nonstandard multigrid scheme. (5.3) with (b l , c l ) #r l,# . Since L#r l,# can be computed e#ciently using exponential weighting, so can the elemental block matrices or stencils l, , L#r l,# . In fact, these stencils can be computed and stored prior to any multigrid cycling so that the same relaxation routine can be used for all levels and angles. Now, assuming that the stencils have been computed and stored before any multi-grid cycling occurs, the computational cost of such a multiple coarsening algorithm is generally not excessive. In fact, one such multilevel V-cycle requires only several fine grid work units (that is, the cost of a fine grid relaxation). For example, consider the worst case, when four rays are introduced on level 2. (The need for more rays would signal a level h so coarse relative to k that the discretization accuracy would be almost meaningless; in fact, the need for four rays on level 2 is already a sign that level 1 accuracy is quite poor.) Then, for level j # 2, there would be a total of 2 j level j problems. Also, for each j, if it is assumed that the stepsize is doubled at each coarsening, then the number of fine grid work units required for relaxation is 2 -2j+2 . Omitting the cost of ray interpolation (which is substantial but proportional to the number of points) and other more minor operations, the total number of fine level work units for a nonstandard V (1, 0)-cycle is then which is only about double the cost of a standard V (1, 0)-cycle. 6. Numerical examples. For simplicity and to facilitate comparisons, the examples treated here are only for the unit square. The lack of a Dirichlet boundary is in fact a more severe test of the methodology in some sense. The results will include multigrid performance for both the standard and the nonstandard schemes described 1946 LEE, MANTEUFFEL, MCCORMICK, AND RUGE Table Standard coarsening: V (1, 0)-cycles. Table Standard coarsening: two-level cycles. in the previous section. We also assess accuracy of the discretization based on ray-type elements. To facilitate assessment of the iterative solver, we first restrict ourselves to the homogeneous boundary value problem (2.2) with The exact solution is of course but this is useful for measuring asymptotic algebraic convergence factors of stationary linear iterative methods such as multigrid methods since it avoids the limiting stagnation caused by machine representation. The unit square is partitioned into rectangles, and the functional F is minimized over the space of piecewise continuous bilinear functions (possibly involving rays) that satisfy the boundary condition on #. In most cases, the finest grid is chosen to satisfy 8 , and rays are first introduced on level 8 (that is, typically when The relaxation scheme is nodal Gauss-Seidel relaxation (that is, block Gauss-Seidel where a block consists of all of the unknowns (u h # ) at each node #). To assess the worst-case convergence factors, twenty multigrid V(1,0)-cycles (that is, one relaxation before and none after coarsening) were performed starting from a random initial guess ). The convergence measure # is then defined as where the superscript in parentheses denotes the iteration number. Tables 6.1 and 6.2, respectively, summarize the results of twenty V (1, 0)- and two-level cycles for standard bilinear elements. Table 6.1 confirms that convergence degrades as a function of k due to the troublesome oscillatory error components present on the coarser levels. Table 6.2 shows that Poisson-like (that is, are obtained when kh is not too large and, comparing it with Table 6.1, indicates that rays should be introduced as soon as kh > 1 Tables 6.3 and 6.4 show that multigrid convergence factors improve substantially by introducing ray functions. Here, "exact" rays (that is, are intro- FOSLS FOR HELMHOLTZ 1947 Table Exact ray (# 0)-cycles. Table Exact ray (# cycles. .3 Table Approximate ray (# 0)-cycles. Table Approximate ray (# delayed until kh > 1 0)-cycles. duced when kh > 1using the algorithm described in the previous section. These factors are fairly uniform in k and compare favorably to the two-level factors for standard coarsening. Moreover, they degrade only slightly when the exact rays are approximated using the recursive scheme described in the previous section (see Table 6.5). To test the e#ect of introducing the rays late in the multigrid solver, we reran these examples but delayed the use of rays until kh > 1. As expected, the rates degrade noticeably, as Tables 6.6 and 6.7 show. As a measure of discretization accuracy, we ran tests to see if our FOSLS scheme has any pollution e#ects. We specified the exact solution 5 +y sin #) , which solves (2.2) with except that the boundary condition is inhomogeneous: 5 +y sin #) 5 +y sin #) on #. 1948 LEE, MANTEUFFEL, MCCORMICK, AND RUGE Table Approximate ray (# delayed until kh > 1 0)-cycles. Table Discretization error in the least-squares functional norm. Table Relative discretization error in the scaled H 1,k norm. kh\h 11111 We then used our FOSLS scheme to approximate p and the scaled gradient u =k #p. To isolate possible pollution, we chose various h, then varied k so that kh is constant 1/64). The errors were measured by comparing the discrete approximation (obtained after applying several multigrid cycles) to the exact solution in a relative sense in the least-squares and product H 1,k norms. Tables 6.8 and 6.9 essentially show constant discretization errors for constant kh, with about a factor of 2 decrease as kh is halved, which is consistent with the assertion that our approach is pollution-free and O(kh), respectively. --R New York Estimates near the boundary for solutions of elliptic partial di A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution The partition of unity method Is the pollution e Expansions in Eigenfunctions of Selfadjoint Operators Finite Elements Theory A least-squares approach based on a discrete minus one inner product for first order systems Stages in developing multigrid solutions Finite element method for the solution of Maxwell's equations in multiple media Inverse Acoustic and Electromagnetic Scattering Theory Partial Di On numerical methods for acoustic problems Finite Element Methods for Navier-Stokes Equations Finite element solution of the Helmholtz equation with high wave number. Multilevel first-order system least-squares (FOSLS) for Helmholtz equations Multilevel Projection Methods for Partial Di A mixed method for approximating Maxwell's equations On electric and magnetic problems for vector fields in anisotropic nonhomogeneous media Iterative methods by space decomposition and subspace correction --TR --CTR Jan Mandel , Mirela O. Popa, Iterative solvers for coupled fluid-solid scattering, Applied Numerical Mathematics, v.54 n.2, p.194-207, July 2005 Jan Mandel, An iterative substructuring method for coupled fluid-solid acoustic problems, Journal of Computational Physics, v.177 n.1, p.95-116, March 20, 2002

first-order system least-squares;helmholtz equation;nonstandard multigrid

351604

Robust Real-Time Periodic Motion Detection, Analysis, and Applications.

AbstractWe describe new techniques to detect and analyze periodic motion as seen from both a static and a moving camera. By tracking objects of interest, we compute an object's self-similarity as it evolves in time. For periodic motion, the self-similarity measure is also periodic and we apply Time-Frequency analysis to detect and characterize the periodic motion. The periodicity is also analyzed robustly using the 2D lattice structures inherent in similarity matrices. A real-time system has been implemented to track and classify objects using periodicity. Examples of object classification (people, running dogs, vehicles), person counting, and nonstationary periodicity are provided.

Introduction Object motions that repeat are common in both nature and the man-made environment in which we live. Perhaps the most prevalent periodic motions are the ambulatory motions made by humans and animals in their gaits (commonly referred to as "biological motion" [16]). Other examples include a person walking, a waving hand, a rotating wheel, ocean waves, and a flying bird. Knowing that an object's motion is periodic is a strong cue for object and action recognition [16, 11]. In addition, periodic motion can also aid in tracking objects. Furthermore, the periodic motion of people can be used to recognize individuals [20]. 1.1 Motivation Our work is motivated by the ability of animals and insects to utilize oscillatory motion for action and object recognition and navigation. There is behavioral evidence that pigeons are well adapted to recognize the types of oscillatory movements that represent components of the motor behavior shown by many living organisms [9]. There is also evidence that certain insects use oscillatory motion for navigational purposes (hovering above flowers during feeding) [17]. Humans can recognize biological motion from viewing lights placed on the joints of moving people [16]. Humans can also recognize periodic movement of image sequences at very low resolutions, even when point correspondences are not possible. For example, Figure 1 shows such a sequence. The effective resolution of this sequence is 9x15 pixels (it was created by resampling a 140x218 (8-bit, 30fps) image sequence to 9x15 and back to 140x218 using bicubic interpolation). In this sequence, note the similarity between frames 0 and 15. We will use image similarity to detect and analyze periodic motion. Figure 1. Low resolution image sequences of a periodic motion (a person walking on a treadmill). The e#ective resolution is 9x15 pixels. 1.2 Periodicity and motion symmetries We define the motion of a point # X(t), at time t, periodic if it repeats itself with a constant period p, i.e.: T (t) is a translation of the point. The period p is the smallest p > 0 that satisfies (1); the frequency of the motion is 1/p. If p is not constant, then the motion is cyclic. In this work, we analyze locally (in time) periodic motion, which approximates many natural forms of cyclic motion. Periodic motion can also be defined in terms of symmetry. Informally, spatial symmetry is self-similarity under a class of transformations, usually the group of Euclidean transformations in the plane (translations, rotations, and reflections)[36]. Periodic motion has a temporal (and sometimes spatial) symmetry. For example, Figures 3(a), 4(a), 5(a), and 6(a) show four simple dynamic systems (pendu- lums). For each system, the motion is such that # X(t) for a point # X(t) on the pendulum. However, each system exhibits qualitatively different types of periodic motion. Figure 5(a) is a simple planar pendulum with a fixed rod under a gravitational field. The motion of this system gives it a temporal mirror symmetry along the shown vertical axis. The system in Figure 4(a) is a similar pendulum, but with a sufficient initial velocity such that it always travels in one angular direction. The motion of this system gives it a temporal mirror symmetry along the shown vertical axis. The system in Figure 3(a) is a similar pendulum, but in zero gravity; note it has an infinite number of axes of symmetry that pass through the pivot of the pendulum. The system in Figure 6(a) consists of a pair of uncoupled and 180 # out of phase pendulums, a system which is often used to model the upper leg motion of humans [24]. This system has a temporal mirror symmetry along the shown vertical axis, as well as an approximate spatial mirror symmetry along the same vertical axis (it is approximate because the pendulums are not identical). The above examples illustrate that while eq. 1 can be used to detect periodicity, it is not sufficient to classify different types of periodic motion. For classification purposes, it is necessary to exploit the dynamics of the system of interest, which we do in Section 3.4. 1.3 Assumptions In this work, we make the following assumptions: (1) the orientation and apparent size of the segmented objects do not change significantly during several periods (or do so periodically); (2) the frame rate is sufficiently fast for capturing the periodic motion (at least double the highest frequency in the periodic motion). Contributions The main contribution of this work is the introduction of novel techniques to robustly detect and analyze periodic motion. We have demonstrated these techniques with video of the quality typically found in both ground and airborne surveillance systems. Of particular interest is the utilization of the symmetries of motion exhibited in nature, which we use for object classification. We also provide several other novel applications of periodic motion, all related to automating a surveillance system. Organization of the Paper In Section 2, we review and critique the related work. The methodology is described in Section 3. Examples and applications of periodic motion, particularly for the automated surveillance domain, are given in Section 4. A real-time implementation of the methods is discussed in Section 5, followed by a summary of the paper in Section 6. Related Work There has been recent interest in segmenting and analyzing periodic or cyclic motion. Existing methods can be categorized as those requiring point correspondences [33, 35]; those analyzing periodicities of pixels [21, 30]; those analyzing features of periodic motion [27, 10, 14]; and those analyzing the periodicities of object similarities [6, 7, 33]. Related work has been done in analyzing the rigidity of moving objects [34, 25]. Below we review and critique each of these methods. Due to some similarities with the presented method, [33, 21, 30] are described in more detail than the other related work. Seitz and Dyer [33] compute a temporal correlation plot for repeating motions using different image comparison functions, dA and d I . The affine comparison function dA allows for view-invariant analysis of image motion, but requires point correspondences (which are achieved by tracking reflectors on the analyzed objects). The image comparison function d I computes the sum of absolute differences between images. However, the objects are not tracked, and thus must have non-translational periodic motion in order for periodic motion to be detected. Cyclic motion is analyzed by computing the period- trace, which are curves that are fit to the surface d. Snakes are used to fit these curves, which assumes that d is well-behaved near zeros so that near-matching configurations show up as local minima of d. The K-S test is utilized to classify periodic and non-periodic motion. The samples used in the K-S test are the correlation matrix M and the hypothesized period-trace PT . The null hypothesis is that the motion is not periodic, i.e., the cumulative distribution function M and PT not are significantly different. The K-S test rejects the null hypothesis when periodic motion is present. However, it also rejects the null hypothesis if M is non-stationary. For example, when M has a trend, the cumulative distribution function of M and PT can be significantly different, resulting in classifying the motion as periodic (even if no periodic motion present). This can occur if the viewpoint of the object or lighting changes significantly during evaluation of M (see Figure 19(a)). The basic weakness of this method is it uses a one-sided hypothesis test which assumes stationarity. A stronger test is needed to detect periodicity in non-stationary data, which we provide in Section 3.4. Polana and Nelson [30] recognize periodic motions in an image sequence by first aligning the frames with respect to the centroid of an object so that the object remains stationary in time. Reference curves, which are lines parallel to the trajectory of the motion flow centroid, are extracted and the spectral power is estimated for the image signals along these curves. The periodicity measure of each reference curve is defined as the normalized difference between the sum of the spectral energy at the highest amplitude frequency and its multiples, and the sum of the energy at the frequencies half way between. et. al [35] analyze the periodic motion of a person walking parallel to the image plane. Both synthetic and real walking sequences are analyzed. For the real images, point correspondences were achieved by manually tracking the joints of the body. Periodicity was detected using Fourier analysis of the smoothed spatio-temporal curvature function of the trajectories created by specific points on the body as it performs periodic motion. A motion based recognition application is described, in which one complete cycle is stored as a model, and a matching process is performed using one cycle of an input trajectory. Allmen [1] used spatio-temporal flow curves of edge image sequences (with no background edges present) to analyze cyclic motion. Repeating patterns in the ST flow curves are detected using curvature scale-space. A potential problem with this technique is that the curvature of the ST flow curves is sensitive to noise. Such a technique would likely fail on very noisy sequences, such as that shown in Figure 15. Niyogi and Adelson [27] analyze human gait by first segmenting a person walking parallel to the image plane using background subtraction. A spatio-temporal surface is fit to the XYT pattern created by the walking person. This surface is approximately periodic, and reflects the periodicity of the gait. Related work [26] used this surface (extracted differently) for gait recognition. Liu and Picard [21] assume a static camera and use background subtraction to segment motion. Foreground objects are tracked, and their path is fit to a line using a Hough transform (all examples have motion parallel to the image plane). The power spectrum of the temporal histories of each pixel is then analyzed using Fourier analysis, and the harmonic energy cause by periodic motion is estimated. An implicit assumption in [21] is that the background is homogeneous (a sufficiently non-homogeneous background will swamp the harmonic energy). Our work differs from [21] and [30] in that we analyze the periodicities of the image similarities of large areas of an object, not just individual pixels aligned with an object. Because of this difference (and the fact that we use a smooth image similarity metric) our Fourier analysis is much simpler, since the signals we analyze do not have significant harmonics of the fundamental frequency. The harmonics in [21] and [30] are due to the large discontinuities in the signal of a single pixel; our self-similarity metric does not have such discontinuities. Fujiyoshi and Lipton [10] segment moving objects from a static camera and extract the object bound- aries. From the object boundary, a "star" skeleton is produced, which is then Fourier analyzed for periodic motion. This method requires accurate motion segmentation, which is not always possible (e.g., see Figure 16). Also, objects must be segmented individually; no partial occlusions are allowed (as shown in Figure 21(a)). In addition, since only the boundary of the object is analyzed for periodic change (and not the interior of the object), some periodic motions may not be detected (e.g., a textured rolling ball, or a person walking directly toward the camera). Selinger and Wixson [34] track objects and compute self-similarities of that object. A simple heuristic using the peaks of the 1-D similarity measure is used to classify rigid and non-rigid moving objects, which in our tests fails to classify correctly for noisy images (e.g., the sequence in Figure 15). Heisele and Wohler [14] recognize pedestrians using color images from a moving camera. The images are segmented using a color/position feature space, and the resulting clusters are tracked. A quadratic polynomial classifier extracts those clusters which represent the legs of pedestrians. The clusters are then classified by a time delay neural network, with spatio-temporal receptive fields. This method requires accurate object segmentation. A 3-CCD color camera was used to facilitate the color clustering, and pedestrians are approximately 100 pixels in height. These image qualities and resolutions are typically not found in surveillance applications. There has also been some work done in classifying periodic motion. Polana and Nelson [30] use the dominant frequency of the detected periodicity to determine the temporal scale of the motion. A temporally scaled XYT template, where XY is a feature based on optical flow, is used to match the given motion. The periodic motions include walking, running, swinging, jumping, skiing, jumping jacks, and a toy frog. This technique is view dependent, and has not been demonstrated to generalize across different subjects and viewing conditions. Also, since optical flow is used, it will be highly susceptible to image noise. Cohen et. al [5] classifies oscillatory gestures of a moving light by modeling the gestures as simple one-dimensional ordinary differential equations. Six classes of gestures are considered (all circular and linear paths). This technique requires point correspondences, and has not been shown to work on arbitrary oscillatory motions. Area-based techniques, such as the present method, have several advantages over pixel-based tech- niques, such as [30, 21]. Specifically, area-based techniques allow the analysis of the dynamics of the entire object, which is not achievable by pixel based techniques. This allows for classification of different types of periodic motion, such as those given in Section 4.1 and Section 4.4. In addition, area-based techniques allow detection and analysis of periodic motion that is not parallel to the image plane. All examples given in [30, 21] have motion parallel to the image plane, which ensures there is sufficient periodic pixel variation for the techniques to work. However, since area-based methods compute object similarities which span many pixels, the individual pixel variations do not have to be large. For exam- ple, our method can detect periodic motion from video sequences of people walking directly toward the camera. A related benefit is that area-based techniques allow the analysis of low S/N images, such as that shown in Figure 16, since the S/N of the object similarity measure (such as (5)) is higher than that of a single pixel. The algorithm for periodicity detection and analysis consists of two parts. First, we segment the motion and track objects in the foreground. We then align each object along the temporal axis (using the object's tracking results) and compute the object's self-similarity as it evolves in time. For periodic motions, the self-similarity metric is periodic, and we apply Time-Frequency analysis to detect and characterize the periodicity. The periodicity is also analyzed robustly using the 2-D lattice structures inherent in similarity matrices. 3.1 Motion Segmentation and Tracking Given an image sequence I t from a moving camera, we segment regions of independent motion. The images I t are first Gaussian filtered to reduce noise, resulting in I # . The image I # is then stabilized [12] with respect to image I # t-# , resulting in V t,t-# . The images V t,t-# and I # are differenced and thresholded to detect regions of motion, resulting in a binary motion image: (2) where TM is a threshold. In order to eliminate false motion at occlusion boundaries (and help filter spurious noise), the motion images M t,# and M t,-# are logically and'ed together: An example of M t is shown in Figure 21(b). Note that for large values of # , motion parallax will cause false motion in M t . In our examples (for a moving camera), #=300 ms was used. Note that in many surveillance applications, images are acquired using a camera with automatic gain, shutter, and exposure. In these cases, normalizing the image mean before comparing images I t 1 and I t 2 will help minimize false motion due to a change in the gain, shutter, or exposure. A morphological open operation is performed on M t (yielding M # t ), which reduces motion due to image noise. The connected components for M # t are computed, and small components are eliminated (further reducing image noise). The connected components which are spatially similar (in distance) are then merged, and the merged connected components are added to a list of objects O t to be tracked. An object has the following attributes: area, centroid, bounding box, velocity, ID number, and age (in frames). Objects in O t and O t+k , k > 0, are corresponded using spatial and temporal coherency. It should be noted that the tracker is not required to be very accurate, as the self-similarity metric we use is robust and can handle tracking errors of several pixels (as measured in our examples). Also note that when the background of a tracked object is sufficiently homogeneous, and the tracked object does not change size significantly during several periods, then accurate object segmentation is not necessary. In these cases, we can allow O t to include both the foreground and background. Examples of such backgrounds include grassy fields, dirt roads, or parking lots. An example of such a sequence is given in Figure 15. 3.2 Periodicity Detection and Analysis The output of the motion segmentation and tracking algorithm is a set of foreground objects, each of which has a centroid and size. To detect periodicity for each object, we first align the segmented object (for each frame) using the object's centroid, and resize the objects (using a Mitchell filter [32]) so that they all have the same dimensions. The scaling is required to account for apparent size change due to change in distance from the object to the camera. Because the object segmentation can be noisy, the object dimensions are estimated using the median of N frames (where N is the number of frames we analyze the object over). The object O t 's self-similarity is then computed at times t 1 and t 2 . While many image similarity metrics can be defined (e.g., normalized cross-correlation, Hausdorff distance [15], color indexing [2]), perhaps the simplest is absolute correlation: is the bounding box of object O t 1 . In order to account for tracking errors, the minimal S is found by translating over a small search radius r: |dx,dy|<r For periodic motions, S # will also be periodic. For example, Figure 8(a) shows a plot of S # for all combinations of t 1 and t 2 for a walking sequence (the similarity values have been linearly scaled to the grayscale intensity range [0,255]; dark regions show more similarity). Note that a similarity plot should be symmetric along the main diagonal; however, if substantial image scaling is required, this will not be the case. In addition, there will always be a dark line on the main diagonal (since an object is similar to itself at any given time), and periodic motions will have dark lines (or curves if the period is not constant) parallel to the diagonal. To determine if an object exhibits periodicity, we estimate the 1-D power spectrum of S # a fixed t 1 and all values of t 2 (i.e., the columns of S # ). In estimating the spectral power, the columns of S # are linearly detrended and a Hanning filter is applied. A more accurate spectrum is estimated by averaging the spectra of multiple t 1 's [31] to get a final power estimate P (f i ), where f i is the frequency. Periodic motion will show up as peaks in this spectrum at the motion's fundamental frequencies. A peak at frequency f i is significant if where K is a threshold value (typically 3), P is the mean of P , and # P is the standard deviation of P . Note that multiple peaks can be significant, as we will see in the examples. In the above test, we assume that the period is locally constant. The locality is made precise using Time-Frequency analysis given in Section 3.3. We also assume that there are only linear amplitude modulations to the columns of S # (so that linear detrending is sufficient to make the data stationary), and that any additive noise to S # is Gaussian. Both of these assumption are relaxed in the method given in Section 3.4. 3.2.1 Fisher's Test If we assume that columns of S # are stationary and contaminated with white noise, and that any periodicity present consists of a single fundumental frequency, then we can apply the well known Fisher's test [29, 3]. Fisher's test will reject the null hypothesis (that S # is only white noise) if P (f i ) is substantially larger than the average value. Assuming N is even, let . (7) To apply the test, we compute the realized value x of E q from S # , and then compute the probability: 0). If this probability is less than #, then we reject the null hypothesis at level (in practice we use This test is optimal if there exists a single periodic component at a Fourier frequency f i in white noise stationary data [29]. To test for periodicities containing multiple frequencies, Seigel's test [29] can be applied. In practice, Fisher's test, like the K-S test used by [33], works well if the periodic data is stationary with white noise. However, in most of our non-periodic test data (e.g., Figure 19(a)), which is not stationary, both Fisher's and the K-S test yield false periodicities with high confidence. 3.2.2 Recurrence Matrices It is interesting to note that S # is a recurrence matrix [8, 4], without using time-delayed embedded dimensions. Recurrence matrices are a qualitative tool used to perform time series analysis of non-linear dynamical systems (both periodic and non-periodic). Recurrence matrices make no assumptions on the stationarity of the data, and do not require many data points to be used (a few cycles of periodic data is sufficient). The input for a recurrence matrix is a multi-dimensional temporally sampled signal. In our use, the input signal is the tracked object image sequence O t , and the distance measure is image similarity. Given a recurrence matrix, the initial trajectory # X(t) of a point on an object can be recovered up to an isometry [23]. Therefore, the recurrence plot encodes the spatiotemporal dynamics of the moving object. The similarity plot encodes a projection of the spatiotemporal dynamics of the moving object. 3.3 Time-Frequency Analysis For stationary periodicity (i.e., periodicity with statistics that don't change with time), the above analysis is sufficient. However, for non-stationary periodicity, Fourier analysis is not appropriate. Instead, we use Time-Frequency analysis and the Short-Time Fourier Transform (STFT) [28]: -# is a short-time analysis window, and x(u) is the signal to analyze (S # in our case). The short-time analysis window effectively suppresses the signal x(u) outside a neighborhood around the analysis time point t. Therefore, the STFT is a "local" spectrum of the signal x(u) around t. We use a Hanning windowing function as the short-time analysis window. The window length should be chosen to be long enough to achieve a good power spectrum estimate, but short enough to capture a local change in the periodicity. In practice, a window length equal to several periods works well for typical human motions. An example of non-stationary periodicity is given in Section 4.7. 3.4 Robust Periodicity Analysis In Sections 3.2 and 3.3, we used a hypothesis test on the 1-D power spectrum of S # to determine if contained any periodic motion. The null hypothesis is that there is only white noise in the spectrum, which is rejected by eq. 6 if significant periodic motion is present. However, the null hypothesis can also be rejected if S # contains significant non-Gaussian noise, or if the period is locally non-constant, or if S # is amplitude modulated non-linearly. We seek a technique that minimizes the number of false periodicities, while maximizing the number of true periodicities. Toward this end, we devise a test that performs well when the assumptions stated in Section 3.2 are satisfied, but does not yield false periodicities when these assumptions are violated. An alternative technique to Fourier analysis of the 1-D columns of S is to analyze the 2-D power spectrum of S # . However, as noted in [19], the autocorrelation of S # for regular textures has more prominent peaks than those in the 2-D Fourier spectrum. Let A be the normalized autocorrelation of the A(d x , d y where S # R is the mean of S # over the region R, # RL is the mean of S # over the region R shifted by the lag (dx, dy), and the regions R and R L cover S # and the lagged S # . If S # is periodic, then A will have peaks regularly spaced in a planar lattice M d , where d is the distance between the lattice points. In our examples, we will consider two lattices, a square lattice M S,d (Figure 2(a)), and a 45 # rotated square lattice M R,d (Figure 2(b)). The peaks P in A are matched to M d using the match error measure e: is the closest peak to the lattice point M d,i , TD (T D < d/2) is the maximum distance P i can deviate from M d,i , and is the minimum autocorrelation value that the matched peak may have. M d matches P if all the following are satisfied: min where T e is a match thresholds; [d 1 , d 2 ] is the range of d; TM is the minimum number of points in M d to match. In practice, we let 0.25. The range determines the possible range of the expected period, with the requirement 0 < d 1 < d 2 < L, where L is the maximum lag used in computing A. The number of points in MR and M S can be based on the period of the expected periodicity, and frame-rate of the camera. The period is the sampling interval (e.g., ms for NTSC video). Peaks in A are determined by first smoothing A with Gaussian filter G, yielding A # . A # (i, j) is a peak if A # (i, j) is a strict maximum in a local neighborhood with radius N . In our examples, G is a 5. Lin et. al [19] provides an automatic method for determining the optimal size of G. Square Lattice d (a) 2d (b) Figure 2. Lattices used to match the peaks of the autocorrelation of S # . (a) Square lattice (b) rotated square lattice. 4 Examples and Applications 4.1 Synthetic Data In this section, we demonstrate the methods on synthetic data examples. We generated images of a periodic planar pendulum, with different initial conditions, parameters, and configurations. Note that the equation of motion for a simple planar pendulum is sin where g is the gravitational acceleration, L is the length of the rigid rod, and # is the angle between the pendulum rod and vertical axis [22]. In the first example (see Figure 3(a)), we set so that the pendulum has a circular motion with a constant angular velocity. The diagonal lines in the similarity plot ( Figure are formed due to the self-similarity of the pendulum at every complete cycle. The autocorrelation (Figure 3(c)) has no peaks. (a) TSimilarity of Image T 1 and T 2 50 100 150 200 250 300 350 400100200300400 (b) Autocorrelation of Similarity TLag -252575125(c) Figure 3. (a) Pendulum in zero gravity with a constant angular velocity. The arrows denote the direction of motion. (b) Similarity plot for pendulum. Darker pixels are more similar. (c) Autocorrelation of similarity plot. In the next example, we use the same configuration, but set g > 0 and the initial angular velocity to be sufficient so that the pendulum still has a single angular direction. However, in this configuration the angular velocity is not constant, which is reflected in the qualitatively different similarity plot Figure 4(b)) and autocorrelation (Figure 4(c)). Note that the peaks in A match the lattice structure in Figure 2(a). By decreasing the initial angular velocity, the pendulum will oscillate with a changing angular di- rection, as shown in Figure 5(a). The similarity plot for this system is shown in Figure 5(b), and the autocorrelation in Figure 5(c). Note that the peaks in A match the lattice structure in Figure 2(b). (a) TSimilarity of Image T 1 and T 2 100 200 300 400 500 600200400600 (b) Autocorrelation of Similarity TLag (c) Figure 4. (a) Pendulum in gravity with single angular direction. The arrows denote the direction and magnitude of motion; the pendulum travels faster at the bottom of its trajectory than at the top. (b) Similarity plot for pendulum. (c) Autocorrelation of similarity plot. The peaks are denoted by '+' symbols. (a) TSimilarity of Image T 1 and T 2 100 200 300 400 500 600200400600 (b) Autocorrelation of Similarity TLag (c) Figure 5. (a) Pendulum in gravity with an oscillating angular direction. The arrows denote the direction of motion. (b) Similarity plot for pendulum. (c) Autocorrelation of similarity plot. The peaks are denoted by '+' symbols. Finally, for the system of two pendulums 180 # out of phase shown in Figure 6(a), the similarity plot is shown in Figure 6(b), and the autocorrelation is shown in Figure 6(c). Note that the peaks in A match the lattice structure in Figure 2(b). Also note the lower measures of similarity for the diagonal lines and the cross-diagonal lines S(t, and the corresponding effect on A. (a) TSimilarity of Image T 1 and T 2 100 200 300 400 500 600200400600 (b) Autocorrelation of Similarity TLag (c) Figure 6. (a) Two pendulum out of phase 180 # in gravity. The arrows denote the direction of motion. (b) Similarity plot for pendulums. (c) Autocorrelation of similarity plot. The peaks are denoted by '+' symbols. 4.2 The Symmetry of a Walking Person In this example we first analyze periodic motion with no (little) translational motion, a person walking on a treadmill (Figure 7). This sequence was captured using a static JVC KY-F55B color camera at 640x480 @ 30fps, deinterlaced, and scaled to 160x120. Since the camera is static and there is no translational motion, background subtraction was used to segment the motion [6]. The similarity plot S # for this sequence is shown in Figure 8(a). The dark lines correspond to two images in the sequence that are similar. The darkest line is the main diagonal, since S # (t, dark lines parallel to the main diagonal are formed since S # (t, kp/2+ t) # 0, where p is the period, and k is an integer. The dark lines perpendicular to the main diagonal are formed since S # (t, kp/2- t) # 0, and is due to the symmetry of human walking (see Figure 10). It is interesting to note that at the intersections of these lines, these images are similar to either (a), (b), or (c) in Figure 7 (see Figure 8(b)). That is, S # encodes the phase of the person walking, not just the period. This fact is exploited in the example in Section 4.5. The autocorrelation A of S # is shown in Figure 9(b). The peaks in A form a rotated square lattice Figure 2(b)), which is used for object classification (Section 4.4). Note that the magnitude of the peaks in A ( Figure 9(b)) have a pattern similar to the A in Figure 6(c). (a) (b) (c) Figure 7. Person walking on a treadmill. TSimilarity of Image T 1 and T 2 (a) TSimilarity of Image T 1 and T 2 AC AA (b) Figure 8. (a) Similarity plot for the person walking in Figure 7. (b) Lattice structure for the upper left quadrant of (a). At the intersections of the diagonal and cross diagonal lines are images similar to (a), (b), (c) in Figure 7. This can be used to determine the phase of the walking person. Frequency (Hz) Power Spectral Power Power Mean (a) Autocorrelation of Similarity TLag (b) Figure 9. (a) Power spectrum of similarity of a walking person. (b) Autocorrelation of the similarity of the walking person in Figure 7 (smoothed with a 5 5 filter). The peaks (shown with white '+' symbols) are used to fit the rotated square lattice in Figure 2(b). Figure 10. Cycle of a person walking (p = 32). Note the similarity of frame t and p/2 - t, and the similarity of frame t and p/2 t. We next analyze the motion of a person who is walking at an approximately 25 # offset to the camera's image plane from a static camera. (Figure 11(a)). The segmented person is approximately 20 pixels in height, and is shown in Figure 12(a). The similarity plot (Figure 11(b)) shows dark diagonal lines at a period of approximately 1 second (32 frames), which correspond to the period of the person's walking. The lighter diagonal lines shown with a period of approximately 0.5 seconds (16 frames) are explained by first noting that the person's right arm swing is not fully visible (due to the 25 # offset to the image plane). Therefore, it takes two steps for the body to be maximally self-similar, while the legs become very self-similar at every step. The effect of this is that the similarity measure S # is the composition of two periodic signals, with periods differing by a factor of two. This is shown in Figure 12(b), where the aligned object image is partitioned in the three segments (the upper 25%, next 25%, and lower 50% of the body), and S # is computed for each segment. The upper 25%, which includes the head and shoulders, shows no periodic motion; the next 25%, which includes the one visible arm, has a period double that of the lower 50% (which includes the legs). Figure 12(c) shows the average power spectrum for all the columns in S # . (a) TSimilarity of Image T 1 and T 2 (b) Figure 11. (a) First image of a 100-image walking sequence (the subject is walking approx. 25 # o#set from the camera's image plane). (b) Walking sequence similarity plot, which shows the similarity of the object (person) at times t 1 and t 2 . Dark regions show greater degrees of similarity. Similarity Similarity of Image 1 with Image T (a) Similarity Separated Similarity of Image 1 to Image T upper 25% next 25% lower 50% (b) Power Spectral Power of Similarity Power Mean (c) Figure 12. (a) Column 1 of Figure 11(b), with the corresponding segmented object for the local minima. (b) Image similarity for upper 25%, next 25%, and lower 50% of body. (c) Average power spectrum of all columns of Figure 11(b). 4.3 The Symmetry of a Running Dog In this example, we look at the periodicity of a running dog from a static camera. Figure 13 shows a complete cycle of a dog (a Black Labrador). Unlike the symmetry of a walking/running person, a running dog has a lack of similarity for S # (t, kp - t). This results in the similarity plot (Figure 14(a)) having dark lines parallel to the main diagonal, formed by S # (t, kp + t), but no lines perpendicular to the main diagonal (as with a walking/running person). The similarity plot has peaks (Figure 14(a)) that correspond to poses of the dog at frame 0 in Figure 13. The autocorrelation A of S # is shown in Figure 14(b); the peaks of the A form a square lattice (Figure 2(a)), which is used in Section 4.4 for object classification. Figure 13. Cycle of a running dog (p = 12). Note the lack of similarity for any two frames t 1 and t 2 , 4.4 Object Classification Using Periodicity A common task in an automated surveillance system is to classify moving objects. In this example, we classify three types of moving objects: people, dogs, and other. We use the lattice fitting method described in Section 3.4 for the classification, which is motivated by texture classification methods. Specifically, the square lattice M S (Figure 2(a)) is used to classify running dogs, and the 45 # square lattice MR (Figure 2(b)) is used to classify walking or running people. Note that M R,d is a subset of M S,d , so if both M S,d and M R,d match, MR is declared the winner. If neither lattice provides a good match to A, then the moving object is classified as other. The video database used to test the classification consists of video from both airborne surveillance (people and vehicles), and ground surveillance (people, vehicles, and dogs). The database consists of vehicle sequences (25 from airborne video); 55 person sequences (50 from airborne video); and 4 Similarity of Image T 1 and T 2 (a) Autocorrelation of Similarity TLag (b) Figure 14. (a) Similarity plot of the running dog in Figure 13. Note there there are no dark lines perpendicular to the main diagonal, as shown in Figure 8(a). (b) Autocorrelation of the similarity plot (smoothed with a 5 5 filter). The peaks (shown with white '+' symbols) are used to fit the square lattice in Figure 2(a). dog sequences (all from ground video). For the airborne video and dog sequences, the background was not segmented from the foreground object. For these sequences, the background was sufficiently homogeneous (e.g., dirt roads, parking lots, grassy fields) for this method to work. For the other sequences (taken with a static camera), the background was segmented as described in [6]. The airborne video in was recorded from a Sony XC-999 camera (640x240 @ 30fps) at an altitude of about 1500 feet. There is significant motion blur due to a slow shutter speed and fast camera motion. Additional noise is induced by the analog capture of the video from duplicated SVHS tape. Figure 15 shows a person running across a parking lot. The person is approximately 12x7 pixels in size (Fig- ure 16). The similarity plot in Figure 17(a) shows a clearly periodic motion, which corresponds to the person running. Figure 18 shows that the person is running with a frequency of 1.3Hz; the second peak at 2.6Hz is due to the symmetry of the human motion described in Section 4.2. The autocorrelation of S # is shown in Figure 17(b). Figure 19(b) shows the similarity plot for the vehicle in Figure 19(a), which has no periodicity. The spectral power for the vehicle (Figure 20(b)) is flat. The autocorrelation of S # has only 2 peaks (Figure 20(a)). The results of the classifications are shown in Table 1. The thresholds used for the lattice matching are those given in Section 3.4. Each sequence is 100 images (30 fps); a lag time of images (1 second) is used to compute A. Other Person Dog Other Person Table 1. Confusion matrix for person, dog, and other classification. 4.5 Counting People Another common task in an automated surveillance system is to count the number of people entering and leaving an area. This task is difficult, since when people are close to each other, it is not always Figure 15. Person running across a parking lot, viewed from a moving camera at an altitude of 1500'. (a) (b) (c) Figure 16. Zoomed images of the person in Figure 15, which correspond to the poses in Figure 7. The person is 12x7 pixels in size. Similarity of Image T 1 and T 2 Autocorrelation of Similarity TLag (b) Figure 17. (a) Similarity plot of the running person in Figure 15. (b) Autocorrelation of upper quadrant of S # . The peaks are used to fit the rotated square lattice in Figure 2(b). Power Spectral Power of Similarity Power Mean Figure 18. Spectral power of the running person in Figure 15. (a) Similarity of Image T 1 and T 2 (b) Figure 19. (a) Vehicle driving across a parking lot. (b) Similarity plot of the vehicle. Autocorrelation of Similarity TLag (a) Power Spectral Power power mean (b) Figure 20. (a) Spectral power of the vehicle in Figure 19(a). (b) Autocorrelation of S # of the vehicle in Figure 19(a) (smoothed with a 5 5 filter). The peaks are denoted by '+' symbols. simple to distinguish the individuals. For example, Figure 21(a) is a frame from an airborne video sequence that shows three people running along a road, and the result of the motion segmentation Figure 21(b)). Simple motion blob counting will give an inaccurate estimate of the number of people. However, if we know the approximate location of the airplane (via GPS) and have an approximate site model (a ground plane), we can estimate what the expected image size an "average" person should be. This size is used to window a region with motion for periodic detection. In this example, three non-overlapping windows were found to have periodic motion, each corresponding to a person. The similarity plots and spectral powers are shown in Figure 22. The similarity plots in Figure 22 can also be used to extract the phase angle of the running person. The phase angle is encoded in the position of the cross diagonals of S # . In this example, the phase angles are all significantly different from one another, giving further evidence that we have not over-counted the number of people. (a) (b) Figure 21. (a) Three people running, viewed from a moving camera at an altitude of 1500'. (b) Segmented motion. Person 1 Person 2 Person 3 Frequency (Hz) Power Frequency (Hz) Frequency (Hz) Figure 22. Similarity plots and spectral power for 3 people in Figure 21(a). Note that the frequency resolution is not as high as in Figure 18, since fewer frames are used to estimate the power. 4.6 Simple Event Detection In this example, we show how periodicity can be used as input for event detection. Figure 23(a) shows a person walking through a low contrast area (in a shadow) toward the camera; half way through the 200 image sequence the person stops swinging his arms and puts them into his pockets. This action is shown on the similarity plots for the upper and lower portions of the body. Specifically, in Figure 23(c), a periodic pattern for the upper part of the body is visible for the images [1,100], but not for [101,200]. This is further shown by the significant peak in the power spectrum for the images Figure 24(a)) and the lack of significant peaks in the power spectrum for the images [101,200] Figure 24(b)). Thus, while the image of the person is only 37 pixels high in this sequence and we are not tracking his body parts, we can deduce that he stopped swinging his arms at about frame 100. An automated surveillance system can use this technique to help decide if someone is carrying an object. In [13], we combine periodicity and shape analysis to detect if someone is carrying an object. 4.7 Non-Stationary Periodicity In this example, a person is walking, and roughly half way through the sequence, starts to run (see Figure 25(a)). The similarity plot (Figure 25(b)) clearly shows this transition. Using a short-time analysis windowing Hanning function of length 3300 ms (100 frames), the power is estimated in the (a) TSimilarity of Image T 1 and T 2 (Lower 40% of Body) (b) TSimilarity of Image T 1 and T 2 (Upper 60% of Body) (c) Figure 23. (a) Frame 100 from a low contrast 200 frame sequence; the subject (marked with a white arrow) puts his hands in his pockets halfway through the sequence. (b) Similarity plot of the lower 40% of the body. (c) Similarity plot of the upper 60% of the body. The periodicity ceases after the middle of the sequence. 0.20.40.60.8Frequency (Hz) Power Spectral Power of Similarity - Upper - Q2 Power Mean (a) Power Spectral Power of Similarity - Upper - Q4 Power Mean (b) Figure 24. (a) Power spectra of the upper left quadrant of Figure 23(c). (b) Power spectra of the lower right quadrant of Figure 23(c). walking and running stages (Figure 26). 4.8 Estimating Human Stride Using Periodicity In [26] and [20], human gait was used for person recognition. In this example, we do not analyze the gait (which is how people walk or run), but rather estimate the stride length of a walking or running person. The stride itself can be useful for person recognition, particularly during tracking. For example, stride length can help object (person) correspondence after occlusions. Stride length can also be used for input to a surveillance system to detect auto-theft in a parking area (e.g., a person of different size and stride length drove off with a car than the person who drove in with the car). Assume the area of surveillance has a site model, and the camera is calibrated. The estimated stride length is is the ground velocity of the person, and p is the period. For best results, v g and p should be filtered to reduce the inherent noise in the tracking and period estimation. For example, in Figure 25(a), the estimated stride of the person is 22" when walking, and 42" when running, which is within 2" of the person's actual stride. (a) TSimilarity of Image T 1 and T 2 50 100 150 200 250 300100200300 (b) Figure 25. (a) Person walking, then running. (b) Similarity plot of walking/running sequence. Frequency (Hz) Power Walking power mean Frequency (Hz) Power Running power mean Figure 26. Spectral power for walking/running sequence in Figure 25(a). 5 Real-Time System A real-time system has been implemented to track and classify objects using periodicity. The system uses a dual Processor 550MHz Pentium III Xeon-based PC, and runs at 15Hz with 640x240 grayscale images captured from an airborne video camera. The system uses the real-time stabilization results from [12]. We will briefly discuss how the method can be efficiently implemented to run on a real-time system. In computing S # , for each new frame, only a single column that corresponds to the new frame needs to be recomputed; the remaining entries can be reused (shifted) for the updated S # . Therefore, for each new frame, only O(N) S # (i, computations need to be done, where N is the number of rows and columns in S # . For computing A, the 2D FFT can be utilized to greatly decrease the computational cost [18]. Finally, SIMD instructions, such as those available on the Pentium III, can be utilized for computing as well as A (either directly or using the FFT). 6 Conclusions We have described new techniques to detect and analyze periodic motion as seen from both a static and moving camera. By tracking objects of interest, we compute an object's self-similarity as it evolves in time. For periodic motion, the self-similarity measure is also periodic, and we apply Time-Frequency analysis to detect and characterize the periodic motion. The periodicity is also analyzed robustly using the 2-D lattice structures inherent in similarity matrices. Future work includes using alternative independent motion algorithms for moving camera video, which could make the analysis more robust for non-homogeneous backgrounds for the case of a moving camera. Further use of the symmetries of motion for use in classification of additional types of periodic motion is also being investigated. Acknowledgments The airborne video was provided by the DARPA Airborne Video Surveillance project. This paper was written under the support of contract DAAL-01-97-K-0102 (ARPA Order E653), DAAB07-98-C- J019, and AASERT Grant DAAH-04-96-1-0221. --R Image sequence description using spatiotemporal flow curves: Toward Motion-Based Recog- nition Color indexing. Time Series: Theory and Methods. Recurrence plots revisited. Dynamic system representation Recurrence plots of dynamical systems. The pigeon's discrimination of movement patterns (lissajous figures) and contour-dependent rotational invariance The interpretation of visual motion: recognizing moving light displays. Backpack: Detection of people carrying objects using silhouettes. Comparing images using the hausdorff distance. Visual motion perception. Visual position stabilization in the hummingbird hawk moth Fast normalized cross-correlation Extracting periodicity of a regular texture based on autocorrelation functions. 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Aliaga, Efficient multi-viewpoint acquisition of 3D objects undergoing repetitive motions, Proceedings of the 2007 symposium on Interactive 3D graphics and games, April 30-May 02, 2007, Seattle, Washington Enrica Dente , Anil Anthony Bharath , Jeffrey Ng , Aldert Vrij , Samantha Mann , Anthony Bull, Tracking hand and finger movements for behaviour analysis, Pattern Recognition Letters, v.27 n.15, p.1797-1808, November, 2006 Enrica Dente , Anil Anthony Bharath , Jeffrey Ng , Aldert Vrij , Samantha Mann , Anthony Bull, Tracking hand and finger movements for behaviour analysis, Pattern Recognition Letters, v.27 n.15, p.1797-1808, November 2006 Tao Zhao , Ram Nevatia, Tracking Multiple Humans in Complex Situations, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.26 n.9, p.1208-1221, September 2004 S. Ioffe , D. A. 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motion-based recognition;periodic motion;motion segmention;object classification;person detection;motion symmetries

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A Bayesian Computer Vision System for Modeling Human Interactions.

AbstractWe describe a real-time computer vision and machine learning system for modeling and recognizing human behaviors in a visual surveillance task [1]. The system is particularly concerned with detecting when interactions between people occur and classifying the type of interaction. Examples of interesting interaction behaviors include following another person, altering one's path to meet another, and so forth. Our system combines top-down with bottom-up information in a closed feedback loop, with both components employing a statistical Bayesian approach [2]. We propose and compare two different state-based learning architectures, namely, HMMs and CHMMs for modeling behaviors and interactions. The CHMM model is shown to work much more efficiently and accurately. Finally, to deal with the problem of limited training data, a synthetic Alife-style training system is used to develop flexible prior models for recognizing human interactions. We demonstrate the ability to use these a priori models to accurately classify real human behaviors and interactions with no additional tuning or training.

Introduction We describe a real-time computer vision and machine learning system for modeling and recognizing human behaviors in a visual surveillance task. The system is particularly concerned with detecting when interactions between people occur, and classifying the type of interaction. Over the last decade there has been growing interest within the computer vision and machine learning communities in the problem of analyzing human behavior in video ([10],[3],[20], [8], [17], [14],[9], [11]). Such systems typically consist of a low- or mid-level computer vision system to detect and segment a moving object - human or car, for example -, and a higher level interpretation module that classifies the motion into 'atomic' behaviors such as, for example, a pointing gesture or a car turning left. However, there have been relatively few efforts to understand more human behaviors that have substantial extent in time, particularly when they involve interactions between people. This level of interpretation is the goal of this paper, with the intention of building systems that can deal with the complexity of multi-person pedestrian and highway scenes. This computational task combines elements of AI/machine learning and computer vision, and presents challenging problems in both domains: from a Computer Vision viewpoint, it requires real-time, accurate and robust detection and tracking of the objects of interest in an unconstrained environment; from a Machine Learning and Artificial Intelligence perspective behavior models for interacting agents are needed to interpret the set of perceived actions and detect eventual anomalous behaviors or potentially dangerous situations. Moreover, all the processing modules need to be integrated in a consistent way. Our approach to modeling person-to-person interactions is to use supervised statistical learning techniques to teach the system to recognize normal single-person behaviors and common person-to-person inter- actions. A major problem with a data-driven statistical approach, especially when modeling rare or anomalous behaviors, is the limited number of examples of those behaviors for training the models. A major emphasis of our work, therefore, is on efficient Bayesian integration of both prior knowledge (by the use of synthetic prior models) with evidence from data (by situation-specific parameter tuning). Our goal is to be able to successfully apply the system to any normal multi-person interaction situation without additional training. Another potential problem arises when a completely new pattern of behavior is presented to the system. After the system has been trained at a few different sites, previously unobserved behaviors will be (by definition) rare and unusual. To account for such novel behaviors the system should be able to recognize such new behaviors, and to build models of the behavior from as as little as a single example. We have pursued a Bayesian approach to modeling that includes both prior knowledge and evidence from data, believing that the Bayesian approach provides the best framework for coping with small data sets and novel behaviors. Graphical models [6], such as Hidden Markov Models (HMMs) [21] and Coupled Hidden Markov Models (CHMMs) [5, 4], seem most appropriate for modeling and classifying human behaviors because they offer dynamic time warping, a well-understood training algorithm, and a clear Bayesian semantics for both individual (HMMs) and interacting or coupled (CHMMs) generative processes. To specify the priors in our system, we have found it useful to develop a framework for building and training models of the behaviors of interest using synthetic agents. Simulation with the agents yields synthetic data that is used to train prior models. These prior models are then used recursively in a Bayesian frame-work to fit real behavioral data. This approach provides a rather straightforward and flexible technique to the design of priors, one that does not require strong analytical assumptions to be made about the form of the priors 1 . In our experiments we have found that by combining such synthetic priors with limited real data we can easily achieve very high accuracies of recognition of different human-to-human interactions. Thus, our system is robust to cases in which there are only a few examples of a certain behavior (such as in interaction described in section 5.1) or even no examples except synthetically-generated ones. The paper is structured as follows: section 2 presents an overview of the system, section 3 describes the computer vision techniques used for segmentation and tracking of the pedestrians, and the statistical models used for behavior modeling and recognition are described in section 4. Section 5 contains experimental results with both synthetic agent data and real video data, and section 6 summarizes the main conclusions and sketches our future directions of research. Finally a summary of the CHMM formulation is presented in the appendix. System Overview Our system employs a static camera with wide field-of- view watching a dynamic outdoor scene (the extension to an active camera [1] is straightforward and planned for the next version). A real-time computer vision system segments moving objects from the learned scene. The scene description method allows variations in lighting, weather, etc., to be learned and accurately discounted. 1 Note that our priors have the same form as our posteriors, namely they are Markov models. For each moving object an appearance-based description is generated, allowing it to be tracked though temporary occlusions and multi-object meetings. An Extended Kalman filter tracks the objects location, coarse shape, color pattern, and velocity. This temporally ordered stream of data is then used to obtain a behavioral description of each object, and to detect interactions between objects. Figure 1 depicts the processing loop and main functional units of our ultimate system. 1. The real-time computer vision input module detects and tracks moving objects in the scene, and for each moving object outputs a feature vector describing its motion and heading, and its spatial relationship to all nearby moving objects. 2. These feature vectors constitute the input to stochastic state-based behavior models. Both HMMs and CHMMs, with varying structures depending on the complexity of the behavior, are then used for classifying the perceived behaviors. Figure 1: Top-down and bottom-up processing loop Note that both top-down and bottom-up streams of information are continuously managed and combined for each moving object within the scene. Consequently our Bayesian approach offers a mathematical frame-work for both combining the observations (bottom-up) with complex behavioral priors (top-down) to provide expectations that will be fed back to the perceptual system. 3 Segmentation and Tracking The first step in the system is to reliably and robustly detect and track the pedestrians in the scene. We use 2-D blob features for modeling each pedes- trian. The notion of "blobs" as a representation for image features has a long history in computer vision [19, 15, 2, 25, 18], and has had many different mathematical definitions. In our usage it is a compact set of pixels that share some visual properties that are not shared by the surrounding pixels. These properties could be color, texture, brightness, motion, shading, a combination of these, or any other salient spatio-temporal property derived from the signal (the image sequence). 3.1 Segmentation by Eigenbackground subtraction In our system the main cue for clustering the pixels into blobs is motion, because we have a static background with moving objects. To detect these moving objects we build an adaptive eigenspace that models the background. This eigenspace model describes the range of appearances (e.g., lighting variations over the day, weather variations, etc.) that have been ob- served. The eigenspace can also be generated from a site model using standard computer graphics techniques The eigenspace model is formed by taking a sample of N images and computing both the mean - b background image and its covariance matrix C b . This covariance matrix can be diagonalized via an eigenvalue decomposition b , where \Phi b is the eigenvector matrix of the covariance of the data and L b is the corresponding diagonal matrix of its eigen- values. In order to reduce the dimensionality of the space, in principal component analysis (PCA) only M eigenvectors (eigenbackgrounds) are kept, corresponding to the M largest eigenvalues to give a \Phi M matrix. A principal component feature vector I then formed, where is the mean normalized image vector. Note that moving objects, because they don't appear in the same location in the N sample images and they are typically small, do not have a significant contribution to this model. Consequently the portions of an image containing a moving object cannot be well described by this eigenspace model (except in very unusual cases), whereas the static portions of the image can be accurately described as a sum of the the various eigenbasis vectors. That is, the eigenspace provides a robust model of the probability distribution function of the background, but not for the moving objects. Once the eigenbackground images (stored in a matrix called hereafter) are obtained, as well as their mean - b , we can project each input image I i onto the space expanded by the eigenbackground images X i to model the static parts of the scene, pertaining to the background. Therefore, by comput- Figure 2: Background mean image, blob segmentation image and input image with blob bounding boxes ing and thresholding the Euclidean distance (distance from feature space DFFS [16]) between the input image and the projected image we can detect the moving objects present in the scene: D where t is a given threshold. Note that it is easy to adaptively perform the eigenbackground subtraction, in order to compensate for changes such as big shad- ows. This motion mask is the input to a connected component algorithm that produces blob descriptions that characterize each person's shape. We have also experimented with modeling the background by using a mixture of Gaussian distributions at each pixel, as in Pfinder [26]. However we finally opted for the eigenbackground method because it offered good results and less computational load. 3.2 Tracking The trajectories of each blob are computed and saved into a dynamic track memory. Each trajectory has associated a first order Extended Kalman filter that predicts the blob's position and velocity in the next frame. Recall that the Kalman Filter is the 'best linear unbiased estimator' in a mean squared sense and that for Gaussian processes, the Kalman filter equations corresponds to the optimal Bayes' estimate. In order to handle occlusions as well as to solve the correspondence between blobs over time, the appearance of each blob is also modeled by a Gaussian PDF in RGB color space. When a new blob appears in the scene, a new trajectory is associated to it. Thus for each blob the Kalman-filter-generated spatial PDF and the Gaussian color PDF are combined to form a joint image space and color space PDF. In subsequent frames the Mahalanobis distance is used to determine the blob that is most likely to have the same identity. 4 Behavior Models In this section we develop our framework for building and applying models of individual behaviors and person-to-person interactions. In order to build effective computer models of human behaviors we need to address the question of how knowledge can be mapped onto computation to dynamically deliver consistent interpretations From a strict computational viewpoint there are two key problems when processing the continuous flow of feature data coming from a stream of input video: (1) Managing the computational load imposed by frame-by-frame examination of all of the agents and their interactions. For example, the number of possible interactions between any two agents of a set of N agents is N (N \Gamma 1)=2. If naively managed this load can easily become large for even moderate N ; (2) Even when the frame-by-frame load is small and the representation of each agent's instantaneous behavior is compact, there is still the problem of managing all this information over time. Statistical directed acyclic graphs (DAGs) or probabilistic inference networks (PINs) [7, 13] can provide a computationally efficient solution to these problems. HMMs and their extensions, such as CHMMs, can be viewed as a particular, simple case of temporal PIN or DAG. PINs consist of a set of random variables represented as nodes as well as directed edges or links between them. They define a mathematical form of the joint or conditional PDF between the random vari- ables. They constitute a simple graphical way of representing causal dependencies between variables. The absence of directed links between nodes implies a conditional independence. Moreover there is a family of transformations performed on the graphical structure that has a direct translation in terms of mathematical operations applied to the underlying PDF. Finally they are modular, i.e. one can express the joint global PDF as the product of local conditional PDFS. PINs present several important advantages that are relevant to our problem: they can handle incomplete data as well as uncertainty; they are trainable and easier to avoid overfitting; they encode causality in a natural way; there are algorithms for both doing prediction and probabilistic inference; they offer a frame-work for combining prior knowledge and data; and finally they are modular and parallelizable. In this paper the behaviors we examine are generated by pedestrians walking in an open outdoor envi- ronment. Our goal is to develop a generic, compositional analysis of the observed behaviors in terms of states and transitions between states over time in such a manner that (1) the states correspond to our common sense notions of human behaviors, and (2) they are immediately applicable to a wide range of sites and viewing situations. Figure 3 shows a typical image for our pedestrian scenario. Figure 3: A typical image of a pedestrian plaza Observations States O O O' States Observations S' Figure 4: Graphical representation of HMM and CHMM rolled-out in time 4.1 Visual Understanding via Graphical Models: HMMs and CHMMs Hidden Markov models (HMMs) are a popular probabilistic framework for modeling processes that have structure in time. They have a clear Bayesian seman- tics, efficient algorithms for state and parameter esti- mation, and they automatically perform dynamic time warping. An HMM is essentially a quantization of a system's configuration space into a small number of discrete states, together with probabilities for transitions between states. A single finite discrete variable indexes the current state of the system. Any information about the history of the process needed for future inferences must be reflected in the current value of this state variable. Graphically HMMs are often depicted 'rolled-out in time' as PINs, such as in figure 4. However, many interesting systems are composed of multiple interacting processes, and thus merit a compositional representation of two or more variables. This is typically the case for systems that have structure both in time and space. With a single state vari- able, Markov models are ill-suited to these problems. In order to model these interactions a more complex architecture is needed. Extensions to the basic Markov model generally increase the memory of the system (durational model- ing), providing it with compositional state in time. We are interested in systems that have compositional state in space, e.g., more than one simultaneous state variable. It is well known that the exact solution of extensions of the basic HMM to 3 or more chains is intractable. In those cases approximation techniques are needed ([22, 12, 23, 24]). However, it is also known that there exists an exact solution for the case of 2 interacting chains, as it is our case [22, 4]. We therefore use two Coupled Hidden Markov Models (CHMMs) for modeling two interacting processes, in our case they correspond to individual humans. In this architecture state chains are coupled via matrices of conditional probabilities modeling causal (tempo- ral) influences between their hidden state variables. The graphical representation of CHMMs is shown in figure 4. From the graph it can be seen that for each chain, the state at time t depends on the state at time both chains. The influence of one chain on the other is through a causal link. The appendix contains a summary of the CHMM formulation. In this paper we compare performance of HMMs and CHMMs for maximum a posteriori (MAP) state estimation. We compute the most likely sequence of states - S within a model given the observation sequence ng. This most likely sequence is obtained by - In the case of HMMs the posterior state sequence probability P (SjO) is given by Y (1) is the set of discrete states, corresponds to the state at time t. P ijj is the state-to-state transition probability (i.e. probability of being in state a i at time t given that the system was in state a j at time t \Gamma 1). In the following we will write them as P s t js . The prior probabilities for the initial state are P i are the output probabilities for each state, (i.e. the probability of observing given state a i at time t). In the case of CHMMs we need to introduce another set of probabilities, P s t js 0 , which correspond to the probability of state s t at time t in one chain given that the other chain - denoted hereafter by superscript 0 - was in state s 0 These new probabilities express the causal influence (coupling) of one chain to the other. The posterior state probability for CHMMs is given by (2) \Theta Y js t denote states and observations for each of the Markov chains that compose the CHMMs. We direct the reader to [4] for a more detailed description of the MAP estimation in CHMMs. Coming back to our problem of modeling human behaviors, two persons (each modeled as a generative process) may interact without wholly determining each others' behavior. Instead, each of them has its own internal dynamics and is influenced (either weakly or strongly) by others. The probabilities and js describe this kind of interactions and CHMMs are intended to model them in as efficient a manner as is possible. 5 Experimental Results Our goal is to have a system that will accurately interpret behaviors and interactions within almost any pedestrian scene with little or no training. One critical problem, therefore, is generation of models that capture our prior knowledge about human behavior. The selection of priors is one of the most controversial and open issues in Bayesian inference. To address this problem we have created a synthetic agents modeling package which allows us to build flexible prior behavior models. 5.1 Synthetic Agents Behaviors We have developed a framework for creating synthetic agents that mimic human behavior in a virtual en- vironment. The agents can be assigned different behaviors and they can interact with each other as well. Currently they can generate 5 different interacting behaviors and various kinds of individual behaviors (with no interaction). The parameters of this virtual environment are modeled on the basis of a real pedestrian scene from which we obtained (by hand) measurements of typical pedestrian movement. One of the main motivations for constructing such synthetic agents is the ability to generate synthetic data which allows us to determine which Markov model architecture will be best for recognizing a new behavior (since it is difficult to collect real examples of rare behaviors). By designing the synthetic agents models such that they have the best generalization and invariance properties possible, we can obtain flexible prior models that are transferable to real human behaviors with little or no need of additional training. The use of synthetic agents to generate robust behavior models from very few real behavior examples is of special importance in a visual surveillance task, where typically the behaviors of greatest interest are also the most rare. In the experiments reported here, we considered five different interacting behaviors: (1) Follow, reach and walk together (inter1), (2) Approach, meet and go on separately (inter2), (3) Approach, meet and go on together (inter3), (4) Change direction in order to meet, approach, meet and continue together (inter4), and Change direction in order to meet, approach, meet and go on separately (inter5). Note that we assume that these interactions can happen at any moment in time and at any location, provided only that the precondititions for the interactions are satisfied. For each agent the position, orientation and velocity is measured, and from this data a feature vector is constructed which consists of: - d 12 , the derivative of the relative distance between two agents; ff or degree of alignment of the agents, and the magnitude of their velocities. Note that such feature vector is invariant to the absolute position and direction of the agents and the particular environment they are in. Figure 5 illustrates the agents trajectories and associated feature vector for an example of interaction 2, i.e. an 'approach, meet and continue separately' behavior. -0.4 -0.2Alignment Relative distance Derivative of Relative distance Figure 5: Example trajectories and feature vector for interaction 2, or approach, meet and continue separately behavior. 5.1.1 Comparison of CHMM and HMM architectures We built models of the previously described interactions with both CHMMs and HMMs. We used 2 or 3 states per chain in the case of CHMMs, and 3 to 5 states in the case of HMMs (accordingly to the complexity of the various interactions). Each of these architectures corresponds to a different physical hy- pothesis: CHMMs encode a spatial coupling in time between two agents (e.g., a non-stationary process) whereas HMMs model the data as an isolated, stationary process. We used from 11 to 75 sequences for training each of the models, depending on their complexity, such that we avoided overfitting. The optimal number of training examples, of states for each interaction as well as the optimal model parameters were obtained by a 10% cross-validation process. In all cases, the models were set up with a full state-to- state connection topology, so that the training algorithm was responsible for determining an appropriate state structure for the training data. The feature vector was 6-dimensional in the case of HMMs, whereas in the case of CHMMs each agent was modeled by a different chain, each of them with a 3-dimensional feature vector. To compare the performance of the two previously described architectures we used the best trained models to classify 20 unseen new sequences. In order to find the most likely model, the Viterbi algorithm was used for HMMs and the N-heads dynamic programming forward-backward propagation algorithm for CHMMs. Table 5.1.1 illustrates the accuracy for each of the two different architectures and interactions. Note the superiority of CHMMs versus HMMs for classifying the different interactions and, more significantly, identifying the case in which there are no interactions present in the testing data. Accuracy on synthetic data HMMs CHMMs Table 1: Accuracy for HMMs and CHMMs on synthetic data. Accuracy at recognizing when no inter-action occurs ('No inter'), and accuracy at classifying each type of interaction: 'Inter1' is follow, reach and walk together; 'Inter2' is approach, meet and go on; 'Inter3' is approach, meet and continue together; 'In- ter4' is change direction to meet, approach, meet and go together and 'Inter5' is change direction to meet, approach, meet and go on separately Complexity in time and space is an important issue when modeling dynamic time series. The number of degrees of freedom (state-to-state probabili- ties+output means+output covariances) in the largest best-scoring model was 85 for HMMs and 54 for CHMMs. We also performed an analysis of the accuracies of the models and architectures with respect to the number of sequences used for training. Figure 5.1.1 illustrates the accuracies in the case of interaction 4 (change direction for meeting, stop and continue together). Efficiency in terms of training data is specially important in the case of on-line real-time learning systems -such as ours would ultimately be- and/or in domains in which collecting clean labeled data may be difficult. 901030507090number of sequences used for training accurancy Single HMMs Coupled HMMs curve for synthetic data CHMMs False alarm rate Detection rate Figure First figure: Accuracies of CHMMs (solid line) and HMMs (dotted line) for one particular in- teraction. The dashed line is the accuracy on testing without considering the case of no interaction, while the dark line includes this case. Second figure: ROC curve on synthetic data. The cross-product HMMs that result from incorporating both generative processes into the same joint- product state space usually requires many more sequences for training because of the larger number of parameters. In our case, this appears to result in a accuracy ceiling of around 80% for any amount of training that was evaluated, whereas for CHMMs we were able to reach approximately 100% accuracy with only a small amount of training. From this result it seems that the CHMMs architecture, with two coupled generative processes, is more suited to the problem of the behavior of interacting agents than a generative process encoded by a single HMM. In a visual surveillance system the false alarm rate is often as important as the classification accuracy. In an ideal automatic surveillance system, all the targeted behaviors should be detected with a close-to- zero false alarm rate, so that we can reasonably alert a human operator to examine them further. To analyze this aspect of our system's performance, we calculated the system's ROC curve. Figure 5.1.1 shows that it is quite possible to achieve very low false alarm rates while still maintaining good classification accuracy. 5.2 Pedestrian Behaviors Our goal is to develop a framework for detecting, classifying and learning generic models of behavior in a visual surveillance situation. It is important that the models be generic, applicable to many different situa- tions, rather than being tuned to the particular viewing or site. This was one of our main motivations for developing a virtual agent environment for modeling behaviors. If the synthetic agents are 'similar' enough in their behavior to humans, then the same models that were trained with synthetic data should be directly applicable to human data. This section describes the experiments we have performed analyzing real pedestrian data using both synthetic and site-specific models (models trained on data from the site being monitored). 5.2.1 Data collection and preprocessing Using the person detection and tracking system described in section 3 we obtained 2D blob features for each person in several hours of video. Up to 20 examples of following and various types of meeting behaviors were detected and processed. The feature vector - x coming from the computer vision processing module consisted of the 2D (x; y) centroid (mean position) of each person's blob, the Kalman Filter state for each instant of time, consisting of (-x; - y), where -: represents the filter estimation, and the (r; components of the mean of the Gaussian fitted to each blob in color space. The frame-rate of the vision system was of about 20-30 Hz on an SGI R10000 O2 computer. We low-pass filtered the data with a 3Hz cutoff filter and computed for every pair of nearby persons a feature vector consisting of: - derivative of the relative distance between two per- sons, jv i norm of the velocity vector for each person, or degree of alignment of the trajectories of each person. Typical trajectories and feature vectors for an 'approach, meet and continue separately' behavior (interaction 2) are shown in figure 7. This is the same type of behavior as the one displayed in figure 5 for the synthetic agents. Note the similarity of the feature vectors in both cases. 5.2.2 Behavior Models and Results CHMMs were used for modeling three different be- haviors: meet and continue together (interaction 3); meet and split (interaction 2) and follow (interaction 1). In addition, an interaction versus no interaction detection test was also performed. HMMs performed 1.5Velocity Magnitudes 100 200 300 -1.4 -1.2 -0.4 -0.2Alignment Relative distance 100 200 300 Derivative of Relative distance Figure 7: Example trajectories and feature vector for interaction 2, or approach, meet and continue separately behavior. much worse than CHMMs and therefore we omit reporting their results. We used models trained with two types of data: 1. Prior-only (synthetic data) models: that is, the behavior models learned in our synthetic agent environment and then directly applied to the real data with no additional training or tuning of the parameters. 2. Posterior (synthetic-plus-real data) models: new behavior models trained by using as starting points the synthetic best models. We used 8 examples of each interaction data from the specific site. Recognition accuracies for both these 'prior' and 'pos- terior' CHMMs are summarized in table 5.2.2. It is noteworthy that with only 8 training examples, the recognition accuracy on the real data could be raised to 100%. This results demonstrates the ability to accomplish extremely rapid refinement of our behavior models from the initial prior models. Finally the ROC curve for the posterior CHMMs is displayed in figure 8. One of the most interesting results from these experiments is the high accuracy obtained when testing the a priori models obtained from synthetic agent simulations. The fact that a priori models transfer so well to real data demonstrates the robustness of the approach. It shows that with our synthetic agent training system, we can develop models of many different types of behavior - avoiding thus the problem of limited amount of training data - and apply these models to real human behaviors without additional parameter tuning or training. Parameters sensitivity In order to evaluate the sensitivity of our classification accuracy to variations in the model parameters, we trained a set of models where we changed different parameters of the agents' Testing on real pedestrian data Prior Posterior CHMMs CHMMs No-inter 90.9 100 Table 2: Accuracy for both untuned, a priori models and site-specific CHMMs tested on real pedestrian data. The first entry in each row is the interaction vs no-interaction accuracy, the remaining entries are classification accuracies between the different interacting behaviors. Interactions are: 'Inter1' follow, reach and walk together; 'Inter2' approach, meet and go on; 'Inter3' approach, meet and continue together. curve for pedestrian data CHMM False alarm rate Detection rate Figure 8: ROC curve for real pedestrian data dynamics by factors of 2:5 and 5. The performance of these altered models turned out to be virtually the same in every case except for the 'inter1' (follow) inter- action, which seems to be sensitive to people's relative rates of movement. 6 Summary and Conclusions In this paper we have described a computer vision system and a mathematical modeling framework for recognizing different human behaviors and interactions in a visual surveillance task. Our system combines top-down with bottom-up information in a closed feedback loop, with both components employing a statistical Bayesian approach. Two different state-based statistical learning archi- tectures, namely HMMs and CHMMs, have been proposed and compared for modeling behaviors and in- teractions. The superiority of the CHMM formulation has been demonstrated in terms of both training efficiency and classification accuracy. A synthetic agent training system has been created in order to develop flexible and interpretable prior behavior models, and we have demonstrated the ability to use these a priori models to accurately classify real behaviors with no additional tuning or training. This fact is specially important, given the limited amount of training data available. Acknowledgments We would like to sincerely thank Michael Jordan, Tony Jebara and Matthew Brand for their inestimable help and insightful comments. Appendix A: Forward (ff) and Backward (fi) expressions for CHMMs In [4] a deterministic approximation for maximum a posterior (MAP) state estimation is introduced. It enables fast classification and parameter estimation via expectation maximization, and also obtains an upper bound on the cross entropy with the full (combina- toric) posterior which can be minimized using a sub-space that is linear in the number of state variables. An "N-heads" dynamic programming algorithm samples from the O(N ) highest probability paths through a compacted state trellis, with complexity O(T (CN for C chains of N states apiece observing T data points. For interesting cases with limited couplings the complexity falls further to O(TCN 2 ). For HMMs the forward-backward or Baum-Welch algorithm provides expressions for the ff and fi vari- ables, whose product leads to the likelihood of a sequence at each instant of time. In the case of CHMMs two state-paths have to be followed over time for each chain: one path corresponds to the 'head' (represented with subscript 'h') and another corresponds to the 'sidekick' (indicated with subscript 'k') of this head. Therefore, in the new forward-backward algorithm the expressions for computing the ff and fi variables will incorporate the probabilities of the head and sidekick for each chain (the second chain is indicated with 0 ). As an illustration of the effect of maintaining multiple paths per chain, the traditional expression for the ff variable in a single HMM: will be transformed into a pair of equations, one for the full posterior ff and another for the marginalized posterior ff: ff jh ff jh ff The fi variable can be computed in a similar way by tracing back through the paths selected by the forward analysis. After collecting statistics using N-heads dynamic programming, transition matrices within chains are re-estimated according to the conventional HMM expression. The coupling matrices are given by: --R Active perception vs. passive per- ception The representation space paradigm of concurrent evolving object de- scriptions Computers seeing action. Coupled hidden markov models for modeling interacting processes. Alex Pent- land Operations for learning with graphical models. A guide to the literature on learning probabilistic networks from data. Advanced visual surveillance using bayesian networks. What is going on? Active gesture recognition using partially observable markov decision processes. Building qualitative event models automatically from visual input. Factorial hidden Markov models. A tutorial on learning with bayesian networks. Automatic symbolic traffic scene analysis using belief networks. An unsupervised clustering approach to spatial preprocessing of mss imagery. Probabilistic visual learning for object detection. From image sequences towards conceptual descriptions. Lafter: Lips and face tracking. Classification by clustering. Modeling and prediction of human behavior. A tutorial on hidden markov models and selected applications in speech recognition. Boltzmann chains and hidden Markov models. Probabilistic independence networks for hidden Markov probability models. Mean field networks that learn to discriminate temporally distorted strings. --TR --CTR Koichi Sato , Brian L. Evans , J. K. 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Hidden Markov Models;tracking;visual surveillance;human behavior recognition;people detection

351611

Normalized Cuts and Image Segmentation.

AbstractWe propose a novel approach for solving the perceptual grouping problem in vision. Rather than focusing on local features and their consistencies in the image data, our approach aims at extracting the global impression of an image. We treat image segmentation as a graph partitioning problem and propose a novel global criterion, the normalized cut, for segmenting the graph. The normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups. We show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion. We have applied this approach to segmenting static images, as well as motion sequences, and found the results to be very encouraging.

Introduction Nearly launched the Gestalt approach which laid out the importance of perceptual grouping and organization in visual perception. For our purposes, the problem of grouping can be well motivated by considering the set of points shown in the figure (1). Figure 1: How many groups? Typically a human observer will perceive four objects in the image-a circular ring with a cloud of points inside it, and two loosely connected clumps of points on its right. However this is not the unique partitioning of the scene. One can argue that there are three objects- the two clumps on the right constitute one dumbbell shaped object. Or there are only two objects, a dumb bell shaped object on the right, and a circular galaxy like structure on the left. If one were perverse, one could argue that in fact every point was a distinct object. This may seem to be an artificial example, but every attempt at image segmentation ultimately has to confront a similar question-there are many possible partitions of the domain D of an image into subsets D i (including the extreme one of every pixel being a separate entity). How do we pick the "right" one? We believe the Bayesian view is appropriate- one wants to find the most probable interpretation in the context of prior world knowledge. The difficulty, of course, is in specifying the prior world knowledge-some of it is low level such as coherence of brightness, color, texture, or motion, but equally important is mid- or high-level knowledge about symmetries of objects or object models. This suggests to us that image segmentation based on low level cues can not and should not aim to produce a complete final "correct" segmentation. The objective should instead be to use the low-level coherence of brightness, color, texture or motion attributes to sequentially come up with hierarchical partitions. Mid and high level knowledge can be used to either confirm these groups or select some for further attention. This attention could result in further repartitioning or grouping. The key point is that image partitioning is to be done from the big picture downwards, rather like a painter first marking out the major areas and then filling in the details. Prior literature on the related problems of clustering, grouping and image segmentation is huge. The clustering community[12] has offered us agglomerative and divisive algorithms; in image segmentation we have region-based merge and split algorithms. The hierarchical divisive approach that we advocate produces a tree, the dendrogram. While most of these ideas go back to the 70s (and earlier), the 1980s brought in the use of Markov Random Fields[10] and variational formulations[17, 2, 14]. The MRF and variational formulations also exposed two basic questions (1) What is the criterion that one wants to optimize? and (2) Is there an efficient algorithm for carrying out the optimization? Many an attractive criterion has been doomed by the inability to find an effective algorithm to find its minimum-greedy or gradient descent type approaches fail to find global optima for these high dimensional, nonlinear problems. Our approach is most related to the graph theoretic formulation of grouping. The set of points in an arbitrary feature space are represented as a weighted undirected graph E), where the nodes of the graph are the points in the feature space, and an edge is formed between every pair of nodes. The weight on each edge, w(i; j), is a function of the similarity between nodes i and j. In grouping, we seek to partition the set of vertices into disjoint sets where by some measure the similarity among the vertices in a set V i is high and across different sets V i ,V j is low. To partition a graph, we need to also ask the following questions: 1. What is the precise criterion for a good partition? 2. How can such a partition be computed efficiently? In the image segmentation and data clustering community, there has been much previous work using variations of the minimal spanning tree or limited neighborhood set approaches. Although those use efficient computational methods, the segmentation criteria used in most of them are based on local properties of the graph. Because perceptual grouping is about extracting the global impressions of a scene, as we saw earlier, this partitioning criterion often falls short of this main goal. In this paper we propose a new graph-theoretic criterion for measuring the goodness of an image partition- the normalized cut. We introduce and justify this criterion in section 2. The minimization of this criterion can be formulated as a generalized eigenvalue problem; the eigenvectors of this problem can be used to construct good partitions of the image and the process can be continued recursively as desired(section 2.1) Section 3 gives a detailed explanation of the steps of our grouping algorithm. In section 4 we show experimental results. The formulation and minimization of the normalized cut criterion draws on a body of results from the field of spectral graph theory(section 5). Relationship to work in computer vision is discussed in section 6, and comparison with related eigenvector based segmentation methods is represented in section 6.1. We conclude in section 7. Grouping as graph partitioning A graph E) can be partitioned into two disjoint sets, A; B, by simply removing edges connecting the two parts. The degree of dissimilarity between these two pieces can be computed as total weight of the edges that have been removed. In graph theoretic language, it is called the cut: w(u; v): (1) The optimal bi-partitioning of a graph is the one that minimizes this cut value. Although there are exponential number of such partitions, finding the minimum cut of a graph is a well studied problem, and there exist efficient algorithms for solving it. Wu and Leahy[25] proposed a clustering method based on this minimum cut criterion. In particular, they seek to partition a graph into k-subgraphs, such that the maximum cut across the subgroups is minimized. This problem can be efficiently solved by recursively finding the minimum cuts that bisect the existing segments. As shown in Wu & Leahy's work, this globally optimal criterion can be used to produce good segmentation on some of the images. However, as Wu and Leahy also noticed in their work, the minimum cut criteria favors cutting small sets of isolated nodes in the graph. This is not surprising since the cut defined in (1) increases with the number of edges going across the two partitioned parts. Figure (2) illustrates one such case. Assuming the edge weights are inversely proportional to the better cut Figure 2: A case where minimum cut gives a bad partition. distance between the two nodes, we see the cut that partitions out node n 1 or n 2 will have a very small value. In fact, any cut that partitions out individual nodes on the right half will have smaller cut value than the cut that partitions the nodes into the left and right halves. To avoid this unnatural bias for partitioning out small sets of points, we propose a new measure of disassociation between two groups. Instead of looking at the value of total edge weight connecting the two partitions, our measure computes the cut cost as a fraction of the total edge connections to all the nodes in the graph. We call this disassociation measure the normalized cut (Ncut): (2) u2A;t2V w(u; t) is the total connection from nodes in A to all nodes in the graph, and assoc(B; V ) is similarly defined. With this definition of the disassociation between the groups, the cut that partitions out small isolated points will no longer have small Ncut value, since the cut value will almost certainly be a large percentage of the total connection from that small set to all other nodes. In the case illustrated in figure 2, we see that the cut 1 value across node n 1 will be 100% of the total connection from that node. In the same spirit, we can define a measure for total normalized association within groups for a given partition: where assoc(A; are total weights of edges connecting nodes within A and B respectively. We see again this is an unbiased measure, which reflects how tightly on average nodes within the group are connected to each other. Another important property of this definition of association and disassociation of a partition is that they are naturally related: Hence the two partition criteria that we seek in our grouping algorithm, minimizing the disassociation between the groups and maximizing the association within the group, are in fact identical, and can be satisfied simultaneously. In our algorithm, we will use this normalized cut as the partition criterion. Unfortunately minimizing normalized cut exactly is NP-complete, even for the special case of graphs on grids. The proof, due to C. Papadimitriou, can be found in appendix A. However, we will show that when we embed the normalized cut problem in the real value domain, an approximate solution can be found efficiently. 2.1 Computing the optimal partition Given a partition of nodes of a graph, V, into two sets A and B, let x be an dimensional indicator vector, x node i is in A, and \Gamma1 otherwise. Let be the total connection from node i to all other nodes. With the definitions x and d we can rewrite Ncut(A; B) as: Let D be an N \Theta N diagonal matrix with d on its diagonal, W be an N \Theta N symmetrical matrix with , and 1 be an N \Theta 1 vector of all ones. Using the fact 1+xand 1\Gammaxare indicator vectors for respectively, we can rewrite as: k1 T D1 can then further expand the above equation as: Dropping the last constant term, which in this case equals 0, we get Letting , and since Setting is easy to see that since Putting everything together we have, with the condition y(i) 2 f1; \Gammabg and y T Note that the above expression is the Rayleigh quotient[11]. If y is relaxed to take on real values, we can minimize equation (5) by solving the generalized eigenvalue system, (D However, we have two constraints on y, which come from the condition on the corresponding indicator vector x. First consider the constraint y T We can show this constraint on y is automatically satisfied by the solution of the generalized eigensystem. We will do so by first transforming equation (6) into a standard eigensystem, and show the corresponding condition is satisfied there. Rewrite equation (6) as y. One can easily verify that z 1 is an eigenvector of equation (7) with eigenvalue of 0. Furthermore, 2 is symmetric positive semidefinite, since (D \Gamma W), also called the Laplacian matrix, is known to be positive semidefinite[18]. Hence z 0 is in fact the smallest eigenvector of equation (7), and all eigenvectors of equation are perpendicular to each other. In particular, z 1 the second smallest eigenvector is perpendicular to z 0 . Translating this statement back into the general eigensystem (6), we have 1 is the smallest eigenvector with eigenvalue of 0, and 1, where y 1 is the second smallest eigenvector of (6). Now recall a simple fact about the Rayleigh quotient[11]: Let A be a real symmetric matrix. Under the constraint that x is orthogonal to the smallest eigenvectors x 1 the quotient x T Ax x T x is minimized by the next smallest eigenvector x j , and its minimum value is the corresponding eigenvalue - j . As a result, we obtain: z z and consequently, Thus the second smallest eigenvector of the generalized eigensystem (6) is the real valued solution to our normalized cut problem. The only reason that it is not necessarily the solution to our original problem is that the second constraint on y that y(i) takes on two discrete values is not automatically satisfied. In fact relaxing this constraint is what makes this optimization problem tractable in the first place. We will show in section (3) how this real valued solution can be transformed into a discrete form. A similar argument can also be made to show that the eigenvector with the third smallest eigenvalue is the real valued solution that optimally sub-partitions the first two parts. In fact this line of argument can be extended to show that one can sub-divide the existing graphs, each time using the eigenvector with the next smallest eigenvalue. However, in practice because the approximation error from the real valued solution to the discrete valued solution accumulates with every eigenvector taken, and all eigenvectors have to satisfy a global mutual orthogonality constraint, solutions based on higher eigenvectors become unreliable. It is best to restart solving the partitioning problem on each subgraph individually. It is interesting to note that, while the second smallest eigenvector y of (6) only approximates the optimal normalized cut solution, it exactly minimizes the following problem: in real-valued domain, where Roughly speaking, this forces the indicator vector y to take similar values for nodes i and j that are tightly coupled(large w ij ). In summary, we propose using the normalized cut criteria for graph partitioning, and we have shown how this criteria can be computed efficiently by solving a generalized eigenvalue problem. 3 The grouping algorithm Our grouping algorithm consists of the following steps: 1. Given an image or image sequence, set up a weighted graph E), and set the weight on the edge connecting two nodes being a measure of the similarity between the two nodes. 2. Solve (D \Gamma eigenvectors with the smallest eigenvalues. 3. Use the eigenvector with second smallest eigenvalue to bipartition the graph. 4. Decide if the current partition should be sub-divided, and recursively repartition the segmented parts if necessary. The grouping algorithm as well as its computational complexity can be best illustrated by using the following two examples. 3.1 Example 1: Point Set Case Take the example illustrated in figure 3. Suppose we would like to group points on the 2D plane based purely on their spatial proximity. This can be done through the following steps: 1. Define a weighted graph E), by taking each point as a node in the graph, and connecting each pair of nodes by a graph edge. The weight on the graph edge connecting node i and j is set to be w(i; is the Euclidean distance between the two nodes, and oe x controls the scale of the spatial proximity measure. oe x is set to be 2.0 which is 10% of the height of the point set layout. Figure 4 shows the weight matrix for the weighted graph constructed. Figure 3: A point set in the plane. Figure 4: The weight matrix constructed for the point set in (3), using spatial proximity as the similarity measure. The points in figure (3), are numbered as: 1-90, points in the circular ring in counter-clockwise order, 90-100, points in the cluster inside the ring, 100-120, and 120-140, the upper and lower clusters on the right respectively. Notice that the non-zero weights are mostly concentrated in a few blocks around the diagonal. The entries in those square blocks are the connections within each of the clusters. 2. Solve for the eigenvectors with the smallest eigenvalues of the system (D or equivalently the eigenvectors with the largest eigenvalues of the system This can be done by using a generalized eigenvector solver, or by first transforming the generalized eigenvalue system of (11) or (12) into standard eigenvector problems of or with y, and solve it with a standard eigenvector solver. Either way, we will obtain the solution of the eigenvector problem as shown in figure 5. 3. Partition the point set using the eigenvectors computed. As we have shown, the eigenvector with the second smallest eigenvalue is the continuous approximation to the discrete bi-partitioning indicator vector that we seek. For the case that we have constructed, the eigenvector with the second smallest eigenvalue(figure 5.3) is indeed very close to a discrete one. One can just partition the nodes in the graph based on the sign of their values in the eigenvector. Using this rule, we can partition the point set into two sets as shown in figure 6. To recursively subdivide each of the two groups, we can either 1) rerun the above procedure on each of the individual groups, or 2) using the the eigenvectors with the next smallest eigenvalues as approximation to the sub-partitioning indicator vectors for each of the groups. We can see that in this case, the eigenvector with the third and the fourth smallest eigenvalue are also good partitioning indicator vectors. Using zero as the splitting point, one can partition the nodes based on their values in the eigenvector into two halves. The sub-partition of the existing groups based on those two subsequent eigenvectors are shown in figure 7. In this case, we see that the eigenvectors computed from system (11) are very close to the discrete solution that we seek. The second smallest eigenvalue is very close to the optimal eigenvector eigenvalue -0.2 -0.050.10.2 -0.06 -0.06 -0.06 Figure 5: Subplot (1) plots the smallest 10 eigenvalues of the generalized eigenvalue system (11). Subplot (2) - (6) shows the eigenvectors corresponding the the 5 smallest eigenvalues of the system. Note the eigenvector with the smallest eigenvalue (2) is a constant as shown in section 2, and eigenvector with the second smallest eigenvalue (3) is an indicator vector: it takes on only positive values for points in the two clusters to the right. Therefore, using this eigenvector, we can find the first partition of the point set into two clusters: the points on the left with the ring and the cluster inside, and points on the right with the two dumb bell shaped clusters. Furthermore, note that the eigenvectors with the third smallest eigenvalue, subplot (3) and the fourth smallest eigenvalue, subplot(4), are indicator vectors which can be used to partition apart the ring set with the cluster inside of it, and the two clusters on the right. Figure Partition of the point set using the eigenvector with the second smallest eigenvalue. Figure 7: Sub-partitioning of the point sets using the eigenvectors with the third and fourth smallest eigenvalues. normalized cut value. For the general case, although we are not necessarily this lucky, there is a bound on how far the second smallest eigenvalue can deviate from the optimal normalized cut value, as we shall see in section 5. However, there is little theory on how close the eigenvector is to the discrete form that the normalized cut algorithm seeks. In our experience, the eigenvector computed is quite close to the desired discrete solution. 3.2 Example 2: Brightness Images Having studied an example of point set grouping, we turn our attention to the case of static image segmentation based on brightness and spatial features. Figure 8 shows an image that we would like to segment. Figure 8: A gray level image of a baseball game. Just as in the point set grouping case, we have the following steps for image segmentation: 1. Construct a weighted graph, E), by taking each pixel as a node, and connecting each pair of pixels by an edge. The weight on that edge should reflect the likelihood of the two pixels belong to one object. Using just the brightness value of the pixels and their spatial location, we can define the graph edge weight connecting two nodes i and j as: \GammakF (i)\GammaF (j)k 2oe I e \GammakX (i)\GammaX (j)k2 Figure 9 shows the weight matrix W associated with this weighted graph. nr (b) n=nr * nc n=nr * nc (c) (a) (d) Figure 9: The similarity measure between each pair of pixels in (a) can be summarized in a n \Theta n weight matrix W, shown in (b), where n is the number of pixels in the image. Instead of displaying W itself, which is very large, two particular rows, i 1 and i 2 of W are shown in (c) and (d). Each of the rows, is the connection weights from a pixel to all other pixels in the image. The two rows,i 1 and i 2 are reshaped into the size of the image, and displayed. The brightness value in (c) and (d) reflects the connection weights. Note that W contains large number of zeros or near zeros, due to the spatial proximity factor. 2. Solve for the eigenvectors with the smallest eigenvalues of the system (D As we saw above, the generalized eigensystem in (16) can be transformed into a standard eigenvalue problem of Solving a standard eigenvalue problem for all eigenvectors takes O(n 3 ) operations, where n is the number of nodes in the graph. This becomes impractical for image segmentation applications where n is the number of pixels in an image. Fortunately, our graph partitioning has the following properties: 1) the graphs often are only locally connected and the resulting eigensystem are very sparse, 2) only the top few eigenvectors are needed for graph partitioning, and 3) the precision requirement for the eigenvectors is low, often only the right sign bit is re- quired. These special properties of our problem can be fully exploited by an eigensolver called the Lanczos method. The running time of a Lanczos algorithm is O(mn) +O(mM(n))[11], where m is the maximum number of matrix-vector computations required, and M(n) is the cost of a matrix-vector computation of Ax, where . Note that sparse structure in A is identical to that of the weight matrix W. Due to the sparse structure in the weight matrix W, and therefore A, the matrix- vector computation is only of O(n), where n is the number of the nodes. To see why this is the case, we will look at the cost of the inner product of one row of A with a vector x. Let y . For a fixed i, A ij is only nonzero if node j is in a spatial neighborhood of i. Hence there are only a fixed number of operations required for each A i \Delta x, and the total cost of computing Ax is O(n). Figure 10 is graphical illustration of this special inner product operation for the case of the image segmentation. Furthermore, it turns out that we can substantially cut down additional connections from each node to its neighbors by randomly selecting the connections within the neighborhood for the weighted graph as shown in figure 11. Empirically, we have found that one can remove up to 90% of the total connections with each of the neighborhoods when the neighborhoods are large, without effecting the eigenvector solution to the system. nr w(i,j)00 x y x y A Figure 10: The inner product operation between A(i; and x, of dimension n, is a convolution operation in the case of image segmentation. (a) (b) Figure 11: Instead of connecting a node i to all the nodes in its neighborhood (indicated by the shaded area in (a)), we will only connect i to randomly selected nodes(indicated by the open circles in (b)). Putting everything together, each of the matrix-vector computations cost O(n) operations with a small constant factor. The number m depends on many factors[11]. In our experiments on image segmentation, we observed that m is typically less than O(n 1 Figure 12 shows the smallest eigenvectors computed for the generalized eigensystem with the weight matrix defined above. (1) (2) (3) eigenvalue Figure 12: Subplot (1) plots the smallest eigenvectors of the generalized eigenvalue system (11). Subplot (2) - (9) shows the eigenvectors corresponding the 2nd smallest to the 9th smallest eigenvalues of the system. The eigenvectors are reshaped to be the size of the image. 3. Once the eigenvectors are computed, we can partition the graph into two pieces using the second smallest eigenvector. In the ideal case, the eigenvector should only take on two discrete values, and the signs of the values can tell us exactly how to partition the graph. However, our eigenvectors can take on continuous values, and we need to choose a splitting point to partition it into two parts. There are many different ways of choosing such splitting point. One can take 0 or the median value as the splitting point, or one can search for the splitting point such that the resulting partition has the best Ncut(A; B) value. We take the latter approach in our work. Currently, the search is done by checking l evenly spaced possible splitting points, and computing the best Ncut among them. In our experiments, the values in the eigenvectors are usually well separated, and this method of choosing a splitting point is very reliable even with a small l. Figure 13 shows this process. 4. After the graph is broken into two pieces, we can recursively run our algorithm on the two partitioned parts. Or equivalently, we could take advantage of the special properties of the other top eigenvectors as explained in previous section to subdivide the graph based on those eigenvectors. The recursion stops once the Ncut value exceeds certain limit. We also impose a stability criterion on the partition. As we saw earlier, and as we see in the eigenvectors with the 7-9th smallest eigenvalues(figure(12.7-9)), sometimes an eigenvector can take on the shape of a continuous function rather that the discrete indicator function that we seek. From the view of segmentation, such an eigenvector is attempting to subdivide an image region where there is no sure way of breaking it. In fact, if we are forced to partition the image based on this eigenvector, we will see there are many different splitting points which have similar Ncut values. Hence the partition will be highly uncertain and unstable. In our current segmentation scheme, we simply choose to ignore all those eigenvectors which have smoothly varying eigenvector values. We achieve this by imposing a stability criterion which measures the degree of smoothness in the eigenvector values. The simplest measure is based on first computing the histogram of the eigenvector values, and then computing the ratio between the minimum and maximum values in the bins. When the eigenvector values are continuously varying, the values in the histogram bins will stay relatively the same, and the ratio will be relatively high. In our experiments, we find that simple thresholding on the ratio described above can be used to exclude unstable eigenvectors. We have set that value to be 0.06 in all our experiments. Figure 14 shows the final segmentation for the image shown in figure 8. -0.2 -0.10.10.30.50.7(d) (b) (a) (c) Figure 13: The eigenvector in (a) is a close approximation to a discrete partitioning indicator vector. Its histogram, shown in (b), indicates that the values in the eigenvector cluster around two extreme values. (c) and (d) shows the partitioning results with different splitting points indicated by the arrows in (b). The partition with the best normalized cut value is chosen. (a) (b) (c) (d) (e) (f) Figure 14: (a) shows the original image of size 80 \Theta 100. Image intensity is normalized to lie within 0 and 1. Subplot (b) - (h) shows the components of the partition with Ncut value less than 0.04. Parameter setting: oe I = 0:1, oe 3.3 Recursive 2-way Ncut In summary, our grouping algorithm consists of the following steps: 1. Given a set of features, set up a weighted graph E), compute the weight on each edge, and summarize the information into W, and D. 2. Solve (D \Gamma eigenvectors with the smallest eigenvalues. 3. Use the eigenvector with second smallest eigenvalue to bipartition the graph by finding the splitting point such that Ncut is maximized, 4. Decide if the current partition should be sub-divided by checking the stability of the cut, and make sure Ncut is below pre-specified value. Recursively repartition the segmented parts if necessary. The number of groups segmented by this method is controlled directly by the maximum allowed Ncut. 3.4 Simultanous K-way cut with multiple eigenvectors One drawback of the recursive 2-way cut is its treatment of the oscillatory eigenvectors. The stability criteria provides us from cutting oscillatory eigenvectors, but it also prevents us cutting the subsequent eigenvectors which might be perfect partitioning vectors. Also the approach is computationally wasteful; only the second eigenvector is used whereas the next few small eigenvectors also contain useful partitioning information. Instead of finding the partition using recursive 2-way cut as described above, one can use the all the top eigenvectors to simultanously obtain a K-way partition. In this method, the n top eigenvectors are used as n dimensional indicator vectors for each pixel. In the first step, a simple clustering algorithm, such as the k-means algorithm, is used to obtain an over-segmentation of the image into k 0 groups. No attempt is made to identify and exclude oscillatory eigenvectors-they exarcabete the oversegmentation, but that will be dealt with subsequently. In the second step, one can proceed in the following two ways: 1. Greedy pruning: iteratively merge two segments at a time until only k segments are left. At each merge step, those two segments are merged that minimize the k-way Ncut criterion defined as: where A i is the ith subset of whole set V. This computation can be efficiently carried out by iteratively updating the compacted weight matrix W c , with W c (i; 2. Global recursive cut. From the initial k 0 segments we can build a condensed graph segment A i corresponds to a node V c i of the graph. The weight on each graph edge W c (i; j) is defined to be the total edge weights from elements in A i to elements in A j . From this condensed graph, we then recursively bi-partition the graph according the Ncut criteria. This can be carried out either with the generalized eigenvalue system as in section 3.3, or with exhaustive search in the discrete domain. Exhaustive search is possible in this case since k 0 is small, typically k 0 - 100. We have experimented with this simultanous k-way cut method on our recent test images. However, the results presented in this paper are all based on the recursive 2-way partitioning algorithm outlined in the previous subsection 3.3. 4 Experiments We have applied our grouping algorithm to image segmentation based on brightness, color, texture, or motion information. In the monocular case, we construct the graph E) by taking each pixel as a node, and define the edge weight w ij between node i and j as the product of a feature similarity term and spatial proximity term: \GammakF (i)\GammaF (j)k 2oe I e \GammakX (i)\GammaX (j)k 2oe X if where X(i) is the spatial location of node i, and F (i) is a feature vector based on intensity, color, or texture information at that node defined as: in the case of segmenting point sets, I(i), the intensity value, for segmenting brightness images, are the HSV values, for color segmentation, where the f i are DOOG filters at various scales and orientations as used in[16], in the case of texture segmentation. Note that the weight any pair of nodes i and j that are more than r pixels apart. We first tested our grouping algorithm on spatial point sets similar to the one shown in figure (2). Figure (15) shows a point set and the segmentation result. As we can see from the figure, the normalized cut criterion is indeed able to partition the point set in a desirable way as we have argued in section (2). Figure 15: (a) Point set generated by two Poisson processes, with densities of 2.5 and 1.0 on the left and right clusters respectively, (b) 4 and \Theta indicates the partition of point set in (a). Parameter settings: oe Figures (16), (17), (18), and (19) shows the result of our segmentation algorithm on various brightness images. Figure (16), (17) are synthetic images with added noise. Figure (18) and (19) are natural images. Note that the "objects" in figure (19) have rather ill-defined c a Figure image showing a noisy "step" image. Intensity varies from 0 to 1, and Gaussian noise with shows the eigenvector with the second smallest eigenvalue, and subplot (c) shows the resulting partition. a b c d Figure 17: (a) A synthetic image showing three image patches forming a junction. Image intensity varies from 0 to 1, and Gaussian noise with oe = 0:1 is added. (b)-(d) shows the top three components of the partition. a b c d Figure (a) shows a 80x100 baseball scene, image intensity is normalized to lie within 0 and 1. (b)-(h) shows the components of the partition with Ncut value less than 0.04. Parameter setting: oe I = 0.01, oe 5. a b c d Figure 19: (a) shows a 126x106 weather radar image. (b)-(g) show the components of the partition with Ncut value less than 0.08. Parameter setting: oe I = 0:005, oe boundaries which would make edge detection perform poorly. Figure (20) shows the segmentation on a color image, reproduced in gray scale in these transactions. The original image and many other examples can be found at web site http://www.cs.berkeley.edu/~jshi/Grouping. Note that in all these examples the algorithm is able to extract the major components of scene, while ignoring small intra-component variations. As desired, recursive partitioning can be used to further decompose each piece. a b c Figure 20: (a) shows a 77x107 color image. (b)-(e) show the components of the partition with Ncut value less than 0.04. Parameter settings: oe I = 0.01, oe 5. Figure shows preliminary results on texture segmentation for a natural image of a zebra against a background. Note that the measure we have used is orientation-variant, and therefore parts of the zebra skin with different stripe orientation should be marked as separate regions. In the motion case, we will treat the image sequence as a spatiotemporal data set. Given an image sequence, a weighted graph is constructed by taking each pixel in the image sequence as a node, and connecting pixels that are in the spatiotemporal neighborhood of each other. The weight on each graph edge is defined as: e \Gammad a b c d Figure 21: (a) shows an image of a zebra. The remaining images show the major components of the partition. The texture features used correspond to convolutions with DOOG filters[16] at 6 orientations and 5 scales. where d(i; j) is the "motion distance" between two pixels i and j. Note that X i in this case represents the spatial-temporal position of pixel i. To compute this "motion distance", we will use a motion feature called motion profile. By motion profile we seek to estimate the probability distribution of image velocity at each pixel. Let I t (X) denote a image window centered at the pixel at location X 2 R 2 at time t. We denote by P i (dx) the motion profile of an image patch at node i, I t (X i ), at time t corresponding to another image patch I t+1 (X i +dx) at time t can be estimated by first computing the similarity S i (dx) between I t (X i ) and I t+1 (X i +dx), and normalizing it to get a probability distribution: There are many ways one can compute similarity between two image patches; we will use a measure that is based on the sum of squared differences(SSD): ssd local neighborhood of image patch I t (X i ). The "motion distance" between two image pixels is then defined as one minus the cross-correlation of the motion profiles: dx In figure (22) and (23) we show results of the normalized cut algorithm on a synthetic random dot motion sequence and a indoor motion sequence respectively. For more elaborate discussion on motion segmentation using normalized cut, as well as how to segment and track over long image sequences, readers might want to refer to our paper[21]. 4.1 Computation time As we saw from section 3.2, the running time of the normalized cut algorithm is O(mn), where n is the number of pixels, and m is the number of steps Lanczos takes to converge. On the 100 \Theta 120 test images shown here, the normalized cut algorithm takes about 2 minutes on a Intel Pentium 200MHz machines. Figure 22: Row (a) of this plot shows the six frames of a synthetic random dot image sequence. Row (b) shows the outlines of the two moving patches in this image sequence. The outlines shown here are for illustration purposes only. Row (1)-(3) shows the top three partitions of this image sequence that have Ncut values less than 0.05. The segmentation algorithm produces 3D space-time partitions of the image sequence. Cross-sections of those partitions are shown here. The original image size is 100 \Theta 100, and the segmentation is computed using image patches(superpixels) of size 3 \Theta 3. Each image patch is connected to other image patches that are less than 5 superpixels away in spatial distance, and 3 frames away in temporal distance. Figure 23: Subimage (a) and (b) shows two frames of an image sequence. Segmentation results on this two frame image sequence are shown in subimage (1) to (5). Segments in (1) and (2) correspond to the person in the foreground, and segments in (3) to (5) correspond to the background. The reason that the head of the person is segmented away from the body is that although they have similar motion, their motion profiles are different. The head region contains 2D textures and the motion profiles are more peaked, while in the body region the motion profiles are more spread out. Segment (3) is broken away from (4) and (5) for the same reason. An multi-resolution implementation can be used to reduce this running time further on larger images. In our current experiments, with this implementation, the running time on a 300 \Theta 400 image can be reduced to about 20 seconds on Intel Pentium 300MHz machines. Furthermore, the bottle neck of the computation, a sparse matrix-vector multiplication step, can be easily parallelized taking advantage of future computer chip designs. In our current implementation, the sparse eigenvalue decomposition is computed using the LASO2 numerical package developed by D. Scott. 4.2 Choice of graph edge weight In the examples shown here, we used an exponential function of the form of on the weighted graph edge with feature similarity of d(x). The value of oe is typically set to of the total range of the feature distance function d(x). The exponential weighting function is chosen here for its relatively simplicity as well as neutrality, since the focus of this paper is on developing a general segmentation procedure, given a feature similarity measure. We found this choice of weight function is quite adequate for typical image and feature spaces. Section 6.1 shows the effect of using different weighting functions and parameters on the output of the normalized cut algorithm. However, the general problem of defining feature similarity incorporating a variety of cues is not a trivial one. The grouping cues could be of different abstraction levels and types, and they could be in conflict with each other. Furthermore, the weighting function could vary from image region to image region, particularly in a textured image. Some of these issues are addressed in [15]. 5 Relationship to Spectral Graph Theory The computational approach that we have developed for image segmentation is based on concepts from spectral graph theory. The core idea to use matrix theory and linear algebra to study properties of the incidence matrix, W and the Laplacian matrix, D \Gamma W, of the graph and relate them back to various properties of the original graph. This is a rich area of mathematics, and the idea of using eigenvectors of the Laplacian for finding partitions of graphs can be traced back to Cheeger[4], Donath & Hoffman[7], and Fiedler[9]. This area has also seen contributions by theoretical computer scientists[1, 3, 22, 23]. It can be shown that our notion of normalized cut is related by a constant factor to the concept of conductance in[22]. For a tutorial introduction to spectral graph theory, we recommend the recent monograph by Fan Chung[5]. In this monograph, Chung[5] proposes a "normalized" definition of the Laplacian, as . The eigenvectors for this "normalized" Laplacian, when multiplied by 2 , are exactly the generalized eigenvectors we used to compute normalized cut. Chung points out that the eigenvalues of this "normalized" Laplacian relate well to graph invariants for general graph in ways that eigenvalues of the standard Laplacian has failed to do. Spectral graph theory provides us some guidance on the goodness of the approximation to the normalized cut provided by the second eigenvalue of the normalized Laplacian. One way is through bounds on the normalized Cheeger constant[5] which in our terminology can be defined as The eigenvalues of (6) are related to the Cheeger constant by the inequality[5]: G Earlier work on spectral partitioning used the second eigenvectors of the Laplacian of the graph defined as D \Gamma W to partition a graph. The second smallest eigenvalue of D \Gamma W is sometimes known as the Fiedler value. Several results have been derived relating the ratio cut, and the Fiedler value. A ratio cut of a partition of V , which in fact is the standard definition of the Cheeger constant, is defined as cut(A;V \GammaA) . It was shown that if the Fiedler value is small, partitioning graph based on the Fiedler vector will lead to good ratio cut[1][23]. Our derivation in section 2.1 can be adapted (by replacing the matrix D in the denominators by the identity matrix I) to show that the Fiedler vector is a real valued solution to the problem of minAaeV cut(A;V \GammaA) , which we can call the average cut. Although average cut looks similar to the normalized cut, average cut does not have the important property of having a simple relationship to the average association, which can be analogously defined as assoc(A;A) Consequently, one can not simultaneously minimize the disassociation across the partitions, while maximizing the association within the groups. When we applied both techniques to the image segmentation problem, we found that the normalized cut produces better results in practice. There are also other explanations why the normalized cut has better behavior from graph theoretical point of view, as pointed out by Chung[5]. As far as we are aware, our work, first presented in[20], represents the first application of spectral partitioning to computer vision or image analysis. There is however one application area that has seen substantial application of spectral partitioning-the area of parallel scientific computing. The problem there is to balance the workload over multiple processors taking into account communication needs. One of the early papers is [18]. The generalized eigenvalue approach was first applied to graph partitioning by [8] for dynamically balancing computational load in a parallel computer. Their algorithm is motivated by [13]'s paper on representing a hypergraph in a Euclidean Space. 5.1 A physical interpretation As one might expect, a physical analogy can be set up for the generalized eigenvalue system (6) that we used to approximate the solution of normalized cut. We can construct a spring-mass system from the weighted graph by taking graph nodes as physical nodes and graph edges as springs connecting each pair of nodes. Furthermore, we will define the graph edge weight as the spring stiffness, and the total edge weights connecting to a node as its mass. Imagine what would happen if we were to give a hard shake to this spring-mass system forcing the nodes to oscillate in the direction perpendicular to the image plane. Nodes that have stronger spring connections among them will likely oscillate together. As the shaking become more violent, weaker springs connecting to this group of node will be over-stretched. Eventually the group will "pop" off from the image plane. The overall steady state behavior of the nodes can be described by its fundamental mode of oscillation. In fact, it can be shown that the fundamental modes of oscillation of this spring mass system are exactly the generalized eigenvectors of (6). ij be the spring stiffness connecting nodes i and j. Define K to be the n \Theta n stiffness matrix, with K(i; . Define the diagonal n \Theta n mass matrix M as Let x(t) be the n \Theta 1 vector describing the motion of each node. This spring-mass dynamic system can be described by: Assuming the solution take the form of the steady state solutions of this spring-mass system satisfy: analogous to equation (6) for normalized cut. Each solution pair describes a fundamental mode of the spring-mass system. The eigenvectors v k give the steady state displacement of the oscillation in each mode, and the eigenvalues ! 2 k give the energy required to sustain each mode of oscillation. Therefore, finding graph partitions that have small normalized cut values is, in effect, the same as finding a way to "pop" off image regions with minimal effort. 6 Relationship to other graph theoretic approaches to image segmentation In the computer vision community, there has been some been previous work on image segmentation formulated as a graph partition problem. Wu&Leahy[25] use the minimum cut criterion for their segmentation. As mentioned earlier, our criticism of this criterion is that it tends to favor cutting off small regions which is undesirable in the context of image segmen- tation. In an attempt to get more balanced partitions, Cox et.al. [6] seek to minimize the ratio cut(A;V \GammaA) some function of the set A. When weight(A) is taken to the be the sum of the elements in A, we see that this criterion becomes one of the terms in the definition of average cut above. Cox et. al. use an efficient discrete algorithm to solve their optimization problem assuming the graph is planar. Sarkar & Boyer[19] use the eigenvector with the largest eigenvalue of the system -x for finding the most coherent region in an edge map. Using a similar derivation as in section (2.1), we can see that the first largest eigenvector of their system approximates , and the second largest eigenvector approximates minAaeV;BaeV assoc(A;A) However, the approximation is not tight, and there is no guarantee that As we should see later in the section , this situation can happen quite often in practice. Since this algorithm is essentially looking for clusters that have tight within-grouping similarity, we will call this criteria average association. 6.1 Comparison with related eigenvector based methods The normalized cut formulation has certain resemblance to the average cut, the standard spectral graph partitioning, as well as average association formulation. All these three algorithms can be reduced to solving certain eigenvalue systems. How are they related to each other? Figure summarizes the relationship between these three algorithms. On one hand, both the normalized cut and the average cut algorithm are trying to find a "balanced partition" of a weighted graph, while on the other hand, the normalized association and the average association are trying to find "tight" clusters in the graph. Since the normalized association is exactly the normalized cut value, the normalized cut formulation seeks a balance between the goal of clustering and segmentation. It is, therefore, not too surprising to see that the normalized cut vector can be approximated with the generalized eigenvector of (D well as that of Judging from the discrete formulations of these three grouping criterion, it can be seen that the average association, assoc(A;A) , has a bias for finding tight clusters. Therefore it runs the risk of becoming too greedy in finding small but tight clusters in the data. This might be perfect for data that are Gaussian distributed. However for typical data in the real world that are more likely to be made up of a mixture of various different types of Wx='l x (D-W) or Average cut Normalized Cut or Discrete formulation Continuous solution |A| |B| |A| |B| asso(B,V) Average association Finding clumps Finding splits Figure 24: Relationship between normalized cut and other eigenvector based partitioning techniques. Compared to the average cut and average association formulation, normalized cut seeks a balance between the goal of finding clumps and finding splits. distributions, this bias in grouping will have undesired consequences, as we shall illustrate in the examples below. For average cut, cut(A;B) , the opposite problem arises - one can not ensure the two partitions computed will have tight within-group similarity. This becomes particularly problematic if the dissimilarity among the different groups varies from one to another, or if there are several possible partitions all with similar average cut values. To illustrate these points, let us first consider a set of randomly distributed data in 1D shown in figure 25. The 1D data is made up by two subsets of points, one randomly distributed from 0 to 0.5, and the other from 0.65 to 1.0. Each data point is taken as a node in the graph, and the weighted graph edge connecting two points is defined to be inversely proportional to the distance between two nodes. We will use three monotonically decreasing weighting functions, defined on the distance function, d(x), with different rate of fall-off. The three weighting functions are plotted in figure 26(a), 27(a), and 28(a). The first function, 0:1 , plotted in figure 26(a), has the fastest decreasing rate among the three. With this weighting function, only close-by points are connected, as shown in the graph weight matrix W plotted in figure 26(b). In this case, average association fails to find the right partition. Instead it focuses on finding small clusters in each of the two main subgroups. The second function, plotted in figure 27(a), has the slowest decreasing rate among the three. With this weighting function, most points have some non-trivial connections to the rest. To find a cut of the graph, a number of edges with heavy weights have to be removed. In addition, the cluster on the right has less within-group similarity comparing with the cluster on the left. In this case, average cut has trouble deciding on where to cut. The third function, 0:2 , plotted in figure 28(a), has a moderate decreasing rate. With this weighting function, the nearby point connections are balanced against far-away point connections. In this case, all three algorithms performs well with normalized cut producing a clearer solution than the two other methods. These problems illustrated in figure 26, 27 and 28, in fact are quite typical in segmenting real natural images. This is particularly true in the case of texture segmentation. Different Figure 25: A set of randomly distributed points in 1D. The first 20 points are randomly distributed from 0.0 to 0.5, and the remaining 12 points are randomly distributed from 0.65 to 1.0. Segmentation result of these points with different weighting functions are show in figure 26, 27, and 28. texture regions often have very different within-group similarity, or coherence. It is very difficult to pre-determine the right weighting function on each image region. Therefore it is important to design a grouping algorithm that is more tolerant to a wide range of weighting functions. The advantage of using normalized cut becomes more evident in this case. Figure 29 illustrates this point on a natural texture image shown previously in figure 21. 7 Conclusion In this paper, we developed a grouping algorithm based on the view that perceptual grouping should be a process that aims to extract global impressions of a scene and provides a hierarchical description of it. By treating the grouping problem as a graph partitioning problem, we proposed the normalized cut criteria for segmenting the graph. Normalized cut is an unbiased measure of disassociation between sub-groups of a graph, and it has the nice property that minimizing normalized cut leads directly to maximizing the normalized association which is an unbiased measure for total association within the sub-groups. In finding an efficient algorithm for computing the minimum normalized cut, we showed that a generalized eigenvalue system provides a real valued solution to our problem. 0.1Average Cut: (D -0.2 -0.10.10.3 Average Association: -0.4 -0.3 -0.2 -0.4 -0.3 -0.2 Figure weighting function with fast rate of fall-off: 0:1 , shown in subplot (a) in solid line. The dotted lines show the two alternative weighting functions used in figure 27 and 28. Subplot (b) shows the corresponding graph weight matrix W . The two columns (1) and (2) below show the first, and second extreme eigenvectors for the Normalized cut(row 1), Average cut(row 2), and Average association(row 3). For both normalized cut, and average cut, the smallest eigenvector is a constant vector as predicted. In this case, both normalized cut and average cut perform well, while the average association fails to do the right thing. Instead it tries to pick out isolated small clusters. Average Cut: (D -0.50.5Average Association: -0.4 -0.20.2 Figure 27: A weighting function with slow rate of fall-off: shown in subplot (a) in solid line. The dotted lines show the two alternative weighting functions used in figure 26 and 28. Subplot (b) shows the corresponding graph weight matrix W . The two columns (1) and (2) below show the first, and second extreme eigenvectors for the Normalized cut(row 1), Average cut(row 2), and Average association(row 3). In this case, both normalized cut and average association give the right partition, while the average cut has trouble deciding on where to cut. Average Cut: (D -0.4 -0.20.2Average Association: -0.4 -0.20.2 Figure 28: A weighting function with medium rate of fall-off: 0:2 , shown in subplot (a) in solid line. The dotted lines show the two alternative weighting functions used in figure 26 and 28. Subplot (b) shows the corresponding graph weight matrix W . The two columns (1) and (2) below show the first, and second extreme eigenvectors for the Normalized cut(row 1), Average cut(row 2), and average association(row 3). All these three algorithms perform satisfactorily in this case, with normalized cut producing a clearer solution than the other two cuts. (b) (c) 50 100 150 200 250 300 350100200 50 100 150 200 250 300 350100200 (d) (e) 50 100 150 200 250 300 350100200 50 100 150 200 250 300 350100200 Figure 29: Normalized cut and average association result on the zebra image in figure 21. Subplot (a) shows the second largest eigenvector of approximating the normalized cut vector. Subplot (b) - (e) shows the first to fourth largest eigenvectors of approximating the average association vector, using the same graph weight matrix. In this image, pixels on the zebra body have, on average, lower degree of coherence than the pixels in the background. The average association, with its tendency to find tight clusters, partitions out only small clusters in the background. The normalized cut algorithm, having to balance the goal of clustering and segmentation, finds the better partition in this case. A computational method based on this idea has been developed, and applied to segmentation of brightness, color, and texture images. Results of experiments on real and synthetic images are very encouraging, and illustrate that the normalized cut criterion does indeed satisfy our initial goal of extracting the "big picture" of a scene. Acknowledgment This research was supported by (ARO)DAAH04-96-1-0341, and an NSF Graduate Fellowship to J. Shi. We thank Christos Papadimitriou for supplying the proof of NP-completeness for normalized cuts on a grid. In addition, we wish to acknowledge Umesh Vazirani and Alistair for discussions on graph theoretic algorithms and Inderjit Dhillon and Mark Adams for useful pointers to numerical packages. 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graph partitioning;image segmentation;grouping

352648

A Comparison of Prediction Accuracy, Complexity, and Training Time of Thirty-Three Old and New Classification Algorithms.

Twenty-two decision tree, nine statistical, and two neural network algorithms are compared on thirty-two datasets in terms of classification accuracy, training time, and (in the case of trees) number of leaves. Classification accuracy is measured by mean error rate and mean rank of error rate. Both criteria place a statistical, spline-based, algorithm called POLYCLSSS at the top, although it is not statistically significantly different from twenty other algorithms. Another statistical algorithm, logistic regression, is second with respect to the two accuracy criteria. The most accurate decision tree algorithm is QUEST with linear splits, which ranks fourth and fifth, respectively. Although spline-based statistical algorithms tend to have good accuracy, they also require relatively long training times. POLYCLASS, for example, is third last in terms of median training time. It often requires hours of training compared to seconds for other algorithms. The QUEST and logistic regression algorithms are substantially faster. Among decision tree algorithms with univariate splits, C4.5, IND-CART, and QUEST have the best combinations of error rate and speed. But C4.5 tends to produce trees with twice as many leaves as those from IND-CART and QUEST.

Introduction There is much current research in the machine learning and statistics communities on algorithms for decision tree classifiers. Often the emphasis is on the accuracy of the algorithms. One study, called the StatLog Project (Michie, Spiegelhalter and Taylor, 1994), compares the accuracy of several decision tree algorithms against some non-decision tree algorithms on a large number of datasets. Other studies that are smaller in scale include Brodley and Utgoff (1992), Brown, Corruble and Pittard (1993), Curram and Mingers (1994), and Shavlik, Mooney and Towell (1991). Recently, comprehensibility of the tree structures has received some attention. Comprehensibility typically decreases with increase in tree size and complexity. If two trees employ the same kind of tests and have the same prediction accuracy, the one with fewer leaves is usually preferred. Breslow and Aha (1997) survey methods of tree simplification to improve their comprehensibility. A third criterion that has been largely ignored is the relative training time of the algorithms. The StatLog Project finds that no algorithm is uniformly most accurate over the datasets studied. Instead, many algorithms possess comparable accuracy. For such algorithms, excessive training times may be undesirable (Hand, 1997). The purpose of our paper is to extend the results of the StatLog Project in the following ways: 1. In addition to classification accuracy and size of trees, we compare the training times of the algorithms. Although training time depends somewhat on imple- mentation, it turns out that there are such large differences in times (seconds versus days) that the differences cannot be attributed to implementation alone. 2. We include some decision tree algorithms that are not included in the StatLog Project, namely, S-Plus tree (Clark and Pregibon, 1993), Holte and Maass, 1995), OC1 (Murthy, Kasif and Salzberg, 1994), LMDT (Brodley and Utgoff, 1995), and QUEST (Loh and Shih, 1997). 3. We also include several of the newest spline-based statistical algorithms. Their classification accuracy may be used as benchmarks for comparison with other algorithms in the future. 4. We study the effect of adding independent noise attributes on the classification accuracy and (where appropriate) tree size of each algorithm. It turns out that except possibly for three algorithms, all the others adapt to noise quite well. 5. We examine the scalability of some of the more promising algorithms as the sample size is increased. Our experiment compares twenty-two decision tree algorithms, nine classical and modern statistical algorithms, and two neural network algorithms. Many datasets are taken from the University of California, Irvine (UCI), Repository of Machine Learning Databases (Merz and Murphy, 1996). Fourteen of the datasets are from real-life domains and two are artificially constructed. Five of the datasets were used in the StatLog Project. To increase the probability of finding statistically significant differences between algorithms, the number of datasets is doubled by the addition of noise attributes. The resulting total of number of datasets is thirty-two. Section 2 briefly describes the algorithms and Section 3 gives some background to the datasets. Section 4 explains the experimental setup used in this study and Section 5 analyzes the results. The issue of scalability is studied in Section 6 and conclusions and recommendations are given in Section 7. 2. The algorithms Only a short description of each algorithm is given. Details may be found in the cited references. If an algorithm requires class prior probabilities, they are made proportional to the training sample sizes. 2.1. Trees and rules CART: We use the version of CART (Breiman, Friedman, Olshen and Stone, 1984) implemented in the cart style of the IND package (Buntine and Caru- ana, 1992) with the Gini index of diversity as the splitting criterion. The trees based on the 0-SE and 1-SE pruning rules are denoted by IC0 and IC1 respec- tively. The software is obtained from http://ic-www.arc.nasa.gov/ic/projects/bayes-group/ind/IND-program.html. S-Plus tree: This is a variant of the CART algorithm written in the S language (Becker, Chambers and Wilks, 1988). It is described in Clark and Pregibon (1993). It employs deviance as the splitting criterion. The best tree is chosen by ten-fold cross-validation. Pruning is performed with the p.tree() function in the treefix library (Venables and Ripley, 1997) from the StatLib S Archive at http://lib.stat.cmu.edu/S/. The 0-SE and 1-SE trees are denoted by ST0 and ST1 respectively. C4.5: We use Release 8 (Quinlan, 1993; Quinlan, 1996) with the default settings including pruning (http://www.cse.unsw.edu.au/~quinlan/). After a tree is constructed, the C4.5 rule induction program is used to produce a set of rules. The trees are denoted by C4T and the rules by C4R. FACT: This fast classification tree algorithm is described in Loh and Vanichse- takul (1988). It employs statistical tests to select an attribute for splitting each node and then uses discriminant analysis to find the split point. The size of the tree is determined by a set of stopping rules. The trees based on univariate splits (splits on a single attribute) are denoted by FTU and those based on linear combination splits (splits on linear functions of attributes) are denoted by FTL. The Fortran 77 program is obtained from http://www.stat.wisc.edu/~loh/. QUEST: This new classification tree algorithm is described in Loh and Shih (1997). QUEST can be used with univariate or linear combination splits. A unique feature is that its attribute selection method has negligible bias. If all the attributes are uninformative with respect to the class attribute, then each has approximately the same chance of being selected to split a node. Ten-fold cross-validation is used to prune the trees. The univariate 0-SE and 1-SE trees are denoted by QU0 and QU1, respectively. The corresponding trees with linear combination splits are denoted by QL0 and QL1, respectively. The results in this paper are based on version 1.7.10 of the program. The software is obtained from http://www.stat.wisc.edu/~loh/quest.html. IND: This is due to Buntine (1992). We use version 2.1 with the default settings. IND comes with several standard predefined styles. We compare four Bayesian styles in this paper: bayes, bayes opt, mml, and mml opt (denoted by IB, IBO, IM, and IMO, respectively). The opt methods extend the non-opt methods by growing several different trees and storing them in a compact graph struc- ture. Although more time and memory intensive, the opt styles can increase classification accuracy. OC1: This algorithm is described in Murthy et al. (1994). We use version 3 (http://www.cs.jhu.edu/~salzberg/announce-oc1.html) and compare three styles. The first one (denoted by OCM) is the default that uses a mixture of univariate and linear combination splits. The second one (option -a; denoted by OCU) uses only univariate splits. The third one (option -o; denoted by OCL) uses only linear combination splits. Other options are kept at their default values. LMDT: The algorithm is described in Brodley and Utgoff (1995). It constructs a decision tree based on multivariate tests that are linear combinations of the attributes. The tree is denoted by LMT. We use the default values in the software from http://yake.ecn.purdue.edu/~brodley/software/lmdt.html. CAL5: This is from the Fraunhofer Society, Institute for Information and Data Processing, Germany (M-uller and Wysotzki, 1994; M-uller and Wysotzki, 1997). We use version 2. CAL5 is designed specifically for numerical-valued attributes. However, it has a procedure to handle categorical attributes so that mixed attributes (numerical and categorical) can be included. In this study we optimize the two parameters which control tree construction. They are the predefined threshold S and significance level ff. We randomly split the training set into two parts, stratified by the classes: two-thirds are used to construct the tree and one-third is used as a validation set to choose the optimal parameter configura- tion. We employ the c-shell program that comes with the CAL5 package to choose the best parameters by varying ff between 0.10 and 0.90 and S between and 0.95 in steps of 0.05. The best combination of values that minimize the error rate on the validation set is chosen. The tree is then constructed on all the records in the training set using the chosen parameter values. It is denoted by CAL. T1: This is a one-level decision tree that classifies examples on the basis of only one split on a single attribute (Holte, 1993). A split on a categorical attribute 5with b categories can produce up to b being reserved for missing attribute values). On the other hand, a split on a continuous attribute can yield up to J leaves, where J is the number of classes (one leaf is again reserved for missing values). The software is obtained from http://www.csi.uottawa.ca/~holte/Learning/other-sites.html. 2.2. Statistical algorithms LDA: This is linear discriminant analysis, a classical statistical method. It models the instances within each class as normally distributed with a common covariance matrix. This yields linear discriminant functions. QDA: This is quadratic discriminant analysis. It also models class distributions as normal, but estimates each covariance matrix by the corresponding sample covariance matrix. As a result, the discriminant functions are quadratic. Details on LDA and QDA can be found in many statistics textbooks, e.g., Johnson and (1992). We use the SAS PROC DISCRIM (SAS Institute, Inc., 1990) implementation of LDA and QDA with the default settings. NN: This is the SAS PROC DISCRIM implementation of the nearest neighbor method. The pooled covariance matrix is used to compute Mahalanobis distances LOG: This is logistic discriminant analysis. The results are obtained with a poly- tomous logistic regression (see, e.g., Agresti (1990)) Fortran 90 routine written by the first author (http://www.stat.wisc.edu/~limt/logdiscr/). FDA: This is flexible discriminant analysis (Hastie, Tibshirani and Buja, 1994), a generalization of linear discriminant analysis that casts the classification problem as one involving regression. Only the MARS (Friedman, 1991) nonparametric regression procedure is studied here. We use the S-Plus function fda from the mda library of the StatLib S Archive. Two models are used: an additive model (degree=1, denoted by FM1) and a model containing first-order interactions (degree=2 with penalty=3, denoted by FM2). PDA: This is a form of penalized LDA (Hastie, Buja and Tibshirani, 1995). It is designed for situations in which there are many highly correlated attributes. The classification problem is cast into a penalized regression framework via optimal scoring. PDA is implemented in S-Plus using the function fda with method=gen.ridge. MDA: This stands for mixture discriminant analysis (Hastie and Tibshirani, 1996). It fits Gaussian mixture density functions to each class to produce a classifier. MDA is implemented in S-Plus using the library mda. POL: This is the POLYCLASS algorithm (Kooperberg, Bose and Stone, 1997). It fits a polytomous logistic regression model using linear splines and their tensor products. It provides estimates for conditional class probabilities which can then be used to predict class labels. POL is implemented in S-Plus using the function poly.fit from the polyclass library of the StatLib S Archive. Model selection is done with ten-fold cross-validation. 2.3. Neural networks LVQ: We use the learning vector quantization algorithm in the S-Plus class library (Venables and Ripley, 1997) at the StatLib S Archive. Details of the algorithm may be found in Kohonen (1995). Ten percent of the training set are used to initialize the algorithm, using the function lvqinit. Training is carried out with the optimized learning rate function olvq1, a fast and robust LVQ algorithm. Additional fine-tuning in learning is performed with the function lvq1. The number of iterations is ten times the size of the training set in both olvq1 and lvq1. We use the default values of 0.3 and 0.03 for ff, the learning rate parameter, in olvq1 and lvq1, respectively. RBF: This is the radial basis function network implemented in the SAS tnn3.sas macro (Sarle, 1994) for feedforward neural networks (http://www.sas.com). The network architecture is specified with the ARCH=RBF argument. In this study, we construct a network with only one hidden layer. The number of hidden units is chosen to be 20% of the total number of input and output units [2.5% (5 hidden units) only for the dna and dna+ datasets and 10% (5 hidden units) for the tae and tae+ datasets because of memory and storage limitations]. Although the macro can perform model selection to choose the optimal number of hidden units, we did not utilize this capability because it would have taken too long for some of the datasets (see Table 6 below). Therefore the results reported here for this algorithm should be regarded as lower bounds on its performance. The hidden layer is fully connected to the input and output layers but there is no direct connection between the input and output layers. At the output layer, each class is represented by one unit taking the value of 1 for that particular category and 0 otherwise, except for the last one which is the reference category. To avoid local optima, ten preliminary trainings were conducted and the best estimates used for subsequent training. More details on the radial basis function network can be found in Bishop (1995) and Ripley (1996). 3. The datasets We briefly describe the sixteen datasets used in the study as well as any modifications that are made for our experiment. Fourteen of them are from real domains while two are artificially created. Thirteen are from UCI. breast cancer (bcw). This is one of the breast cancer databases at UCI, collected at the University of Wisconsin by W. H. Wolberg. The problem is to predict whether a tissue sample taken from a patient's breast is malignant or benign. There are two classes, nine numerical attributes, and 699 observations. Sixteen instances contain a single missing attribute value and are removed from the analysis. Our results are therefore based on 683 records. Error rates are estimated using ten-fold cross-validation. A decision tree analysis of a subset of the data using the FACT algorithm is reported in Wolberg, Tanner, Loh and (1987), Wolberg, Tanner and Loh (1988), and Wolberg, Tanner and Loh (1989). The dataset has also been analyzed with linear programming methods (Mangasarian and Wolberg, 1990). Contraceptive method choice (cmc). The data are taken from the 1987 National Indonesia Contraceptive Prevalence Survey. The samples are married women who were either not pregnant or did not know if they were pregnant at the time of the interview. The problem is to predict the current contraceptive method choice (no use, long-term methods, or short-term methods) of a woman based on her demographic and socio-economic characteristics (Lerman, Molyneaux, Pangemanan and Iswarati, 1991). There are three classes, two numerical attributes, seven categorical attributes, and 1473 records. The error rates are estimated using ten-fold cross-validation. The data are obtained from http://www.stat.wisc.edu/p/stat/ftp/pub/loh/treeprogs/datasets/. StatLog DNA (dna). This UCI dataset in molecular biology was used in the Project. Splice junctions are points in a DNA sequence at which "su- perfluous" DNA is removed during the process of protein creation in higher organisms. The problem is to recognize, given a sequence of DNA, the boundaries between exons (the parts of the DNA sequence retained after splicing) and introns (the parts of the DNA sequence that are spliced out). There are three classes and sixty categorical attributes each having four categories. The sixty categorical attributes represent a window of sixty nucleotides, each having one of four categories. The middle point in the window is classified as one of exon/intron boundaries, intron/exon boundaries, or neither of these. The 3186 examples in the database were divided randomly into a training set of size 2000 and a test set of size 1186. The error rates are estimated from the test set. StatLog heart disease (hea). This UCI dataset is from the Cleveland Clinic Foundation, courtesy of R. Detrano. The problem concerns the prediction of the presence or absence of heart disease given the results of various medical tests carried out on a patient. There are two classes, seven numerical attributes, six categorical attributes, and 270 records. The StatLog Project employed unequal misclassification costs. We use equal costs here because some algorithms do not allow unequal costs. The error rates are estimated using ten-fold cross-validation Boston housing (bos). This UCI dataset gives housing values in Boston suburbs (Harrison and Rubinfeld, 1978). There are three classes, twelve numerical attributes, one binary attribute, and 506 records. Following Loh and Vanichse- takul (1988), the classes are created from the attribute median value of owner-occupied homes as follows: class otherwise. The error rates are estimated using ten-fold cross-validation. LED display (led). This artificial domain is described in Breiman et al. (1984). It contains seven Boolean attributes, representing seven light-emitting diodes, and ten classes, the set of decimal digits. An attribute value is either zero or one, according to whether the corresponding light is off or on for the digit. Each attribute value has a ten percent probability of having its value inverted. The class attribute is an integer between zero and nine, inclusive. A C program from UCI is used to generate 2000 records for the training set and 4000 records for the test set. The error rates are estimated from the test set. BUPA liver disorders (bld). This UCI dataset was contributed by R. S. Forsyth. The problem is to predict whether or not a male patient has a liver disorder based on blood tests and alcohol consumption. There are two classes, six numerical attributes, and 345 records. The error rates are estimated using ten-fold cross-validation. PIMA Indian diabetes (pid). This UCI dataset was contributed by V. Sigillito. The patients in the dataset are females at least twenty-one years old of Pima Indian heritage living near Phoenix, Arizona, USA. The problem is to predict whether a patient would test positive for diabetes given a number of physiological measurements and medical test results. There are two classes, seven numerical attributes, and 532 records. The original dataset consists of 768 records with eight numerical attributes. However, many of the attributes, notably serum in- sulin, contain zero values which are physically impossible. We remove serum insulin and records that have impossible values in other attributes. The error rates are estimated using ten-fold cross validation. StatLog satellite image (sat). This UCI dataset gives the multi-spectral values of pixels within 3 \Theta 3 neighborhoods in a satellite image, and the classification associated with the central pixel in each neighborhood. The aim is to predict the classification given the multi-spectral values. There are six classes and thirty-six numerical attributes. The training set consists of 4435 records while the test set consists of 2000 records. The error rates are estimated from the test set. Image segmentation (seg). This UCI dataset was used in the StatLog Project. The samples are from a database of seven outdoor images. The images are hand-segmented to create a classification for every pixel as one of brickface, sky, foliage, cement, window, path, or grass. There are seven classes, nineteen numerical attributes and 2310 records in the dataset. The error rates are estimated using ten-fold cross-validation. The algorithm T1 could not handle this dataset without modification, because the program requires a large amount of memory. Therefore for T1 (but not for the other algorithms) we discretize each attribute except attributes 3, 4, and 5 into one hundred categories. Attitude towards smoking restrictions (smo). This survey dataset (Bull, 1994) is obtained from http://lib.stat.cmu.edu/datasets/csb/. The problem is to predict attitude toward restrictions on smoking in the workplace (prohibited, restricted, or unrestricted) based on bylaw-related, smoking-related, and sociodemographic covariates. There are three classes, three numerical attributes, and five categorical attributes. We divide the original dataset into a training set of size 1855 and a test set of size 1000. The error rates are estimated from the test set. Thyroid disease (thy). This is the UCI ann-train.datacontributed by R. Werner. The problem is to determine whether or not a patient is hyperthyroid. There are three classes (normal, hyperfunction, and subnormal functioning), six numerical attributes, and fifteen binary attributes. The training set consists of 3772 records and the test set has 3428 records. The error rates are estimated from the test set. StatLog vehicle silhouette (veh). This UCI dataset originated from the Turing Institute, Glasgow, Scotland. The problem is to classify a given silhouette as one of four types of vehicle, using a set of features extracted from the silhouette. Each vehicle is viewed from many angles. The four model vehicles are double decker bus, Chevrolet van, Saab 9000, and Opel Manta 400. There are four classes, eighteen numerical attributes, and 846 records. The error rates are estimated using ten-fold cross-validation. Congressional voting records (vot). This UCI dataset gives the votes of each member of the U. S. House of Representatives of the 98th Congress on sixteen issues. The problem is to classify a Congressman as a Democrat or a Republican based on the sixteen votes. There are two classes, sixteen categorical attributes with three categories each ("yea", "nay", or neither), and 435 records. rates are estimated by ten-fold cross-validation. Waveform (wav). This is an artificial three-class problem based on three wave- forms. Each class consists of a random convex combination of two waveforms sampled at the integers with noise added. A description for generating the data is given in Breiman et al. (1984) and a C program is available from UCI. There are twenty-one numerical attributes, and 600 records in the training set. Error rates are estimated from an independent test set of 3000 records. evaluation (tae). The data consist of evaluations of teaching performance over three regular semesters and two summer semesters of 151 teaching assistant (TA) assignments at the Statistics Department of the University of Wisconsin-Madison. The scores are grouped into three roughly equal-sized categories ("low", "medium", and "high") to form the class attribute. The predictor attributes are (i) whether or not the is a native English speaker (bi- nary), (ii) course instructor (25 categories), (iii) course (26 categories), (iv) summer or regular semester (binary), and (v) class size (numerical). This dataset is first reported in Loh and Shih (1997). It differs from the other datasets in that there are two categorical attributes with large numbers of categories. As a result, decision tree algorithms such as CART that employ exhaustive search usually take much longer to train than other algorithms. (CART has to evaluate 2 splits for each categorical attribute with c values.) Error rates are estimated using ten-fold cross-validation. The data are obtained from http://www.stat.wisc.edu/p/stat/ftp/pub/loh/treeprogs/datasets/. A summary of the attribute features of the datasets is given in Table 1. Table 1. Characteristics of the datasets. The last three columns give the number and type of added noise attributes for each dataset. The notation "N(0,1)" denotes the standard normal distribution, "UI(m,n)" denotes a uniform distribution over the integers m through n inclusive, and "U(0,1)" denotes a uniform distribution over the unit interval. Training No. of original attributes No. and type of noise attributes Data sample No. of Num. Categorical Total Numerical Categorical Total set size classes 2 3 4 5 25 26 dna 2000 3 led 2000 bld pid smo thy veh 846 4 vot 435 2 tae 4. Experimental setup Some algorithms are not designed for categorical attributes. In these cases, each categorical attribute is converted into a vector of 0-1 attributes. That is, if a categorical attribute X takes k values fc g, it is replaced by a 1)-dimensional vector (d otherwise, for , the vector consists of all zeros. The affected algorithms are all the statistical and neural network algorithms as well as the tree algorithms FTL, OCU, OCL, OCM, and LMT. In order to increase the number of datasets and to study the effect of noise attributes on each algorithm, we created sixteen new datasets by adding independent noise attributes. The numbers and types of noise attributes added are given in the right panel of Table 1. The name of each new dataset is the same as the original dataset except for the addition of a '+' symbol. For example, the bcw dataset with noise added is denoted by bcw+. For each dataset, we use one of two different ways to estimate the error rate of an algorithm. For large datasets (size much larger than 1000 and test set of size at least 1000), we use a test set to estimate the error rate. The classifier is constructed using the records in the training set and then it is tested on the test set. Twelve of the thirty-two datasets are analyzed this way. For the remaining twenty datasets, we use the following ten-fold cross-validation procedure to estimate the error rate: 1. The dataset is randomly divided into ten disjoint subsets, with each containing approximately the same number of records. Sampling is stratified by the class labels to ensure that the subset class proportions are roughly the same as those in the whole dataset. 2. For each subset, a classifier is constructed using the records not in it. The classifier is then tested on the withheld subset to obtain a cross-validation estimate of its error rate. 3. The ten cross-validation estimates are averaged to provide an estimate for the classifier constructed from all the data. Because the algorithms are implemented in different programming languages and some languages are not available on all platforms, three types of UNIX workstations are used in our study. The workstation type and implementation language for each algorithm are given in Table 2. The relative performance of the workstations according to SPEC marks is given in Table 3. The floating point SPEC marks show that a task that takes one second on a DEC 3000 would take about 1.4 and 0.8 seconds on a SPARCstation 5 (SS5) and SPARCstation 20 (SS20), respectively. Therefore, to enable comparisons, all training times are reported here in terms of 3000-equivalent seconds-the training times recorded on a SS5 and a SS20 are divided by 1.4 and 0.8, respectively. 5. Results The error rates and training times for the algorithms are given in a separate table for each dataset in the Appendix. The tables also report the error rates of the 'naive' plurality rule, which ignores the information in the covariates and classifies every record to the majority class in the training sample. 5.1. Exploratory analysis of error rates Before we present a formal statistical analysis of the results, it is helpful to study the summary in Table 4. The mean error rate for each algorithm over the datasets is given in the second row. The minimum and maximum error rates and that of Table 2. Hardware and software platform for each algorithm. The workstations are DEC 3000 Alpha Model 300 (DEC), Sun SPARCstation 20 Model 61 (SS20), and Sun SPARCstation 5 (SS5). Algorithm Platform Algorithm Platform Tree & Rules ST1 Splus tree, 1-SE DEC/S QU0 QUEST, univariate 0-SE DEC/F90 LMT LMDT, linear DEC/C QU1 QUEST, univariate 1-SE DEC/F90 CAL CAL5 SS5/C++ QL0 QUEST, linear 0-SE DEC/F90 single split DEC/C QL1 QUEST, linear 1-SE DEC/F90 FTU FACT, univariate DEC/F77 Statistical linear DEC/F77 LDA Linear discriminant anal. DEC/SAS C4T C4.5 trees DEC/C QDA Quadratic discriminant anal. DEC/SAS C4R C4.5 rules DEC/C NN Nearest-neighbor DEC/SAS IB IND bayes style SS5/C LOG Linear logistic regression DEC/F90 IBO IND bayes opt style SS5/C FM1 FDA, degree 1 SS20/S IM IND mml style SS5/C FM2 FDA, degree 2 SS20/S IMO IND mml opt style SS5/C PDA Penalized LDA SS20/S IC0 IND cart, 0-SE SS5/C MDA Mixture discriminant anal. SS20/S IC1 IND cart, 1-SE SS5/C POL POLYCLASS SS20/S OCU OC1, univariate SS5/C OCL OC1, linear SS5/C Neural Network OCM OC1, mixed SS5/C LVQ Learning vector quantization SS20/S ST0 Splus tree, 0-SE DEC/S RBF Radial basis function network DEC/SAS Table 3. SPEC benchmark summary Workstation SPECfp92 SPECint92 Source DEC DEC 3000 Model 300 91.5 66.2 SPEC Newsletter (150MHz) Vol. 5, Issue 2, June 1993 SS20 Sun SPARCstation 20 102.8 88.9 SPEC Newsletter Model 61 (60MHz) Vol. 6, Issue 2, June 1994 SS5 Sun SPARCstation 5 47.3 57.0 SPEC Newsletter (70MHz) Vol. 6, Issue 2, June 1994 the plurality rule are given for each dataset in the last three columns. Let p denote the smallest observed error rate in each row (i.e., dataset). If an algorithm has an error rate within one standard error of p, we consider it to be close to the best and indicate it by a p in the table. The standard error is estimated as follows. If p is from an independent test set, let n denote the size of the test set. Otherwise, if p is a cross-validation estimate, let n denote the size of the training set. The standard error of p is estimated by the formula p)=n. The algorithm with the largest error rate within a row is indicated by an X. The total numbers of p and X-marks for each algorithm are given in the third and fourth rows of the table. The following conclusions may be drawn from the table: Table 4. Minimum, maximum, and 'naive' plurality rule error rates for each dataset. A ` p '-mark indicates that the algorithm has an error rate within one standard error of the minimum for the dataset. A 'X'-mark indicates that the algorithm has the worst error rate for the dataset. The mean error rate for each algorithm is given in the second row. Decision trees and rules Statistical algorithms Nets Error rates Naive Mean bcw cmc pp pp dna dna+ hea bos led bld pid seg smo thy ppp ppp pp ppp pp vot tae X tae+ pp 1. Algorithm POL has the lowest mean error rate. An ordering of the other algorithms in terms of mean error rate is given in the upper half of Table 5. 2. The algorithms can also be ranked in terms of total number of p - and X-marks. By this criterion, the most accurate algorithm is again POL, which has fifteen p Table 5. Ordering of algorithms by mean error rate and mean rank of error rate Mean POL LOG MDA QL0 LDA QL1 PDA IC0 FM2 IBO IMO error .195 .204 .207 .207 .208 .211 .213 .215 .218 .219 .219 rate C4R IM LMT C4T QU0 QU1 OCU IC1 IB OCM ST0 Mean POL FM1 LOG FM2 QL0 LDA QU0 C4R IMO MDA PDA rank 8.3 12.2 12.2 12.2 12.4 13.7 13.9 14.0 14.0 14.3 14.5 of C4T QL1 IBO IM IC0 FTL QU1 OCU IC1 ST0 ST1 error 14.5 14.6 14.7 14.9 15.0 15.4 16.6 16.6 16.9 17.0 17.7 rate LMT OCM IB RBF FTU QDA LVQ OCL CAL NN marks and no X-marks. Eleven algorithms have one or more X-marks. Ranked in increasing order of number of X-marks (in parentheses), they are: FTL(1), OCM(1), ST1(1), FM2(1), MDA(1), FM1(2), OCL(3), QDA(3), NN(4), LVQ(4), T1(11). Excluding these, the remaining algorithms rank in order of decreasing number of p -marks (in parentheses) as: POL(15), LOG(13), QL0(10), LDA(10), PDA(10), QL1(9), OCU(9), (1) QU0(8), QU1(8), C4R(8), IBO(8), RBF(8), C4T(7), IMO(6), IM(5), IC1(5), ST0(5), FTU(4), IC0(4), CAL(4), IB(3), LMT(1). The top four algorithms in (1) also rank among the top five in the upper half of Table 5. 3. The last three columns of the table show that a few algorithms are sometimes less accurate than the plurality rule. They are NN (at cmc, cmc+, smo+), bld+), QDA (smo, thy, thy+), FTL (tae), and ST1 (tae+). 4. The easiest datasets to classify are bcw, bcw+, vot, and vot+; the error rates all lie between 0.03 and 0.09. 5. The most difficult to classify are cmc, cmc+, and tae+, with minimum error rates greater than 0.4. 6. Two other difficult datasets are smo and smo+. In the case of smo, only T1 has a (marginally) lower error rate than that of the plurality rule. No algorithm has a lower error rate than the plurality rule for smo+. 7. The datasets with the largest range of error rates are thy and thy+, where the rates range from 0.005 to 0.890. However, the maximum of 0.890 is due to QDA. If QDA is ignored, the maximum error rate drops to 0.096. 8. There are six datasets with only one p -mark each. They are bld+ (POL), sat (LVQ), sat+ (FM2), seg+ (IBO), veh and veh+ (QDA both times). 9. Overall, the addition of noise attributes does not appear to increase significantly the error rates of the algorithms. 5.2. Statistical significance of error rates 5.2.1. Analysis of variance A statistical procedure called mixed effects analysis of variance can be used to test the simultaneous statistical significance of differences between mean error rates of the algorithms, while controlling for differences between datasets (Neter, Wasserman and Kutner, 1990, p. 800). Although it makes the assumption that the effects of the datasets act like a random sample from a normal distribution, it is quite robust against violation of the assumption. For our data, the procedure gives a significance probability less than 10 \Gamma4 . Hence the hypothesis that the mean error rates are equal is strongly rejected. Simultaneous confidence intervals for differences between mean error rates can be obtained using the Tukey method (Miller, 1981, p. 71). According to this procedure, a difference between the mean error rates of two algorithms is statistically significant at the 10% level if they differ by more than 0.058. To visualize this result, Figure 1(a) plots the mean error rate of each algorithm versus its median training time in seconds. The solid vertical line in the plot is units to the right of the mean error rate for POL. Therefore any algorithm lying to the left of the line has a mean error rate that is not statistically significantly different from that of POL. The algorithms are seen to form four clusters with respect to training time. These clusters are roughly delineated by the three horizontal dotted lines which correspond to training times of one minute, ten minutes, and one hour. Figure 1(b) shows a magnified plot of the eighteen algorithms with median training times less than ten minutes and mean error rate not statistically significantly different from POL. 5.2.2. Analysis of ranks To avoid the normality assumption, we can instead analyze the ranks of the algorithms within datasets. That is, for each dataset, the algorithm with the lowest error rate is assigned rank one, the second lowest rank two, etc., with average ranks assigned in the case of ties. The lower half of Table 5 gives an ordering of the algorithms in terms of mean rank of error rates. Again POL is first and last. Note, however, that the mean rank of POL is 8.3. This shows that it is far from being uniformly most accurate across datasets. Comparing the two methods of ordering in Table 5, it is seen that POL, LOG, QL0, and LDA are the only algorithms with consistently good performance. Three algorithms that perform well by one criterion but not the other are MDA, FM1, and FM2. In the case of MDA, its low mean error rate is due to its excellent performance in four datasets (veh, veh+, wav, and wav+) where many other algorithms do poorly. These domains concern shape identification and the datasets contain only numerical Mean error rate Median sec. FTU C4R IB IBO IM IMO OCU OCL OCM ST0 ST1 LDA QDA PDA MDA POL RBF 1hr 10min 1min (a) All thirty-three methods Mean error rate Median sec. FTU C4R IB IM OCU LDA PDA MDA 1min 5min (b) Less than 10min., accuracy not sig. different from POL Figure 1. Plots of median training time versus mean error rate. The vertical axis is in log-scale. The solid vertical line in plot (a) divides the algorithms into two groups: the mean error rates of the algorithms in the left group do not differ significantly (at the 10% simultaneous significance level) from that of POL, which has the minimum mean error rate. Plot (b) shows the algorithms that are not statistically significantly different from POL in terms of mean error rate and that have median training time less than ten minutes. attributes. MDA is generally unspectacular in the rest of the datasets and this is the reason for its tenth place ranking in terms of mean rank. The situation for FM1 and FM2 is quite different. As its low mean rank indicates, FM1 is usually a good performer. However, it fails miserably in the seg and seg+ datasets, reporting error rates of more than fifty percent when most of the other algorithms have error rates less than ten percent. Thus FM1 seems to be less robust than the other algorithms. FM2 also appears to lack robustness, although to a lesser extent. Its worst performance is in the bos+ dataset, where it has an error rate of forty-two percent, compared to less than thirty-five percent for the other algorithms. The number of X-marks against an algorithm in Table 4 is a good predictor of erratic if not poor performance. MDA, FM1, and FM2 all have at least one X-mark. The Friedman (1937) test is a standard procedure for testing statistical significance in differences of mean ranks. For our experiment, it gives a significance probability less than 10 \Gamma4 . Therefore the null hypothesis that the algorithms are equally accurate on average is again rejected. Further, a difference in mean ranks greater than 8.7 is statistically significant at the 10% level (Hollander and Wolfe, 1999, p. 296). Thus POL is not statistically significantly different from the twenty other algorithms that have mean rank less than or equal to 17.0. Figure 2(a) shows a plot of the median training time versus the mean ranks of the algorithms. Those algorithms that lie to the left of the vertical line are not statistically significantly different from POL. A magnified plot of the subset of algorithms that are not significantly different from POL and that have median training time less than ten minutes is given in Figure 2(b). The algorithms that differ statistically significantly from POL in terms of mean error rate form a subset of those that differ from POL in terms of mean ranks. Thus the rank test appears to be more powerful than the analysis of variance test for this experiment. The fifteen algorithms in Figure 2(b) may be recommended for use in applications where good accuracy and short training time are desired. 5.3. Training time Table 6 gives the median DEC 3000-equivalent training time for each algorithm and the relative training time within datasets. Owing to the large range of training times, only the order relative to the fastest algorithm for each dataset is reported. The fastest algorithm is indicated by a '0'. An algorithm that is between 10 x\Gamma1 to times as slow is indicated by the value of x. For example, in the case of the dna+ dataset, the fastest algorithms are C4T and T1, each requiring two seconds. The slowest algorithm is FM2, which takes more than three million seconds (almost forty days) and hence is between 10 6 to 10 7 times as slow. The last two columns of the table give the fastest and slowest times for each dataset. Table 7 gives an ordering of the algorithms from fastest to slowest according to median training time. Overall, the fastest algorithm is C4T, followed closely by FTU, FTL, and LDA. There are two reasons for the superior speed of C4T compared to the other decision tree algorithms. First, it splits each categorical attribute into as Mean rank Median sec. FTU C4R IB IBO IM IMO OCU OCL OCM LDA QDA PDA MDA POL RBF 1hr 10min 1min (a) All thirty-three methods Mean rank Median sec. C4R IM OCU LDA PDA MDA 1min 5min (b) Less than 10min., accuracy not sig. different from POL Figure 2. Plots of median training time versus mean rank of error rates. The vertical axis is in log-scale. The solid vertical line in plot (a) divides the algorithms into two groups: the mean ranks of the algorithms in the left group do not differ significantly (at the 10% simultaneous significance level) from that of POL. Plot (b) shows the algorithms that are not statistically significantly different from POL in terms of mean rank and that have median training time less than ten minutes. Table 6. DEC 3000-equivalent training times and relative times of the algorithms. The second and third rows give the median training time and rank for each algorithm. An entry of 'x' in the each of the subsequent rows indicates that an algorithm is between times slower than the fastest algorithm for the dataset. The fastest algorithm is denoted by an entry of '0'. The minimum and maximum training times are given in the last two columns. 's', `m', 'h', `d' denote seconds, minutes, hours, and days, respectively. Decision trees and rules Statistical algorithms Nets CPU time Median CPU 3.2m 3.2m 5.9m 5.9m 7s 8s 5s 20s 34s 27.5m 34s 33.9m 52s 47s 46s 14.9m 13.7m 15.1m 14.4m 5.7m 1.3h 36s 10s 15s 20s 4m 15.6m 3.8h 56s 3m 3.2h 1.1m 11.3h 5s Rank Table 7. Ordering of algorithms by median training time 5s 7s 8s 10s 15s 20s 20s 34s 34s 36s 46s 47s 52s 56s 1.1m 3m 3.2m 3.2m 4m 5.7m 5.9m 5.9m OCM 13.7m 14.4m 14.9m 15.1m 15.6m 27.5m 33.9m 1.3h 3.2h 3.8h 11.3h many subnodes as the number of categories. Therefore it wastes no time in forming subsets of categories. Second, its pruning method does not require cross-validation, which can increase training time several fold. The classical statistical algorithms QDA and NN are also quite fast. As expected, decision tree algorithms that employ univariate splits are faster than those that use linear combination splits. The slowest algorithms are POL, FM2, and RBF; two are spline-based and one is a neural network. Although IC0, IC1, ST0 and ST1 all claim to implement the CART algorithm, the IND versions are faster than the S-Plus versions. One reason is that IC0 and IC1 are written in C whereas ST0 and ST1 are written in the S language. Another reason is that the IND versions use heuristics (Buntine, personal communication) instead of greedy search when the number of categories in a categorical attribute is large. This is most apparent in the tae+ dataset where there are categorical attributes with up to twenty-six categories. In this case IC0 and IC1 take around forty seconds versus two and a half hours for ST0 and ST1. The results in Table 4 indicate that IND's classification accuracy is not adversely affected by such heuristics; see Aronis and Provost (1997) for another possible heuristic. Since T1 is a one-level tree, it may appear surprising that it is not faster than algorithms such as C4T that produce multi-level trees. The reason is that splits each continuous attribute into J is the number of classes. On the other hand, C4T always splits a continuous attribute into two intervals only. Therefore when J ? 2, T1 has to spend a lot more time to search for the intervals. 5.4. Size of trees Table 8 gives the number of leaves for each tree algorithm and dataset before noise attributes are added. In the case that an error rate is obtained by ten-fold cross- validation, the entry is the mean number of leaves over the ten cross-validation trees. Table 9 shows how much the number of leaves changes after addition of noise attributes. The mean and median of the number of leaves for each classifier are given in the last columns of the two tables. IBO and IMO clearly yield the largest trees by far. Apart from T1, which is necessarily short by design, the algorithm with the shortest trees on average is QL1, followed closely by FTL and OCL. A ranking of the algorithms with univariate splits (in increasing median number of leaves) is: T1, IC1, ST1, QU1, FTU, IC0, ST0, OCU, QU0, and C4T. Algorithm C4T tends to produce trees with many more leaves than the other algorithms. One reason may be due to under-pruning (although its error rates are quite good). Another is that, unlike the binary-tree algorithms, C4T splits each categorical attribute into as many nodes as the number of categories. Addition of noise attributes typically decreases the size of the trees, except for C4T and CAL which tend to grow larger trees, and IMO which seems to fluctuate rather wildly. These results complement those of Oates and Jensen (1997) who looked at the effect of sample size on the number of leaves of decision tree algorithms and found a significant relationship between tree size and training sample size for C4T. They observed that tree algorithms which employ cost-complexity pruning are better able to control tree growth. 6. Scalability of algorithms Although differences in mean error rates between POL and many other algorithms are not statistically significant, it is clear that if error rate is the sole criterion, POL would be the method of choice. Unfortunately, POL is one of the most compute-intensive 1algorithms. To see how training times increase with sample size, a small scalability study was carried out with the algorithms QU0, QL0, FTL, C4T, C4R, IC0, LOG, FM1, and POL. Training times are measured for these algorithms for training sets of size Four datasets are used to generate the samples-sat, smo+, tae+, and a new, very large UCI dataset called adult which has two classes and six continuous and seven categorical attributes. Since the first three datasets are not large enough for the experiment, bootstrap re-sampling is employed to generate the training sets. That is, N samples are randomly drawn with replacement from each dataset. To avoid getting many replicate records, the value of the class attribute for each sampled case is randomly changed to another value with probability 0.1. (The new value is selected from the pool of alternatives with equal probability.) Bootstrap sampling is not carried out for the adult dataset because it has more than 32,000 records. Instead, the nested training sets are obtained by random sampling without replacement. The times required to train the algorithms are plotted (in log-log scale) in Figure 3. With the exception of POL, FM1 and LOG, the logarithms of the training times seem to increase linearly with log(N ). The non-monotonic behavior of POL and FM1 is puzzling and might be due to randomness in their use of cross-validation for model selection. The erratic behavior of LOG in the adult dataset is caused by convergence problems during model fitting. Many of the lines in Figure 3 are roughly parallel. This suggests that the relative computational speed of the algorithms is fairly constant over the range of sample sizes considered. QL0 and C4R are two exceptions. Cohen (1995) had observed that C4R does not scale well. 7. Conclusions Our results show that the mean error rates of many algorithms are sufficiently similar that their differences are statistically insignificant. The differences are also probably insignificant in practical terms. For example, the mean error rates of the top ranked algorithms POL, LOG, and QL0 differ by less than 0.012. If such a small difference is not important in real applications, the user may wish to select an algorithm based on other criteria such as training time or interpretability of the classifier. Unlike error rates, there are huge differences between the training times of the algorithms. POL, the algorithm with the lowest mean error rate, takes about fifty times as long to train as the next most accurate algorithm. The ratio of times is roughly equivalent to hours versus minutes, and Figure 3 shows that it is maintained over a wide range of sample sizes. For large applications where time is a factor, it may be advantageous to use one of the quicker algorithms. It is interesting that the old statistical algorithm LDA has a mean error rate close to the best. This is surprising because (i) it is not designed for binary-valued attributes (all categorical attributes are transformed to 0-1 vectors prior to application of LDA), and (ii) it is not expected to be effective when class densities CPU time CPU time tae+ CPU time adult CPU time Figure 3. Plots of training time versus sample size in log-log scale for selected algorithms. are multi-modal. Because it is fast, easy to implement, and readily available in statistical packages, it provides a convenient benchmark for comparison against future algorithms. The low error rates of LOG and LDA probably account for much of the performance of the better algorithms. For example, POL is basically a modern version of LOG. It enhances the flexibility of LOG by employing spline-based functions and automatic model selection. Although this strategy is computationally costly, it does produce a slight reduction in the mean error rate-enough to bring it to the top of the pack. The good performance of QL0 may be similarly attributable to LDA. The QUEST linear-split algorithm is designed to overcome the difficulties encountered by LDA in multi-modal situations. It does this by applying a modified form of LDA to partitions of the data, where each partition is represented by a leaf of the decision tree. This strategy alone, however, is not enough, as the higher mean error rate of FTL shows. The latter is based on the FACT algorithm which is a precursor to QUEST. One major difference between the QUEST and FACT algorithms is that the former employs the cost-complexity pruning method of CART whereas the latter does not. Our results suggest that some form of bottom-up pruning may be essential for low error rates. If the purpose of constructing an algorithm is for data interpretation, then perhaps only decision rules or trees with univariate splits will suffice. With the exception of CAL and T1, the differences in mean error rates of the decision rule and tree algorithms are not statistically significant from that of POL. IC0 has the lowest mean error rate and QU0 is best in terms of mean ranks. C4R and C4T are not far behind. Any of these four algorithms should provide good classification accuracy. C4T is the fastest by far, although it tends to yield trees with twice as many leaves as IC0 and QU0. C4R is the next fastest, but Figure 3 shows that it does not scale well. IC0 is slightly faster and its trees have slightly fewer leaves than QU0. However, Loh and Shih (1997) show that CART-based algorithms such as IC0 are prone to produce spurious splits in some situations. Acknowledgments We are indebted to P. Auer, C. E. Brodley, W. Buntine, T. Hastie, R. C. Holte, C. Kooperberg, S. K. Murthy, J. R. Quinlan, W. Sarle, B. Schulmeister, and W. Taylor for help and advice on the installation of the computer programs. We are also grateful to J. W. Molyneaux for providing the 1987 National Indonesia Contraceptive Prevalence Survey data. Finally, we thank W. Cohen, F. 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decision tree;classification tree;statistical classifier;neural net

352664

A Study of Reinforcement Learning in the Continuous Case by the Means of Viscosity Solutions.

This paper proposes a study of Reinforcement Learning (RL) for continuous state-space and time control problems, based on the theoretical framework of viscosity solutions (VSs). We use the method of dynamic programming (DP) which introduces the value function (VF), expectation of the best future cumulative reinforcement. In the continuous case, the value function satisfies a non-linear first (or second) order (depending on the deterministic or stochastic aspect of the process) differential equation called the Hamilton-Jacobi-Bellman (HJB) equation. It is well known that there exists an infinity of generalized solutions (differentiable almost everywhere) to this equation, other than the VF. We show that gradient-descent methods may converge to one of these generalized solutions, thus failing to find the optimal control.In order to solve the HJB equation, we use the powerful framework of viscosity solutions and state that there exists a unique viscosity solution to the HJB equation, which is the value function. Then, we use another main result of VSs (their stability when passing to the limit) to prove the convergence of numerical approximations schemes based on finite difference (FD) and finite element (FE) methods. These methods discretize, at some resolution, the HJB equation into a DP equation of a Markov Decision Process (MDP), which can be solved by DP methods (thanks to a strong contraction property) if all the initial data (the state dynamics and the reinforcement function) were perfectly known. However, in the RL approach, as we consider a system in interaction with some a priori (at least partially) unknown environment, which learns from experience, the initial data are not perfectly known but have to be approximated during learning. The main contribution of this work is to derive a general convergence theorem for RL algorithms when one uses only approximations (in a sense of satisfying some weak contraction property) of the initial data. This result can be used for model-based or model-free RL algorithms, with off-line or on-line updating methods, for deterministic or stochastic state dynamics (though this latter case is not described here), and based on FE or FD discretization methods. It is illustrated with several RL algorithms and one numerical simulation for the Car on the Hill problem.

Introduction This paper is about Reinforcement Learning (RL) in the continuous state-space and time case. RL techniques (see (Kaelbling, Littman, & Moore, 1996) for a survey) are adaptive methods for solving optimal control problems for which only a partial amount of initial data are available to the system that learns. In this paper, we focus on the Dynamic Programming (DP) method which introduces a function, EMI MUNOS called the value function (VF) (or cost function), that estimates the best future cumulative reinforcement (or cost) as a function of initial states. RL in the continuous case is a difficult problem for at least two reasons. Since we consider a continuous state-space, the first reason is that the value function has to be approximated, either by using discretization (with grids or triangulations) or general approximation (such as neural networks, polynomial functions, fuzzy sets, etc.) methods. RL algorithms for continuous state-space have been implemented with neural networks (see for example (Barto, Sutton, & Anderson, 1983), (Barto, 1990), (Gullapalli, 1992), (Williams, 1992), (Lin, 1993), (Sutton & Whitehead, 1993), (Harmon, Baird, & Klopf, 1996), and (Bertsekas & Tsitsiklis, 1996)), fuzzy sets (see (Now'e, 1995), (Glorennec & Jouffe, 1997)), approximators based on state aggregation (see (Singh, Jaakkola, & Jordan, 1994)), clustering (see (Mahadevan sparse-coarse-coded functions (see (Sutton, 1996)) and variable resolution grids (see (Moore, 1991), (Moore & Atkeson, 1995)). However, as it has been pointed out by several authors, the combination of DP methods with function approximators may produce unstable or divergent results even when applied to very simple problems (see (Boyan & Moore, 1995), (Baird, 1995), (Gordon, 1995)). Some results using clever algorithms (like Residual algorithms of (Baird, 1995)) or particular classes of approximation functions (like the Averagers of (Gordon, 1995)) can lead to the convergence to a local or global solution within the class of functions considered. Anyway, it is difficult to define the class of functions (for a neural network, the suitable architecture) within which the optimal value function could be approxi- mated, knowing that we have little prior knowledge of its smoothness properties. The second reason is because we consider a continuous-time variable. Indeed, the value function derived from the DP equation (see (Bellman, 1957)), relates the value at some state as a function of the values at successor states. In the continuous-time limit, as the successor states get infinitely closer, the value at some point becomes a function of its differential, defining a non linear differential equation, called the Hamilton-Jacobi-Bellman (HJB) equation. In the discrete-time case, the resolution of the Markov Decision Process (MDP) is equivalent to the resolution, on the whole state-space, of the DP equation ; this property provides us with DP or RL algorithms that locally solve the DP equation and lead to the optimal solution. With continuous time, it is no longer the case since the HJB equation holds only if the value function is differentiable. And in general, the value function is not differentiable everywhere (even for smooth initial data), thus this equation cannot be solved in the usual sense, because this leads to either no solution (if we look for classical solutions, i.e. differentiable everywhere) or an infinity of solutions (if we look for generalized solutions, i.e. differentiable almost everywhere). This fact, which will be illustrated with a very simple 1-dimensional example, explains why there could be many "bad" solutions to gradient-descent methods for RL. Indeed, such methods intend to minimize the integral of some Hamiltonian. But the generalized solutions of the HJB equation are global optima of this problem, REINFORCEMENT LEARNING BY THE MEANS OF VISCOSITY SOLUTIONS 3 so the gradient-descent methods may lead to approximate any (among an infinity of) generalized solutions giving little chance to reach the desired value function. In order to deal with the problem of integrating the HJB equation, we use the formalism of Viscosity Solutions (VSs), introduced by Crandall and Lions (in (Cran- dall & Lions, 1983) ; see the user's guide (Crandall, Ishii, & Lions, 1992)) in order to define an adequate class (which appears as a weak formulation) of solutions to non-linear first (and second) order differential equation such as HJB equations. The main properties of VSs are their existence, their uniqueness and the fact that the value function is a VS. Thus, for a large class of optimal control problems, there exists a unique VS to the HJB equation, which is the value function. Furthermore, VSs have remarkable stability properties when passing to the limit, from which we can derive proofs of convergence for discretization methods. Our approach here consists in defining a class of convergent numerical schemes, among which are the finite element (FE) and finite difference (FD) approximation schemes introduced in (Kushner, 1990) and (Kushner & Dupuis, 1992) to discretize, at some resolution ffi , the HJB equation into a DP equation for some discrete Markov Decision Process. We apply a result of convergence (from (Barles & Souganidis, 1991)) to prove the convergence of the value function V ffi of the discretized MDP to the value function V of the continuous problem as the discretization step ffi tends to zero. The DP equation of the discretized MDP could be solved by any DP method (because the DP equation satisfies a "strong" contraction property leading successive iterations to converge to the value function, the unique fixed point of this equation), but only if the data (the transition probabilities and the reinforcements) were perfectly known by the learner, which is not the case in the RL approach. Thus, we propose a result of convergence for RL algorithms when we only use "approximations" of these data (in the sense that the approximated DP equation need to satisfy some "weak" contraction property). The convergence occurs as the number of iterations tends to infinity and the discretization step tends to zero. This result applies to model-based or model-free RL algorithms, for off-line or on-line methods, for deterministic or stochastic state dynamics, and for FE or FD based discretization methods. It is illustrated with several RL algorithms and one numerical simulation for the "Car on the Hill" problem. In what follows, we consider the discounted, infinite-time horizon case (for a description of the finite-time horizon case, see (Munos, 1997a)) with deterministic state dynamics (for the stochastic case, see (Munos & Bourgine, 1997) or (Munos, 1997a)). Section 2 introduces the formalism for RL in the continuous case, defines the value function, states the HJB equation and presents a result showing continuity of the VF. Section 3 illustrates the problems of classical solutions to the HJB equation with a simple 1-dimensional example and introduces the notion of viscosity solutions. Section 4 is concerned with numerical approximation of the value function using discretization schemes. The finite element and finite difference methods are EMI MUNOS presented and a general convergence theorem (whose proof is in appendix A) is stated. Section 5 states a convergence theorem (whose proof is in appendix B) for a general class of RL algorithms and illustrates it with several algorithms. Section 6 presents a simple numerical simulation for the "Car on the Hill" problem 2. A formalism for reinforcement learning in the continuous case The objective of reinforcement learning is to learn from experience how to influence the behavior of a dynamic system in order to maximize some payoff function called the reinforcement or reward function (or equivalently to minimize some cost function). This is a problem of optimal control in which the state dynamics and the reinforcement function are, a priori, at least partially unknown. In this paper we are concerned with deterministic problems in which the dynamics of the system is governed by a controlled differential equation. For similar results in the stochastic case, see (Munos & Bourgine, 1997), (Munos, 1997a), for a related work using multi-grid methods, see (Pareigis, 1996). The two possible approaches for optimal control are Pontryagin's maximum principle (for theoretical work, see (Pontryagin, Boltyanskii, Gamkriledze, & Mischenko, 1962) and more recently (Fleming and Rishel1975), for a study of Temporal Differ- ence, see (Doya, 1996), and for an application to the control with neural networks, see (Bersini & Gorrini, 1997)) and the Bellman's Dynamic Programming (DP) (in- troduced in (Bellman, 1957)) approach considered in this paper. 2.1. Deterministic optimal control for discounted infinite-time horizon problems In what follows, we consider infinite-time horizon problems under the discounted framework. In that case, the state dynamics do not depend on the time. For a study of the finite time horizon case (for which there is a dependency in time), see (Munos, 1997a). Let x(t) be the state of the system, which belongs to the state-space O, closure of an open subset O ae IR d . The evolution of the system depends on the current state x(t) and control (or action) u(t) 2 U , where U , closed subset, is the control space ; it is defined by the controlled differential equation : dt where the control u(t) is a bounded, Lebesgue measurable function with values in U . The function f is called the state dynamics. We assume that f is Lipschitzian with respect to the first variable : there exists some constant L f ? 0 such that : For initial state x 0 at time the choice of a control u(t) leads to a unique (because the state dynamics (1) is deterministic) trajectory x(t) (see figure 1). REINFORCEMENT LEARNING BY THE MEANS OF VISCOSITY SOLUTIONS 5 Definition 1. We define the discounted reinforcement functional J , which depends on initial data x 0 and control u(t) for 0 - t - , with - the exit time of x(t) from O (with the trajectory always stays inside O) : with r(x; u) the current reinforcement (defined on O) and R(x) the boundary reinforcement (defined on @O, the boundary of the state-space). fl 2 [0; 1) is the discount factor which weights short-term rewards more than long-term ones (and ensures the convergence of the integral). The objective of the control problem is to find, for any initial state x 0 , the control u (t) that optimizes the reinforcement functional J(x x Figure 1. The state-space O. From initial state x 0 at the choice of control u(t) leads to the trajectory x(t) for 0 - t -; where - is the exit time from the state-space. Remark. Unlike the discrete case, in the continuous case, we need to consider two different reinforcement functions : r is obtained and accumulated during the running of the trajectory, whereas R occurs whenever the trajectory exits from the state-space (if it does). This formalism enables us to consider many optimal control problem, such as reaching a target while avoiding obstacles, viability problems, and many other optimization problems. Definition 2. We define the value function, the maximum value of the reinforcement functional as a function of initial state at time EMI MUNOS Before giving some properties of the value function (HJB equation, continuity and differentiability properties), let us first describe the reinforcement learning frame-work considered here and the constraints it implies. 2.2. The reinforcement learning approach RL techniques are adaptive methods for solving optimal control problems whose data are a priori (at least partially) unknown. Learning occurs iteratively, based on the experience of the interactions between the system and the environment, through the (current and boundary) reinforcement signals. The objective of RL is to find the optimal control, and the techniques used are those of DP. However, in the RL approach, the state dynamics f(x; u), and the reinforcement functions r(x; u); R(x) are partially unknown to the system. Thus RL is a constructive and iterative process that estimates the value function by successive approximations. The learning process includes both a mechanism for the choice of the control, which has to deal with the exploration versus exploitation dilemma (exploration provides the system with new information about the unknown data, whereas exploitation consists in optimizing the estimated values based on the current knowl- edge) (see (Meuleau, 1996)), and a mechanism for integrating new information for refining the approximation of the value function. The latter topic is the object of this paper. The study and the numerical approximations of the value function is of great importance in RL and DP because from this function we can deduce an optimal feed-back controller. The next section shows that the value function satisfies a local property, called the Hamilton-Jacobi-Bellman equation, and points out its relation to the optimal control. 2.3. The Hamilton-Jacobi-Bellman equation Using the dynamic programming principle (introduced in (Bellman, 1957)), we can prove that the value function satisfies a local condition, called the Hamilton-Jacobi- Bellman (HJB) equation (see (Fleming & Soner, 1993) for a complete survey). In the deterministic case studied here, it is a first-order non-linear partial differential equation (in the stochastic case, we can prove that a similar equation of order two holds). Here we assume that U is a compact set. Theorem 1 (Hamilton-Jacobi-Bellman) If the value function V is differentiable at x, let DV (x) be the gradient of V at x, then the Hamilton-Jacobi-Bellman u2U holds at x 2 O. Additionally, V satisfies the following boundary condition : REINFORCEMENT LEARNING BY THE MEANS OF VISCOSITY SOLUTIONS 7 Remark. The boundary condition is an inequality because at some boundary point (for example at x 1 2 @O on figure 1) there may exist a control u(t) such that the corresponding trajectory stays inside O and whose reinforcement functional is strictly superior to the immediate boundary reinforcement R(x 1 ). In such cases, holds with a strict inequality. Remark. Using an equivalent definition, the HJB equation (5) means that V is the solution of the equation : with the Hamiltonian H defined, for any differentiable function W , by : u2U Dynamic programming computes the value function in order to find the optimal control with a feed-back control policy, that is a function -(x) : O ! U such that the optimal control u (t) at time t depends on current state x(t) : u Indeed, from the value function, we deduce the following optimal feed-back control policy u2U Now that we have pointed out the importance of computing the value function for defining the optimal control, we show some properties of V (continuity, dif- ferentiability) and how to integrate (and in what sense) the HJB equation for approximating . 2.4. Continuity of the value function The property of continuity of the value function may be obtained under the following assumption concerning the state dynamics f around the boundary @O (which is assumed smooth, i.e. @O 2 C For all x 2 @O, let \Gamma! n (x) be the outward normal of O at x (for example, see figure 1), we assume that : These hypotheses mean that at any point of the boundary, there ought not be only trajectories tangential to the boundary of the state space. Theorem 2 (Continuity) Suppose that (2) and (9) are satisfied, then the value function is continuous in O. The proof of this theorem can be found in (Barles & Perthame, 1990). EMI MUNOS 3. Introduction to viscosity solutions From theorem 1, we know that if the value function is differentiable then it solves the HJB equation. However, in general, the value function is not differentiable everywhere even when the data of the problem are smooth. Thus, we cannot expect to find classical solutions (i.e. differentiable everywhere) to the HJB equation. Now, if we look for generalized solutions (i.e. differentiable almost everywhere), we find that there are many solutions other than the value function that solve the HJB equation. Therefore, we need to define a weak class of solutions to this equation. Crandall and Lions introduced such a weak formulation by defining the notion of Viscosity Solutions (VSs) in (Crandall & Lions, 1983). For a complete survey, see (Crandall et al., 1992), (Barles, 1994) or (Fleming & Soner, 1993). This notion has been developed for a very broad class of non-linear first and second order differential equations (including HJB equations for the stochastic case of controlled diffusion processes). Among the important properties of viscosity solutions are some uniqueness results, the stability of solutions to approximated equations when passing to the limit (this very important result will be used to prove the convergence of the approximation schemes in section 4.4) and mainly the fact that the value function is the unique viscosity solution of the HJB equation (5) with the boundary condition (6). First, let us illustrate with a simple example the problems raised here when one looks for classical or generalized solutions to the HJB equation. 3.1. 3 problems illustrated with a simple example Let us study a very simple control problem in 1 dimension. Let the state x(t) 2 [0; 1], the control u(t) 2 f\Gamma1; +1g and the state dynamics be : dx Consider a current reinforcement everywhere and a boundary reinforcement defined by R(0) and R(1). In this example, we deduce that the value function is : and the HJB equation is : with the boundary conditions V (0) - R(0) and V (1) - R(1): 1. First problem : there is no classical solution to the HJB equation. 0:3. The corresponding value function is plotted in figure 2. We observe that V is not differentiable everywhere, thus does not satisfy the HJB equation everywhere : there is no classical solution to the HJB equation. REINFORCEMENT LEARNING BY THE MEANS OF VISCOSITY SOLUTIONS 9 x Optimal Value function Figure 2. The value function is not differentiable everywhere. 2. Second problem : there are several generalized solutions. If one looks for generalized solutions that satisfy the HJB equation almost everywhere, we find many functions other than the value function. An example of a function satisfying (11) almost everywhere with the boundary conditions is illustrated in figure 3. x Optimal Value function Figure 3. There are many generalized solutions other than the value function. Remark. This problem is of great importance when one wants to use gradient-descent methods with some general function approximator, like neural networks, to approximate the value function. The use of gradient-descent methods may lead to approximate any of the generalized solutions of the HJB equation and thus fail to find the value function. Indeed, suppose that we use a gradient-descent method for finding a function W minimizing the error : EMI MUNOS Z x2O with H the Hamiltonian defined in section 2.3. Then, the method will converge, in the best case, to any generalized solution V g of (7) (because these functions are global optima of this minimization problem, since their error E(V which are probably different from the value function V . Moreover the control induced by such functions (by the closed loop policy (8)) might be very different from the optimal control (defined by V ). For example, the function plotted in figure 3 generates a control (given by the direction of the gradient) very different from the optimal control, defined by the value function plotted in figure 2. In fact, in such continuous time and space problems, there exists an infinity of global minima for gradient descent methods, and these functions may be very different from the expected value function. In the case of neural networks, we usually use smooth functions (differentiable everywhere), thus neither the value function V (figure 2), nor a generalized solution V g (figure 3) can be exactly represented, but both can be approximated. Let us denote ~ V and ~ , the best approximations of V and V g in the network. Then ~ V and ~ are local minima of the gradient-descent method minimizing E, but nothing proves that ~ V is a global minimum. In this example, it could seem that V is "smoother" than the generalized solutions (because it has only one discontinuity instead of several ones) in the sense that E( ~ E( ~ is not true in general. In any case, in the continuous-time case, when we use a smooth function approximator, there exists an infinity of local solutions for the problem of minimizing the error E and nothing proves that the expected ~ V is a global solution. See (Munos, Baird, & Moore, 1999) for some numerical experiments on simple (one and two dimensional) problems. When time is discretized, this problem disappears, but we still have to be careful when passing to the limit. In this paper, we describe discretization methods that converge to the value function when passing to the limit of the continuous case. 3. Third problem : the boundary condition is an inequality. Here we illustrate the problem of the inequality of the boundary condition. Let 5. The corresponding value function is plotted in figure 4. We observe that V (0) is strictly superior to the boundary reinforcement R(0). This strict inequality occurs at any boundary point x 2 @O for which there exists a control u(t) such that the trajectory goes immediately inside O and generates a better reinforcement functional than the boundary reinforcement R(x) obtained by exiting from O at x: We will give (in definition 4 that follows) a weak (viscosity) formulation of the boundary condition (6). REINFORCEMENT LEARNING BY THE MEANS OF VISCOSITY SOLUTIONS 11 x Value function Figure 4. The boundary condition may hold with a strict inequality condition 3.2. Definition of viscosity solutions In this section, we define the notion of viscosity solutions for continuous functions (a definition for discontinuous functions is given in appendix A). Definition 3 (Viscosity solution). Let W be a continuous real-valued function defined in O. ffl W is a viscosity sub-solution of (7) in O if for all functions ' local maximum of W \Gamma ' such that W ffl W is a viscosity super-solution of (7) in O if for all functions ' local minimum of W \Gamma ' such that W ffl W is a viscosity solution of (7) in O if it is a viscosity sub-solution and a viscosity super-solution of (7) in O: 3.3. Some properties of viscosity solutions The following theorem (whose proof can be found in (Crandall et al., 1992)) states that the value function is a viscosity solution. Theorem 3 Suppose that the hypotheses of Theorem 2 hold. Then the value function V is a viscosity solution of (7) in O: EMI MUNOS In order to deal with the inequality of the boundary condition (6), we define a viscosity formulation in a differential type condition instead of a pure Dirichlet condition. Definition 4 (Viscosity boundary condition). Let W be a continuous real-valued function defined in O, ffl W is a viscosity sub-solution of (7) in O with the boundary condition (6) if it is a viscosity sub-solution of (7) in O and for all functions ' local maximum of W \Gamma ' such that W minfH(x;W;DW ffl W is a viscosity super-solution of (7) in O with the boundary condition (6) if it is a viscosity super-solution of (7) in O and for all functions ' local minimum of W \Gamma ' such that W minfH(x;W;DW ffl W is a viscosity solution of (7) in O with the boundary condition (6) if it is a viscosity sub- and super-solution of (7) in O with the boundary condition (6). Remark. When the Hamiltonian H is related to an optimal control problem (which is the case here), the condition (13) is simply equivalent to the boundary inequality (6). With this definition, theorem 3 extends to viscosity solutions with boundary conditions. Moreover, from a result of uniqueness, we obtain the following theorem (whose proof is in (Crandall et al., 1992) or (Fleming & Soner, Theorem 4 Suppose that the hypotheses of theorem 2 hold. Then the value function V is the unique viscosity solution of (7) in O with the boundary condition (6). Remark. This very important theorem shows the relevance of the viscosity solutions formalism for HJB equations. Moreover this provides us with a very useful framework (as will be illustrated in next few sections) for proving the convergence of numerical approximations to the value function. Now we study numerical approximations of the value function. We define approximation schemes by discretizing the HJB equation with finite element or finite difference methods, and prove the convergence to the viscosity solution of the HJB equation, thus to the value function of the control problem. 4. Approximation with convergent numerical schemes 4.1. Introduction The main idea is to discretize the HJB equation into a Dynamic Programming (DP) equation for some stochastic Markovian Decision Process (MDP). For any resolution ffi , we can solve the MDP and find the discretized value function V ffi by using DP techniques, which are guaranteed to converge since the DP equation is a fixed-point equation satisfying some strong contraction property (see (Puterman, 1994), (Bertsekas, 1987)). We are also interested in the convergence properties of the discretized V ffi to the value function V as ffi decreases to 0. From (Kushner, 1990) and (Kushner & Dupuis, 1992), we define two classes of approximation schemes based on finite difference (FD) (section 4.2) and finite element methods (section 4.3). Section 4.4 provides a very general theorem of convergence (deduced from the abstract formulation of (Barles & Souganidis, 1991) and using the stability properties of viscosity solutions), that covers both FE and FD methods (the only important required properties for the convergence are the monotonicity and the consistency of the scheme). In the following, we assume that the control space U is approximated by finite control spaces U ffi such that : ae U 4.2. Approximation with finite difference methods d be a basis for IR d . The dynamics are Let the positive and negative parts of f i be any discretization step ffi , let us consider the lattices : ffi Z where are any integers, and \Sigma the frontier of \Sigma ffi , denote the set of points f- 2 ffi Z d n O such that at least one adjacent point - \Sigma (see figure 5). Let us denote by jjyjj the 1-norm of any vector y. O S d S d Figure 5. The discretized state-space \Sigma ffi (the dots) and its frontier @ \Sigma ffi (the crosses). EMI MUNOS The FD method consists of replacing the gradient DV (-) by the forward and backward difference quotients of V at - 2 \Sigma ffi in direction e Thus the HJB equation can be approximated by the following equation : Knowing that (\Deltat ln fl) is an approximation of (fl \Deltat \Gamma 1) as \Deltat tends to 0, we deduce the following equivalent approximation equation : for - 2 \Sigma ffi , with p(- 0 j-; for which is a DP equation for a finite Markovian Decision Process whose state-space is space is U ffi and probabilities of transition are p(- 0 j-; u) (see figure 6 for a geometrical interpretation).12 x x x x x x u' x . p( | ,u) . p( | ,u) p( | ,u) x x d d State Control Next state x d Prob. Figure 6. A geometrical interpretation of the FD discretization. The continuous process (on the left) is discretized at some resolution ffi into an MDP (right). The transition probabilities p(- of the MDP are the coordinates of the vector 1 (j \Gamma -) with j the projection of - onto the segment in a direction parallel to f(-; u). From the boundary condition, we define the absorbing terminal states : For REINFORCEMENT LEARNING BY THE MEANS OF VISCOSITY SOLUTIONS 15 By defining F ffi FD the finite difference scheme : DP equation FD This equation states that V ffi is a fixed point of F ffi FD . Moreover, as f is bounded from above (with some value M f FD satisfies the following strong contraction and since - ! 1, there exists a fixed point which is the value function is unique and can be computed by DP iterative methods (see (Puterman, 1994), (Bertsekas, 1987)). Computation of V ffi and convergence. : There are two standard methods for computing the value function V ffi of some MDP : value iteration (V ffi is the limit of a sequence of successive iterations V ffi FD ) and policy iteration (approximations in policy space by alternative policy evaluation steps and policy improvement steps). See (Puterman, 1994), (Bertsekas, 1987) or (Bertsekas & Tsitsiklis, 1996) for more information about DP theory. In section 5, we describe RL methods for computing iteratively the approximated value functions In the following section, we study a similar method for discretizing the continuous process into an MDP by using finite element methods. The convergence of these two methods (i.e. the convergence of the discretized V ffi to the value function V as tends to 0) will be derived from a general theorem in section 4.4. 4.3. Approximations with finite element methods We use a finite element (FE) method (with linear simplexes) based on a triangulation covering the state-space (see figure 7). The value function V is approximated by piecewise linear functions V ffi defined by their values at the vertices f-g of the triangulation \Sigma ffi . The value of V ffi at any point x inside some simplex (- linear combination of V ffi at the vertices d with being the barycentric coordinates of x inside the simplex (- (We recall that the definition of the barycentric coordinates is that - i EMI MUNOS xd d x Figure 7. Triangulation \Sigma ffi of the state-space. V ffi (x) is a linear combination of the V ffi (- i ), for weighted by the barycentric coordinates - i (x). By using a finite element approximation scheme derived from (Kushner, 1990), the continuous HJB equation is approximated by the following equation : where j(-; u) is a point inside \Sigma ffi such that j(-; We require that - satisfies the following condition, for some positive constants k 1 and Remark. It is interesting to notice that this time discretization function -; u) does not need to be constant and can depend on the state - and the control u. This provides us with some freedom on the choice of these parameters, assuming that equation (19) still holds. For a discussion on the choice of a constant time discretization function - according to the space discretization size ffi in order to optimize the precision of the approximations, see (Pareigis, 1997). Let us denote (- 0 ; :::; - d ) the simplex containing j(-; u): As V ffi is linear inside the simplex, this equation can be written : d which is a DP equation for a Markov Decision Process whose state-space is the set of vertices f-g and the probability of transition from (state -, control u) to next states - are the barycentric coordinates of j(-; u) inside simplex 8). The boundary states satisfy the terminal condition : REINFORCEMENT LEARNING BY THE MEANS OF VISCOSITY SOLUTIONS 17 x x x l (h )x xx x x x l l x x x x hState Control Proba. Next state x l (h )x Figure 8. A finite element approximation. Consider a vertex - and j 1 linear combination of V weighted by the barycentric coordinates ). Thus, the probabilities of transition of the MDP are these barycentric coordinates. For By defining F ffi FE the finite element scheme, d the approximated value function V ffi satisfies the DP equation Similarly to the FD scheme, F ffi FE satisfies the following "strong" contraction and since - ! 1, there is a unique solution, V ffi to (23) with (21) which can be computed by DP techniques. 4.4. Convergence of the approximation schemes In this section, we present a convergence theorem for a general class of approximation schemes. We use the stability properties of viscosity solutions (described in (Barles & Souganidis, 1991)) to obtain the convergence. Another kind of convergence result, using probabilistic considerations, can be found in (Kushner & Dupuis, 1992), but such results do not treat the problem with the general boundary condition (9). In fact, the only important required properties for convergence is monotonicity (property (27)) and consistency (properties (30) and (31) below). As a corollary, we deduce that the FE and the FD schemes studied in the previous sections are convergent. EMI MUNOS 4.4.1. A general convergence theorem. Let \Sigma ffi and @ \Sigma ffi be two discrete and finite subsets of IR d . We assume that for all x 2 O; lim ffi#0 dist(x; \Sigma be an operator on the space of bounded functions on \Sigma ffi . We are concerned with the convergence of the solution V ffi to the dynamic programming equation : with the boundary condition : We make the following assumptions on F ffl For any constant c, ffl For any ffi , there exists a solution V ffi to (25) and (26) which is (29) bounded with a constant M V independent of ffi: Consistency : there exists a constant k ? 0 such that : lim inf lim sup Remark. Conditions (30) and (31) are satisfied in the particular case of : lim Theorem 5 (Convergence of the scheme) Assume that the hypotheses of theorem are satisfied. Assume that (27), (28) (30) and (31) hold, then F ffi is a convergent approximation scheme, i.e. the solutions V ffi of (25) and lim \Gamma!x uniformly on any REINFORCEMENT LEARNING BY THE MEANS OF VISCOSITY SOLUTIONS 19 4.4.2. Outline of the proof. We use the procedure described in (Barles and Perthame1988). The idea is to define the largest limit function V and the smallest limit function V prove that they are respectively discontinuous sub- and super viscosity solutions. This proof, based on the general convergence theorem of (Barles & Souganidis, 1991), is given in appendix A. Then we use a comparison result which states that if (9) holds then viscosity sub-solutions are less than viscosity super-solutions, thus V sup - V inf . By definition and the limit function V is the viscosity solution of the HJB equation, thus (from theorem 4) the value function of the problem. 4.4.3. FD and FE approximation schemes converge Corollary 1 The approximation schemes F ffi FD and F ffi FE are convergent. Indeed, for the finite difference scheme, it is obvious that since p(- 0 j-; u) are considered as transition probabilities, the approximation scheme F ffi FD satisfies (27) and (28). As (17) is a DP equation for some MDP, DP theory ensures that (29) is true. We can check that the scheme is also consistent : conditions (30) and (31) hold with FD satisfy the hypotheses of theorem 5. Similarly, for the finite element scheme, from the basic properties of the barycentric coordinates - i (x), the approximation scheme F ffi FE satisfies (27). From (19), condition holds. DP theory ensures that (29) is true. The scheme is consistent and conditions (30) and (31) hold with FE satisfies the hypotheses of theorem 5. 4.5. Summary of the previous results of convergence For any given discretization step ffi , from the "strong" contraction property (18) or (24), DP theory ensures that the values V ffi iterated by some DP algorithm converge to the value V ffi of the approximation scheme F ffi as n tends to infinity. From the convergence of the scheme (theorem 5), the V ffi tend to the value function V of the continuous problem as ffi tends to 0 (see figure 9). d d the "strong" contraction property HJB equation DP equation Figure 9. The HJB equation is discretized, for some resolution ffi; into a DP equation whose solution is V ffi . The convergence of the scheme ensures that V 0: Thanks to the "strong" contraction property, the iterated values V ffi n tend to V ffi as EMI MUNOS Remark. Theorem 5 gives a result of convergence on any that the hypothesis (9), for the continuity of V , is satisfied. However, if this hypothesis is not satisfied, but if the value function is continuous, the theorem still applies. Now, if (9) is not satisfied and the value function is discontinuous at some area, then we still have the convergence on any O where the value function is continuous. 5. Designing convergent reinforcement learning algorithms In order to solve the DP equation (14) or (20), one can use DP off-line methods -such as value iteration, policy iteration, modified policy iteration (see (Puterman, 1994)), with synchronous or asynchronous back-ups, or on-line methods -like Real Time DP (see (Barto, Bradtke, & Singh, 1991), (Bertsekas & Tsitsiklis, 1996)). For example, by introducing the Q-values Q equation (20) can be solved by successive back-ups (indexed by n) of states -; control u (in any order provided that every state and control are updated regularly) by : The values V ffi n of this algorithm converges to the value function V ffi of the discretized MDP as n !1. However, in the RL approach, the state dynamics f and the reinforcement functions are unknown to the learner. Thus, the right side of the iterative rule (33) is unknown and has to be approximated thanks to the available knowledge. In the RL terminology, there are two possible approaches for updating the values : ffl The model-based approach consists of first, learning a model of the state dynamics and of the reinforcement functions, and then using DP algorithms based on such a rule (33) with the approximated model instead of the true values. The learning (the updating of the estimated Q-value Q iteratively during the simulation of trajectories (on-line learning) or at the end (at the exit time) of one or several trajectories (off-line or batch learning). ffl The model-free approach consists of updating incrementally the estimated values n or Q-value Q n of the visited states without learning any model. In what follows, we propose a convergence theorem that applies to a large class of RL algorithms (model-based or model-free, on-line or off-line, for deterministic or stochastic dynamics) provided that the updating rule satisfies some "weak" contraction property with respect to some convergent approximation scheme such as the FD and FE schemes studied previously. 5.1. Convergence of RL algorithms The following theorem gives a general condition for which an RL algorithm converges to the optimal solution of the continuous problem. The idea is that the REINFORCEMENT LEARNING BY THE MEANS OF VISCOSITY SOLUTIONS 21 updated values (by any model-free or model-based method) must be close enough to those of a convergent approximation scheme so that their difference satisfies the "weak" contraction property (34) below. Theorem 6 (Convergence of RL algorithms) For any ffi , let us build finite subsets \Sigma ffi and @ \Sigma ffi satisfying the properties of section 4.4. We consider an algorithm that leads to update every state - 2 \Sigma ffi regularly and every state - 2 @ \Sigma ffi at least once. Let F ffi be a convergent approximation scheme (for example (22)) and V ffi be the solution to (25) and (26). We assume that the values updated at the iteration n satisfy the following properties in the sense of the following "weak" contraction property : for some positive constant k and some function e(ffi) that tends to 0 as in the sense : for some positive constant kR ; then for any ae O; for all " ? 0, there exists \Delta such that for any there exists N; for all n - N , sup This result states that when the hypotheses of the theorem applies (mainly when we find some updating rule satisfying the weak contraction property (34)) then the values V ffi computed by the algorithm converge to the value function V of the continuous problem as the discretization step ffi tends to zero and the number of iterations n tends to infinity. 5.1.1. Outline of the proof The proof of this theorem is given in appendix B. If condition (34) were a strong contraction property such as for some constant - ! 1; then the convergence would be obvious since from (25) and from the fact that all the states are updated regularly, for a fixed would converge to V ffi as From the fact (theorem 5) that V ffi converges to V as could deduce that V ffi EMI MUNOS If it is not the case, we can no longer expect that V ffi if (34) holds, we can prove (this is the object of section B.2 in appendix B) that for any " ? 0; there exists small enough values of ffi such that at some stage N , This result together with the convergence of the scheme leads to the convergence of the algorithm as ffi # 0 and n !1 (see figure 10). contraction property the "weak" RL with d d HJB equation DP equation Figure 10. The values V ffi iterated by an RL algorithm do not converge to V ffi as n !1. However, if the "weak" contraction property is satisfied, the V ffi tend to V as 5.1.2. The challenge of designing convergent algorithms. In general the "strong" contraction property (36) is impossible to obtain unless we have perfect knowledge of the dynamics f and the reinforcement functions r and R. In the RL approach, these components are estimated and approximated during some learning phase. Thus the iterated values V ffi are imperfect, but may be "good enough" to satisfy the weak contraction property (34). Defining such "good" approximations is the challenge for designing convergent RL algorithms. In order to illustrate the method, we present in section 5.2 a procedure for designing model-based algorithms, and in section 5.3, we give a model-free algorithm based on a FE approximation scheme. 5.2. Model-based algorithms The basic idea is to build a model of the state dynamics f and the reinforcement functions r and R at states - from the local knowledge obtained through the simulation of trajectories. So, if some trajectory xn (t) goes inside the neighborhood of - (by defining the neighborhood as an area whose diameter is bounded by kN :ffi for some positive constant kN ) at some time t n and keep a constant control u for a period - n (from )), we can build the model of f(-; u) and r(-; u) : (see figure 11). Then we can approximate the scheme (22), by the following values using the previous model : the Q-values Q are updated according to : (for any function -; u) satisfying (19)), which corresponds to the iterative rule (33) with the model f f n and er instead of f and r. y x x x Figure 11. A trajectory goes through the neighborhood (the grey area) of -. The state dynamics is approximated by ~ -n It is easy to prove (see (Munos & Moore, 1998) or (Munos, 1997a)) that assuming some smoothness assumptions (r; R Lipschitzian), the approximated V ffi n satisfy the condition (34) and theorem 6 applies. Remark. Using the same model, we can build a similar convergent RL algorithm based on the finite difference scheme (22) (see (Munos, 1998)). Thus, it appears quite easy to design model-based algorithms satisfying the condition (34). Remark. This method can also be used in the stochastic case, for which a model of the state dynamics is the average, for several trajectories, of such yn \Gammax n -n , and a model of the noise is their variance (see (Munos & Bourgine, 1997)). Furthermore, it is possible to design model-free algorithms satisfying the condition (34), which is the topic of the following section. 5.3. A Model-free algorithm The Finite Element RL algorithm. Consider a triangulation \Sigma ffi satisfying the properties of section 4.3. The direct RL approach consists of updating on-line the Q-values of the vertices without learning any model of the dynamics. We consider the FE scheme (22) with -; u) being such that j(-; -; u):f(-; u) is the projection of - onto the opposite side of the simplex, in a parallel direction to f(-; u) (see figure 12). If we suppose that the simplexes are EMI MUNOS non degenerated (i.e. 9k ae such that the radius of the sphere inscribed in each simplex is superior to k ae ffi ) then (19) holds. Let us consider that a trajectory x(t) goes through a simplex. Let the input point and be the output point. The control u is assumed to be constant inside the simplex. x x x y x Figure 12. A trajectory going through a simplex. j(-; u) is the projection of - onto the opposite side of the simplex. y\Gammax is a good approximation of j(-; u) \Gamma -. As the values -; u) and j(-; u) are unknown to the system, we make the following estimations (from Thales' ffl -; u) is approximated by - (where - (x) is the - \Gammabarycentric coordinate of x inside the simplex) ffl j(-; u) is approximated by - which only use the knowledge of the state at the input and output points (x and y), the running time - of the trajectory inside the simplex and the barycentric coordinate - (x) (which can be computed as soon as the system knows the vertices of the input side of the simplex). Besides, r(-; u) is approximated by the current reinforcement r(x; u) at the input point. Thanks to the linearity of V ffi inside the simplex, V ffi (j(-; u)) is approximated by the algorithm consists in updating the quality Q with the estimation : and if the system exits from the state-space inside the simplex (i.e. y 2 @O), then update the closest vertex of the simplex with : By assuming some additional regularity assumptions (r and R Lipschitzian, f bounded from below), the values V ffi and (35) which proves the convergence of the model-free RL algorithm based on the FE scheme (see (Munos, 1996) for the proof). In a similar way, we can design a direct RL algorithm based on the finite difference scheme F ffi FD (16) and prove its convergence (see (Munos, 1997b)). 6. A numerical simulation for the "Car on the Hill" problem For a description of the dynamics of this problem, see (Moore & Atkeson, 1995). This problem has a state-space of dimension 2 : the position and the velocity of the car. In our experiments, we chose the reinforcement functions as follows : the current reinforcement r(x; u) is zero everywhere. The terminal reinforcement R(x) is \Gamma1 if the car exits from the left side of the state-space, and varies linearly between depending on the velocity of the car when it exits from the right side of the state-space. The best reinforcement +1 occurs when the car reaches the right boundary with a null velocity (see figure 13). The control u has only 2 possible positive or negative thrust. Thrust Gravitation Resistance Goal Position x R=+1 for null velocity R=-1 for max. velocity Reinforcement Figure 13. The "Car on the Hill" problem. In order to approximate the value function, we used 3 different triangulations composed respectively of 9 by 9, 17 by 17 and 33 by 33 states (see figure 14), and, for each of these, we ran the two algorithms that follows : ffl An asynchronous Real Time DP (based on the updating rule (33)), assuming that we have a perfect model of the initial data (the state dynamics and the reinforcement functions). ffl An asynchronous Finite Element RL algorithm, described in section 5.3 (based on the updating rule (37)), for which the initial data are approximated by parts of trajectories selected at random. EMI MUNOS Velocity Triangulation 2 Triangulation 3 Position Triangulation 1 Figure 14. Three triangulations used for the simulations. In order to evaluate the quality of approximation of these methods, we also computed a very good approximation of the value function V (plotted in figure 15) by using DP (with rule (33)) on a very dense triangulation (of 257 by 257 states) with a perfect model of the initial data. Figure 15. The value function of the "Car on the Hill", computed with a triangulation composed of 257 by 257 states. We have computed the approximation error En being the discretization step of triangulation T k . For this problem, we notice that hypothesis (9) does not hold (because all the trajectories are tangential to the boundary of the state-space at the boundary states of zero velocity), and the value function is discontinuous. A frontier of discontinuity happens because a point beginning just above this frontier can eventually get a positive reward whereas any point below is doomed to exit on the left side of the state-space. Thus, following the remark in section 4.5, in order to compute En (T k ), we chose\Omega to be the whole state-space except some area around the discontinuity. Figures and 17 represent, respectively for the 2 algorithms, the approximation error En (T k ) (for the 3 triangulations function of the number of iterations n. We observe the following points : of approximation Triangulation 1 Triangulation 2 Triangulation 3 Figure 16. The approximation error En (T k ) of the values computed by the asynchronous Real Time DP algorithm as a function of the number of iterations n for several triangulations. of approximation Triangulation 1 Triangulation 3 Triangulation 2 Figure 17. The approximation error En (T k ) of the values computed by the asynchronous Finite Element RL algorithm. ffl Whatever the resolution of the discretization ffi is, the values V ffi computed by RTDP converge, as n increases. Their limit is V ffi , solution of the DP equation EMI MUNOS (20). Moreover, we observe the convergence of the V ffi to the value function V as the resolution ffi tends to zero. These results illustrate the convergence properties showed in figures 9. ffl For a given triangulation, the values V ffi computed by FERL do not converge. For discretization), the error of approximation decreases rapidly, and then oscillates within a large range. For T 2 , the error decreases more slowly (because there are more states to be updated) but then oscillates within a smaller range. And for T 3 (dense discretization), the error decreases still more slowly but eventually gets close to zero (while still oscillating). Thus, we observe that, as illustrated in figure 10, for any given discretization step ffi , the values do not converge. However, they oscillate within a range depending on ffi . Theorem 6 simply states that for any desired precision (8"), there exists a discretization step ffi such that eventually (9N; 8n ? N ), the values will approximate the value function at that precision (supjV ffi "). 7. Conclusion and future work This paper proposes a formalism for the study of RL in the continuous state-space and time case. The Hamilton-Jacobi-Bellman equation is stated and several properties of its solutions are described. The notion of viscosity solution is introduced and used to integrate the HJB equation for finding the value function. We describe discretization methods (by using finite element and finite difference schemes) for approximating the value function, and use the stability properties of the viscosity solutions to prove their convergence. Then, we propose a general method for designing convergent (model-based or model-free) RL algorithms and illustrate it with several examples. The convergence result is obtained by substituting the "strong" contraction property used to prove the convergence of DP method (which cannot hold any more when the initial data are not perfectly known) by some "weak" contraction property, that enables some approximations of these data. The main theorem states a convergence result for RL algorithms as the discretization step ffi tends to 0 and the number of iterations n tends to infinity. For practical applications of this method, we must combine to the learning dynamics (n !1) some structural dynamics which operates on the discretization process. For example, in (Munos, 1997c), an initial rough Delaunay triangulation progressively refined (by adding new vertices) according to a local criterion estimating the irregularities of the value function. In (Munos & Moore, 1999), a Kuhn triangulation embedded in a kd-tree is adaptively refined by a non-local splitting criterion that allows the cells to take into account their impact on other cells when deciding whether to split. Future theoretical work should consider the study of approximation schemes (and the design of algorithms based on these scheme) for adaptive and variable resolution discretizations (like the adaptive discretizations of (Munos & Moore, 1999; Munos, 1997c), the parti-game algorithm of (Moore & Atkeson, 1995), the multi- grid methods of (Akian, 1990) and (Pareigis, 1996), or the sparse grids of (Griebel, 1998)), the study of the rates of convergence of these algorithms (which already exists in some cases, see (Dupuis & James, 1998)), and the study of generalized control problems (with "jumps", generalized boundary conditions, etc. To adequately address practical issues, extensive numerical simulations (and comparison to other methods) have to be conducted, and in order to deal with high dimensional state-spaces, future work should concentrate on designing relevant structural dynamics and condensed function representations. Acknowledgments This work has been funded by DASSAULT-AVIATION, France, and has been carried out at the Laboratory of Engineering for Complex Systems (LISC) of the CEMAGREF, France. I would like to thank Paul Bourgine, Martine Naillon, Guillaume Guy Barles and Andrew Moore. I am also very grateful to my parents, my mentor Daisaku Ikeda and my best friend Guyl'ene. Appendix A Proof of theorem 5 A.1. Outline of the proof We use the Barles and Perthame procedure in (Barles & Perthame, 1988). First we give a definition of discontinuous viscosity solutions. Then we define the largest limit function V sup and the smallest limit function V inf and prove (following (Barles & Souganidis, 1991)), in lemma (1), that V sup (respectively V inf ) is a discontinuous viscosity sub-solution (resp. super-solution). Then we use a strong comparison result (lemma 2) which states that if (9) holds then viscosity sub-solutions are less than viscosity super-solutions, thus V sup - V inf . By definition V sup - V inf , thus and the limit function V is the viscosity solution of the HJB equation, and thus the value function of the problem. A.2. Definition of discontinuous viscosity solutions Let us recall the notions of the upper semi-continuous envelope W and the lower semi-continuous envelope W of a real valued function W : Definition 5. Let W be a locally bounded real valued function defined on O. EMI MUNOS ffl W is a viscosity sub-solution of H(x; W;DW O if for all functions local maximum of W \Lambda \Gamma ' such that W we have : ffl W is a viscosity super-solution of H(x; W;DW O if for all functions local minimum of W \Lambda \Gamma ' such that W we have : ffl W is a viscosity solution of H(x; W;DW O if it is a viscosity sub-solution and a viscosity super-solution of H(x; W;DW A.3. V sup and V inf are viscosity sub- and super-solutions Lemma 1 The two limit functions V sup and are respectively viscosity sub- and super-solutions. Proof: Let us prove that V sup is a sub-solution. The proof that V inf is a supersolution is similar. Let ' be a smooth test function such that V sup \Gamma ' has a maximum (which can be assumed to be strict) at x such that V sup n be a sequence converging to zero. Then V has a maximum at - n which tends to x as tends to 0. Thus, for all - 2 \Sigma By (27), we have : \Theta By (28), we obtain : By \Theta \Theta As tends to 0, the left side of this inequality tends to 0 as Thus, by (31), we have : Thus V sup is a viscosity sub-solution. A.4. Comparison principle between viscosity sub- and super-solutions Assume (9), then (7) and (6) has a weak comparison principle, i.e. for any viscosity sub-solution W and super-solution W of (7) and (6), for all x 2 O we have : For a proof of this comparison result between viscosity sub- and super-solutions see (Barles, 1994), (Barles & Perthame, 1988), (Barles & Perthame, 1990) or for slightly different hypothesis (Fleming & Soner, 1993). A.5. Proof of theorem 5 Proof: From lemma 1, the largest limit function V sup and the smallest limit function V inf are respectively viscosity sub-solution and super-solution of the HJB equation. From the comparison result of lemma 2, V sup - V inf . But by their definition and the approximation scheme V ffi converges to the limit function V , which is the viscosity solution of the HJB equation thus the value function of the problem, and (32) holds true. Appendix Proof of theorem 6 B.1. Outline of the proof We know that from the convergence of the scheme V ffi (theorem 5), for any compact ae O; for any " 1 ? 0; there exists a discretization step ffi such that : sup Let us define : As we have seen in section 5.1.1, if we had the strong contraction property (36), then for any ffi; E would converge to 0 as As we only have the weak contraction property the idea of the following proof is that for any " 2 ? 0; there exists ffi and a stage N; such that for n - N , EMI MUNOS Then we deduce that for any " ? 0; we can find sup B.2. A sufficient condition for Lemma 3 Let us suppose that there exists some constant ff ? 0 such that for any state - updated at stage n, the following condition hold : then there exists N such that for n - N;E ffi Proof: As the algorithm updates every state - 2 \Sigma ffi regularly, there exists an integer m such that at stage n+m all the states - 2 \Sigma ffi have been updated at least once since stage n. Thus, from (B.2) and (B.3) we have : Thus, there exists N 1 such that : 8n - sup Moreover, all states - 2 @ \Sigma ffi are updated at least once, thus there exists N 2 such that for any ffi - Thus from (B.4) and (B.5), for n - Lemma 4 For any " 1 ? 0; there exists \Delta 2 such that for the conditions (B.2) and (B.3) are satisfied. Proof: Let us consider a From the convergence of e(ffi) to 0 when # 0, there exists \Delta 1 such that for Let us prove that (B.2) and (B.3) hold. Let E then from (34), From (B.6), " 2and (B.2) holds for from (34), we have : and condition (B.3) holds. B.3. Convergence of the algorithm Proof: Let us prove theorem 6. For any ae O; for all " ? 0, let us ". From lemma 4, for conditions (B.2) and (B.3) are satisfied, and from lemma 3, there exists N; for all Moreover, from the convergence of the approximation scheme, theorem 5 implies that for any ae O; there exists \Delta 2 such that for all sup Thus for finite discretized state-space \Sigma ffi and @ \Sigma ffi satisfying the properties of section 4.4, there exists N; for all n - N , sup --R M'ethodes multigrilles en contr-ole stochastique Solutions de viscosit'e des 'equations de Hamilton-Jacobi, Vol. <Volume>17</Volume> of Math'ematiques et Applications. time problems in optimal control and vanishing viscosity solutions of hamilton-jacobi equations Comparison principle for dirichlet-type hamilton-jacobi equations and singular perturbations of degenerated elliptic equations Convergence of approximation schemes for fully nonlinear second order equations. Neural networks for control. Neuronlike adaptive elements that can solve difficult learning control problems. Dynamic Programming. A simplification of the back-propagation-through- time algorithm for optimal neurocontrol Dynamic Programming Generalization in reinforcement learning User's guide to viscosity solutions of second order partial differential equations. Viscosity solutions of hamilton-jacobi equations Rates of convergence for approximation schemes in optimal control. Controlled Markov Processes and Viscosity Solutions. Fuzzy q-learning Stable function approximation in dynamic programming. Adaptive sparse grid multilevel methods for elliptic pdes based on finite differences. Reinforcement Learning and its application to control. Reinforcement learning applied to a differential game. Reinforcement learning: a survey. Numerical methods for stochastic control problems in continuous time. Reinforcement Learning for Robots using Neural Networks. Automatic programming of behavior-based robots using reinforcement learning Le dilemme Exploration/Exploitation dans les syst'emes d'apprentissage par renforcement. Variable resolution dynamic programming: Efficiently learning action maps in multivariate real-valued state-spaces The parti-game algorithm for variable resolution reinforcement learning in multidimensional state space A general convergence theorem for reinforcement learning in the continuous case. Gradient descent approaches to neural- net-based solutions of the hamilton-jacobi-bellman equation Reinforcement learning for continuous stochastic control problems. Barycentric interpolators for continuous space and time reinforcement learning. Variable resolution discretization for high-accuracy solutions of optimal control problems Fuzzy reinforcement learning. Adaptive choice of grid and time in reinforcement learning. Neural Information Processing Systems. The Mathematical Theory of Optimal Processes. Markov Decision Processes Reinforcement learning with soft state aggregation. Online learning with random representations. International Conference on Machine Learning. Simple statistical gradient-following algorithms for connectionist reinforcement learning --TR Dynamic programming: deterministic and stochastic models Numerical methods for stochastic control problems in continuous time Connectionist learning for control Automatic programming of behavior-based robots using reinforcement learning Simple Statistical Gradient-Following Algorithms for Connectionist Reinforcement Learning Reinforcement learning and its application to control Numerical methods for stochastic control problems in continuous time Reinforcement learning for robots using neural networks The Parti-game Algorithm for Variable Resolution Reinforcement Learning in Multidimensional State-spaces Rates of Convergence for Approximation Schemes in Optimal Control Reinforcement learning for continuous stochastic control problems Adaptive choice of grid and time in reinforcement learning Barycentric interpolators for continuous space MYAMPERSANDamp; time reinforcement learning Markov Decision Processes Neuro-Dynamic Programming Finite-Element Methods with Local Triangulation Refinement for Continuous Reimforcement Learning Problems A General Convergence Method for Reinforcement Learning in the Continuous Case Variable Resolution Discretization for High-Accuracy Solutions of Optimal Control Problems Dynamic Programming --CTR Rmi Munos , Andrew Moore, Variable Resolution Discretization in Optimal Control, Machine Learning, v.49 n.2-3, p.291-323, November-December 2002

optimal control;reinforcement learning;viscosity solutions;finite difference and finite element methods;dynamic programming;Hamilton-Jacobi-Bellman equation

352748

Robust Plane Sweep for Intersecting Segments.

In this paper, we reexamine in the framework of robust computation the Bentley--Ottmann algorithm for reporting intersecting pairs of segments in the plane. This algorithm has been reported as being very sensitive to numerical errors. Indeed, a simple analysis reveals that it involves predicates of degree 5, presumably never evaluated exactly in most implementations. Within the exact-computation paradigm we introduce two models of computation aimed at replacing the conventional model of real-number arithmetic. The first model (predicate arithmetic) assumes the exact evaluation of the signs of algebraic expressions of some degree, and the second model (exact arithmetic) assumes the exact computation of the value of such (bounded-degree) expressions. We identify the characteristic geometric property enabling the correct report of all intersections by plane sweeps. Verification of this property involves only predicates of (optimal) degree 2, but its straightforward implementation appears highly inefficient. We then present algorithmic variants that have low degree under these models and achieve the same performance as the original Bentley--Ottmann algorithm. The technique is applicable to a more general case of curved segments.

Introduction As is well known, Computational Geometry has traditionally adopted the arithmetic model of exact computation over the real numbers. This model has been extremely productive in terms of algorithmic research, since it has permitted a vast community to focus on the elucidation of the combinatorial (topological) properties of geometric problems, thereby leading to sophisticated and eOEcient algorithms. Such approach, however, has a substantial shortcoming, since all computer calculations have -nite precision, a feature which aoeects not only the quality of the results but even the validity of speci-c algorithms. In other words, in this model algorithm correctness does not automatically translate into program correctness. In fact, there are several reports of failures of implementations of theoretically correct algorithms (see e.g. [For87, Hof89]). This state-of-aoeairs has engendered a vigorous debate within the research community, as is amply documented in the literature. Several proposals have been made to remedy this unsatisfactory situation. They can be split into two broad categories according to whether they perform exact computations (see, e.g., or approximate computations (see, e.g., [Mil88, HHK89, Mil89]). This paper -ne-tunes the exact-computation paradigm. The numerical computations of a geometric algorithm are basically of two types: tests (predicates) and constructions, with clearly distinct roles. Tests are associated with branching decisions in the algorithm that determine the AEow of control, whereas constructions are needed to produce the output data. While approximations in the execution of constructions are often acceptable, approximations in the execution of tests may produce incorrect branching, leading to the inconsistencies which are the object of the criticisms leveled against geometric algorithms. The exact-computation paradigm therefore requires that tests be executed with total accuracy. This will guarantee that the result of a geometric algorithm will be topologically correct albeit geometrically approximate. This also means that robustness is in principle achievable if one is willing to employ the required precision. The reported failures of structurally correct algorithms are entirely attributable to non-compliance with this criterion. Therefore, geometric algorithms can also be characterized on the basis of the complexity of their predicates. The complexity of a predicate is expressed by the degree of a homogeneous polynomial embodying its evaluation. The degree of an algorithm INRIA is the maximum degree of its predicates, and an algorithm is robust if the adopted precision matches the degree requirements. The idegree criterionj is a design principle aimed at developing low-degree algo- rithms. This approach involves re-examining under the degree criterion the rich body of geometric algorithms known today, possibly without negatively aoeecting traditional algorithmic eOEciency. A previous paper [LPT96] considered as an illustration of this approach the issue of proximity queries in two and three dimensions. As an additional case of degree-driven algorithm design, in this paper we confront another class of important geometric problems, which have caused considerable dif- -culties in actual implementations: plane-sweep problems for sets of segments. As we shall see, plane-sweep applications involve a number of predicates of dioeerent degree and algorithmic power. Their analysis will lead not only to new and robust implementations (an outcome of substantial practical interest) but elucidate on a theoretical level some deeper issues pertaining to the structure of several related problems and the mechanism of plane-sweeps. Plane sweep of intersecting segments Given is a -nite set S of line segments in the plane. Each segment is de-ned by the coordinates of its two endpoints. We discuss the three following problems (see Figure report the pairs of segments of S that intersect . construct the arrangement A of S, i.e., the incidence structure of the graph obtained interpreting the union of the segments as a planar graph. Pb3: construct the trapezoidal map T of S. T is obtained by drawing two vertical line segments (walls), one above and one below each endpoint of the segments and each intersection point. The walls are extended either until they meet another segment of S or to in-nity. be the segments of S and let k be the number of intersecting pairs. We say that the segments are in general position if any two intersecting segments intersect in a single point, and all endpoints and intersection points are distinct. RR n# 3270 Figure 1: S, A and T . INRIA The number of intersection points is no more than the number of intersecting pairs of segments and both are equal if the segments are in general position. Therefore, the number of vertices of A is at most k, the number of edges of A is at most and the number of vertical walls in T is at most 2(n + k), the bounds being tight when the segments are in general position. Thus the sizes of both A and T are O(n + k). We didn't consider here the 2-dimensional faces of either A or T . Including them would not change the problems we address. 3 Algebraic degree and arithmetic models It is well known that the eOEcient algorithms that solve Pb1-Pb3 are very unstable when implemented as programs, and several frustrating experiences have been reported [For85]. This motivates us to carefully analyze the predicates involved in those algorithms. We -rst introduce here some terminology borrowed from [LPT96]. We consider each input data (i.e., coordinates of an endpoint of some segment of S) as a variable. An elementary predicate is the sign \Gamma, 0, or + of a homogeneous multivariate polynomial whose arguments are a subset of the input variables. The degree of an elementary predicate is de-ned as the maximum degree of the irreducible factors (over the rationals) of the polynomials that occur in the predicate and that do not have a constant sign. A predicate is more generally a boolean function of elementary predicates. Its degree is the maximum degree of its elementary predicates. The degree of an algorithm A is de-ned as the maximum degree of its predicates. The degree of a problem P is de-ned as the minimum degree of any algorithm that solves P . In most problems in Computational Geometry, However, as d aoeects the speed and/or robustness of an algorithm, it is important to measure d precisely. In the rest of this paper we consider the degree as an additional measure of algorithmic complexity. Note that qualitatively degree and memory requirement are similar, since the arithmetic capabilities demanded by a given degree must be available, albeit they may be never resorted to in an actual run of the algorithm (since the input may be such that predicates may be evaluated reliably with lower precision). RR n# 3270 We will consider two arithmetic models. In the -rst one, called the predicate arithmetic of degree d, the only numerical operations that are allowed are the evaluations of predicates of degree at most d. Algorithms of degree d can therefore be implemented exactly in the predicate arithmetic model of degree d. This model is motivated by recent results that show that evaluating the sign of a polynomial expression may be faster than computing its value (see [ABD This model is however very conservative since the non-availability of the arithmetics required by a predicate is assimilated to an entirely random choice of the value of the predicate. The second model, called the exact arithmetic of degree d, is more demanding. It assumes that values (and not just signs) of polynomials of degree at most d be represented and computed exactly (i.e., roughly as d-fold precision integers). However, higher-degree operations (e.g., a multiplication one of whose factors is a d-fold precision integer) are appropriately rounded. Typical rounding is rounding to the nearest representable number but less accurate rounding can also be adequate as will be demonstrated later. Let A be an algorithm of degree d. If each input data is a b-bit integer, the size of each monomial occuring in a predicate of A is upper bounded by 2 (b+1)d . Moreover, let v be the number of variables that occur in a predicate; for most geometric problems and, in particular, for those considered in this paper, v is a small constant. It follows that an algorithm of degree d requires precision log v) in the exact arithmetic model of degree d. 4 The predicates for Pb1-Pb3 We use the following notations. The coordinates of point A i are denoted x i and y i . means that the x-coordinate of point A i is smaller than the x-coordinate of point A j . Similarly for ! y . [A i A j ] denotes the line segment whose left and right endpoints are respectively A i and A j , while denotes the line containing means that point A i lies below line 4.1 Predicates Pb1 only requires that we check if two line segments intersect (Predicate 2 0 below). INRIA Pb2 requires in addition the ability to sort intersection points along a line segment (Predicate 4 below). Pb3 requires the ability to execute all the predicates listed below : Two other predicates appear in some algorithms that report segment intersections : 4.2 Algebraic degree of the predicates We now analyze the algebraic degree of the predicates introduced above. Proposition 1 The degree of Predicates i and i 0 Proof. We -rst provide explicit formulae for the predicates. Evaluating Predicate 2 is equivalent to evaluating the sign of : can be implemented as follows for the case A 0 ! x A 2 (otherwise we exchange the roles of [A 0 A 1 ] and [A 2 A 3 RR n# 3270 if else return false else if else return false Therefore, in all cases, Predicate 2 0 reduces to Predicate 2. The intersection point I l ] is given by : I I (1) with Predicate 4 reduces to evaluating orient(I; A 0 ; A 1 ) where I is the intersection of It follows from (1) that this is equivalent to evaluating the sign of Explicit formulas for Predicates 3, 4 and 5 can be immediately deduced from the coordinates of the intersection points I which are given by (1). If A 4 A is clear from Equation (1) that is a common factor of x I INRIA do not intersect, Predicate 3 0 reduces to Predicate 2. Otherwise, it reduces to Predicate 3. The above discussion shows that the degree of predicates i and i 0 is at most i. To establish that it is exactly i, we have shown in Appendix that the polynomials of Predicates 2, 3, 4 0 and 5, as well as the factor other than involved in Predicate 4, are irreducible over the rationals. It follows that the proposition is proved for all predicates.Recalling the requirements of the various problems in terms of predicates, we have : Proposition 2 The algebraic degrees of Pb1, Pb2 and Pb3 are respectively 2, 4 and 5. 4.3 Implementation of Predicate 3 with exact arithmetic of degree 2 As it will be useful in the sequel, we show how to implement Predicate 3 (of degree under the exact arithmetic of degree 2. From Equation (1) we know that Predicate 3 can be written as : For convenience, let and x We stipulate to employ AEoating point arithmetic conforming to the ieee 754 standard [Gol91]. In this standard, simple precision allows us to represent b-bit integers with double precision allows us to represent 53. The coordinates of the endpoints of the segments are represented in simple precision and the computations are carried out in double precision. We denote \Phi,\Omega and ff the rounded arithmetic operations +; \Theta and =. In the ieee 754 standard, all four arithmetic operations are exactly rounded, i.e., the computed result is the AEoating point number that best approximates the exact result. RR n# 3270 are polynomials of degree 2, the four terms x 01 , x 21 , A and B in Inequality (2) can be computed exactly and the following monotonicity property is a direct consequence of exact rounding of arithmetic operations Monotonicity property 21\Omega A =) x 01 \Theta B ! x 21 \Theta A: This implies that the comparison between the two computed expressions x and x 21\Omega A evaluates Predicate 3 except when these numbers are equal. In most algorithms, an intersection point is compared with many endpoints. It is therefore more eOEcient to compute and store the coordinates of each intersection point and to perform comparisons with the computed abscissae rather than evaluating (2) repeatedly. We now illustrate an eoeective rounding procedure of the x-coordinates of intersection points. Lemma 3 If the coordinates of the endpoints of the segments are simple precision integers, then the abscissa x I of an intersection point can be rounded to one of its two nearest simple precision integers using only double precision AEoating point arithmetic operations. Proof. We assume that the coordinates of the endpoints of the segments are represented as b-bit integers stored as simple precision AEoating point numbers. The computations are carried out in double precision. The rounded value ~ x I of x I is given by : ~ 21\Omega where bXe denotes the integer nearest to X (with any tie breaking rule). If is a strict bound to the modulus of the relative error of all arithmetic operations, ~ 21\Omega the following relations : As x21 A INRIA 21\Omega We round ~ X to the nearest integer b ~ Xe. Since b ~ Xe and x 1 are (b there is no error in the addition. Therefore, ~ x I is a (b + 2)-bit integer and the absolute error on ~ x I is smaller than 1. 2 It follows that, under the hypothesis of the lemma, if E is an endpoint, I is an intersection point and ~ I the corresponding rounded point, the following monotonicity property holds : Monotonicity property I I Notice that the monotonicity property does not necessarily hold for two intersection points. Remark 1. A result similar to Lemma 3 has been obtained by Priest [Pri92] for points with AEoating point coordinates. More precisely, if the endpoints of the segments are represented as simple precision AEoating-point numbers, Priest [Pri92] has proposed a rather complicated algorithm that uses double precision AEoating point arithmetic and rounds x I to the nearest simple precision AEoating point number. This stronger result also implies the monotonicity property. 4.4 Algebraic degree of the algorithms The naive algorithm for detecting segment intersections (Pb1) evaluates \Theta(n 2 ) Predicates thus is of degree 2, which is degree-optimal by the proposition above. Although the time-complexity of the naive algorithm is worst-case optimal, since 1), it is worth looking for an output sensitive algorithm whose complexity depends on both n and k. Chazelle and Edelsbrunner [CE92] have shown that is a lower bound for Pb1, and therefore also for Pb2 and Pb3. A very recent algorithm of Balaban [Bal95] solves Pb1 optimally in O(n log n time using O(n) space. This algorithm does not solve Pb2 nor Pb3 and, since it uses Predicate 3 0 , its degree is 3. RR n# 3270 Pb2 can be solved by -rst solving Pb1 and subsequently sorting the reported intersection points along each segment. This can easily be done in O((n by a simple algorithm of degree 4 using O(n) space. A direct (and asymptotically more eOEcient) solution to Pb2 has been proposed by Chazelle and Edelsbrunner [CE92]. Its time complexity is O(n log algorithm, which constructs the arrangement of the segments, is of degree 4. A solution to Pb3 can be deduced from a solution to Pb2 in O(n+k) time using a very complicated algorithm of Chazelle [Cha91]. A deterministic and simple algorithm due to Bentley and Ottmann [BO79] solves Pb3 in O((n n) time, which is slightly suboptimal, using O(n) space. This classical algorithm uses the sweep-line paradigm and evaluates O((n n) predicates of all types discussed above, and therefore has degree 5. Incremental randomized algorithms [CS89, BDS construct the trapezoidal map of the segments and thus solve Pb3 and have degree 5. Their time complexity and space requirements are optimal (though only as expected performances). In this paper, we revisit the Bentley-Ottmann algorithm and show that a variant of degree 3 (instead of 5) can solve Pb 1 with no sacri-ce of performance (Section 6.1). Although this algorithm is slightly suboptimal with respect to time complexity, it is much simpler than Balaban's algorithm. We also present two variants of the sweep line algorithms. The -rst one (Section 6.2) uses only predicates of degree at most 2 and applies to the restricted but important special case where the segments belong to two subsets of non intersecting segments. The second one (Section 7) uses the exact arithmetic of degree 2. All these results are based on a (non-eOEcient) lazy sweep-line algorithm (to be presented in Section 5) that solves Pb1 by evaluating predicates of degree at most 2. Remark 2. When the segments are not in general position, the number s of intersection points can be less than the number k of intersecting pairs. In the extreme, Some algorithms can be adapted so that their time complexities depend on s rather than k [BMS94]. However, a lower bound on the degree of such algorithms is 4 since they must be able to detect if two intersection points are identical, therefore to evaluate Predicate 4 0 . INRIA 5 A lazy sweep-line algorithm Let S be a set of n segments whose endpoints are . For a succint review, the standard algorithm -rst sorts increasing x-coordinates and stores the sorted points in a priority queue X. Next, the algorithm begins sweeping the plane with a vertical line L and maintains a data structure Y that represents a sub-set of the segments of S (those currently intersected by L, ordered according to the ordinates of their intersections with L). Intersections are detected in correspondence of adjacencies created in Y , either by insertion/deletion of segment endpoints, or by order exchanges at intersections. An intersection, upon detection, is inserted into X according to its abscissa. Of course, a given intersection may be detected several times. Multiple detections can be resolved by performing a preliminary membership test for an intersection in X and omitting insertion if the intersection has been previously recorded. We stipulate to use another policy to resolve multiple detections, namely to remove from X an intersection point I whose associated segments are no longer adjacent in Y . Event I will be reinserted in X when the segments become again adjacent in Y . This policy has also the advantage of reducing the storage requirement of Bentley-Ottmann's algorithm to O(n) [Bro81]. We now describe a modi-cation of the sweep-line algorithm that does not need to process the intersection points by increasing x-coordinates. First, the algorithm sorts the endpoints of the segments by increasing x-coordinates into an array X. be the sorted list of endpoints. The algorithm uses also a dictionary Y that stores an ordered subset of the line segments. The algorithm rests on the de-nitions of active and prime pairs to be given be- low. We need the following notations. We denote by L(E i ) the vertical line passing denotes the open vertical slab bounded by L(E i ) and denotes the semiclosed slab obtained by adjoining line to the open slab For two segments S k and S l , we denote by A kl their rightmost left endpoint, by B kl their leftmost right endpoint and by I kl their common point when they intersect (see Figure 2). In addition, W kl denotes the set of segment endpoints that belong to the (closed) region bounded by the vertical lines and by the two segments (a double wedge). We denote by kl the most recently processed element of W kl and by F kl the element of W kl to be processed next. (Note that E kl and F kl are always de-ned, since they may res- RR n# 3270 l I kl A kl F kl kl kl Figure 2: For the de-nitions of W \Gamma kl kl pectively coincide with A kl and B kl .) Lastly, we de-ne sets W kl kl as follows. If S k and S l do not intersect, W kl kl consists of all points Otherwise, an endpoint E 2 W kl belongs to W kl (resp., to W \Gamma kl if the slab does (resp., does not) contain I kl . De-nition 4 Let (S k ; S l ) be a pair of segments and assume without loss of generality that S k L(E kl ). The pair is said to be active if the following conditions are satis-ed : INRIA 1. S k and S l are adjacent in Y , 2. 3. F kl kl Observe that the emptiness condition implies that the segments intersect. De-nition 5 A pair of active segments (S k ; S l ) is said to be prime if the next element to be processed belongs to W kl (therefore is F kl kl We say that an intersection (or an active pair) is processed when the algorithm reports it, exchanges its members in Y , and updates the set of active pairs of segments accordingly. After sorting the segment endpoints, the lazy sweep-line algorithm works as follows. While there are active pairs, the algorithm selects any of them and processes it. When there are no more active pairs the algorithm proceeds to the next endpoint, i.e., it inserts or removes the corresponding segment in Y (as appropriate) and updates the set of active pairs. Actually, the next endpoint may be accessed once there are no more prime pairs (a subset of the active pairs), without placing any deadline on the processing of the current active pairs as long as they are not prime. When there are no more active pairs and no more endpoints to be processed, the algorithm stops. For reasons that will be clear below, the algorithm will not be speci-ed in the -nest detail, since several dioeerent implementations are possible. The main issue is the eOEcient detection of active and prime pairs and several solutions, all consistent with the described lazy algorithm, will be discussed in Sections 6 and 7. It should be noted that deciding if a pair of intersecting segments is active or prime reduces to the evaluation of Predicates 2 only. Therefore, the algorithm just described involves only Predicates 1 and 2 and is of degree 2 by Proposition 1. It should also be pointed out that two intersection points or even an intersection point and an endpoint won't necessarily be processed in the order of their x-coordinate. As a consequence, Y won't necessarily represent the ordered set of segments intersecting some vertical line L (as in the standard algorithm). respectively, snapshots of the data structure Y immediately before and after processing event RR n# 3270 only by the segment S that has E i as one of its endpoints. Let ). The order relation in Y is denoted by !. Theorem 6 If Predicates 1 and 2 are evaluated exactly, the described lazy sweep-line algorithm will detect all pairs of segments that intersect. Proof : The algorithm (correctly) sorts the endpoints of the segments by increasing x-coordinates into X. Consequently, the set of segments that intersect L(E) and the set of segments in Y (E) coincide for any endpoint E. The proof of the theorem is articulated now as two lemmas and their implications. Lemma 7 Two segments have exchanged their positions in Y if and only if they intersect and if the pair has been processed. Proof. Let us consider two segments, say S k and S l , that do not intersect. Without loss of generality, let S k ! S l in Y Assume for a contradiction that S l ! S k in positions because they will never form an active pair, Therefore, S l ! S k in Y \Gamma (B kl ) can only happen if there exists a segment l, that at some stage in the execution of the algorithm was present in Y together with S k and S l and caused one of the following two events to occur : 1. Sm ? S l and the positions of Sm and S k are exchanged in Y 2. Sm ! S k and the positions of Sm and S l are exchanged in Y . In both cases, the segments that exchange their positions are not consecutive in Y , violating Condition 1 of De-nition 4. Therefore, two segments can exchange their positions in Y only if they intersect and this can only happen when their intersection is processed. Moreover, when the intersection has been processed, the segments are no longer active and cannot exchange their positions a second time. 2 We say that an endpoint E of S is correctly placed if and only if the subset of the segments that are below E (in the plane) coincides with the subset of the segments INRIA Otherwise, E is said to be misplaced. Lemma 8 If Predicates 1 and 2 are evaluated exactly, both endpoints of every segment are correctly placed. Proof. Assume, for a contradiction, that E of S is the -rst endpoint to be misplaced by the algorithm. can be misplaced only if there exist at least two intersecting segments S k and S l in Y \Gamma (E) such that E belongs to W kl . Proof. First recall that Predicate 2 is the only predicate involved in placing S in Y . Consider -rst the case where E is the left endpoint of S. For any pair (S k ; S l ) of segments in Y \Gamma (E), for which S is either above or below both S k and S l , the relative position of S with respect to S k and S l does not depend on the relative order of S k and S l in Y \Gamma (E). Therefore, E will be correctly placed in Y If E is a right endpoint, since the left endpoint of S has been correctly placed, E can only be misplaced if there exists a segment S intersecting S such that the relative positions of S and S 0 in Y are the same is the rightmost left endpoint of S and S 0 ) while this change has not been executed by the algorithm in Y , i.e. S and S 0 have the same relative position in Y In that l be two segments of Y \Gamma (E) such that E 2 W kl . Assume without loss of generality that S k ! S l in Y (E kl ). Since E is the -rst endpoint to be misplaced, we have S k L(E kl ). For convenience, we will say that two segments S p and S q have been exchanged between E 0 and E 00 , for two events RR n# 3270 The case where E kl cause any diOEculty since S k and S l cannot be active between E kl and E and therefore S k and S l cannot be exchanged between E kl and E, which implies that E is correctly placed with respect to S k and S l . The case where E kl diOEcult. E is not correctly placed only if S k and S l are not exchanged between E kl and E, i.e., S k ! S l in both Y shall prove that this is not possible and therefore conclude that S is correctly placed into Y in this case as well. Assume, for a contradiction, that S k and S l have not been exchanged between E kl and E. As E belongs to W kl l cannot be adjacent in Y \Gamma (E) since otherwise they would constitute a prime pair and they would have been exchanged. be the subsequence of segments of occurring between S k and S l . Assume that (S k ; S l ) is a pair of intersecting segments such that E kl which r is minimal (i.e., for which the above subsequence is shortest). A direct consequence of this de-nition is the following belong to W sects S l , E cannot belong to W We distinguish the two following cases: intersects S l to the left of L(E) because S k i and S l are correctly placed in Y is the -rst endpoint to be misplaced) and misplaced in Y \Gamma (E). It then follows from Claim 2 that E l L(E). As E is the -rst end-point to be misplaced, we have S l since the pair l ) is not active (between E k i l and E), and therefore cannot be exchanged, the same inequality holds in Y \Gamma (E), which contradicts the de-nition of S k i This case is entirely symetric to the previous one. It suOEces to exchange the roles of S k and S l and to reverse the relations ! and ! y . Since a contradiction has been reached in both cases, the lemma is proved. 2 We now complete the proof of the theorem. The previous lemma implies that the endpoints are correctly processed. Indeed let E i be an endpoint. If E i is a right endpoint, we simply remove the corresponding segment from Y and update the set INRIA of active segments. This can be done exactly since predicates of degree - 2 are evaluated correctly. If E i is a left endpoint, it is correctly placed in Y on the basis of the previous lemma. The lemma also implies that all pairs that intersect have been processed. Indeed if are two intersecting segments such that S p L(B pq ), the lemma shows that S p ! S q in Y in Y \Gamma (B pq ), which implies that the pair (S p ; S q ) has been processed (Lemma 7). This concludes the proof of the theorem. 2 Remark 3. Handling the degenerate cases does not cause any diOEculty and the previous algorithm will work with only minor changes. For the initial sorting of the endpoints, we can take any order relation compatible with the order of their x-coordinates, e.g., the lexicographic order. Remark 4. Theorem 6 applies directly to pseudo-segments, i.e., curved segments that intersect in at most one point. Lemmas 7 and 8 also extend to the case of monotone arcs that may intersect in more than one point. To be more precise, in Lemma 7, we have to replace iintersectj by iintersect an odd number of timesj; Lemma 8 and its proof are unchanged provided that we de-ne W kl kl as the subset of W kl consisting of the endpoints E, that the slab E) contains an odd number (resp. none or an even number) of intersection points. As a consequence, the lazy algorithm (which still uses only detect all pairs of arcs that intersect an odd number of times. Remark 5. For line segments, observe that checking whether a pair of segments is active does not require to know (and therefore to maintain) E kl . In fact, we can replace Condition 3 in the de-nition of an active pair by the following condition , the two de-nitions are identical and if I kl ! x E kl , the pair is not active since, by Lemma 8, Condition 2 of the de-nition won't be satis-ed. RR n# 3270 6 EOEcient implementations of the lazy algorithm in the predicate arithmetic model The diOEculty to eOEciently implement the lazy sweep-line algorithm using only predicates of degree at most 2 (i.e., in the predicate arithmetic model of degree 2) is due to veri-cation of the emptiness condition in De-nition 4 and of the condition expressed by De-nition 5. One can easily check that various known implementations of the sweep achieve straightforward veri-cation of the emptiness condition by introducing algorithmic complications. The following subsection describes an eOEcient implementation of the lazy algorithm in the predicate arithmetic model of degree 3. The second subsection improves on this result in a special but important instance of Pb1, namely the case of two sets of non-intersecting segments. The algorithm presented there uses only predicates of degree at most 2. 6.1 Robustness of the standard sweep-line algorithm We shall run our lazy algorithm under the predicate arithmetic model of degree 3. We then have the capability to correctly compare the abscissae of an intersection and of an endpoint. We re-ne the lazy algorithm in the following way. Let E i be the last processed endpoint and let E i+1 be the endpoint to be processed next. An active pair (S k ; S l ) that occurs in Y between Y processed if and only if its intersection point I kl lies to the right of E i and not to the right of As the slab is free of endpoints in its interior, any pair of adjacent segments encountered in Y (between Y that intersect within the slab is active. Moreover the intersection points of all prime pairs belong to the slab. It follows that this instance of the lazy algorithm need not explicitly check whether a pair is active or not, and therefore is much more eOEcient than the lazy algorithm of Section 5. This algorithm is basically what the original algorithm of Bentley- Ottmann becomes when predicates of degree at most 3 are evaluated (recall that the standard algorithm requires the capability to correctly execute predicates of degree up to 5). With the policy concerning multiple detections of intersections that is stipulated at the beginning of Section 5. INRIA We therefore conclude with the following theorem : Theorem 9 If Predicates 1, 2 and 3 are evaluated exactly, the standard sweep-line algorithm will solve Pb1 in O((n It is now appropriate to brieAEy comment on the implementation details of the just described modi-ed algorithm. Data structure Y is implemented as usual as a dic- tionary. Data structure X, however, is even simpler than in the standard algorithm (which uses a priority queue with dictionary access). Here X has a primary component realized as a static dynamic search tree on the abscissae of the endpoints points to a secondary data structure L(E j ) realized as a conventional linked list, containing (in an arbitrary order) adjacent intersecting pairs in Remember that each intersecting pair of L(E j ) is active. Insertion into is performed at one of its ends and so is access for reporting (when the plane sweep reaches slab To eoeect constant-time removal of a pair due to loss of adjacency, however, a pointer could be maintained from a -xed member of the pair (say, the one with smaller left endpoint in lexicographic order) to the record stored in X (notice that the described insertion/removal policy, which guarantees that the elements of X correspond to pairs of adjacent segments in Y , ensures that at most one record is to be pointed to by any member of Y ). We -nally observe that a segment adjacency arising in Y during the execution of the algorithm must be tested for intersection; however, an intersecting pair of adjacent segments is eligible for insertion into X only as long as the plane sweep has not gone beyond the slab containing the intersection in question. As regards the running time, beside the initial sorting of the endpoints and the creation of the corresponding primary tree in time O(n log n), it is easily seen that each intersection uses O(log n) time (amortized), thereby achieving the performance of the standard algorithm. Finally, we note that if only predicates of degree - 2 are evaluated correctly, the algorithm of Bentley-Ottmann may fail to report the set of intersecting pairs of segments. See Figure 3 for an example. Remark 6. The fact that the sweep line algorithm does not need to sort intersection points had already been observed by Myers [Mye85] and Schorn [Sch91]. Myers does not use it for solving robustness problems but for developing an algorithm with an expected running time of O(n log n + k). Schorn uses this fact to decrease RR n# 3270 Figure 3: If the computed x-coordinate of the intersection point of S 1 and S 2 is (erroneously) found to be smaller than the x-coordinate of the left endpoint of S 3 and if S 3 and S 4 are (correctly) inserted below S 2 and above S 1 respectively, then the intersection between S 3 and S 4 will not be detected. Observe that the missed intersection point can be arbitrarily far from the intersection point involved in the wrong decision. INRIA the precision required by the sweep line algorithm from -ve fold to three fold, i.e., Schorn's algorithm uses exact arithmetic of degree 3. Using Theorem 6, we will show in Section 7 that double precision suOEces. 6.2 Reporting intersections between two sets of nonintersecting line segments In this subsection, we consider two sets of line segments in the plane, S b (the blue set) and S r (the red set), where no two segments in S b (similarly, in S r ) intersect. Such a problem arises in many applications, including the union of two polygons and the merge of two planar maps. We denote by n b and n r the cardinalities of S b respectively, and let Mairson and Stol- [MS88] have proposed an algorithm that works for arcs of curve as well as for line segments. Its time complexity is O(n log n + k), which is optimal, and requires O(n space (O(n) in case of line segments). The same asymptotic time-bound has been obtained by Chazelle et al. [CEGS94] and by Chazelle and Edelsbrunner [CE92]. The latter algorithm is not restricted to two sets of nonintersecting line segments. Other algorithms have been proposed by Nievergelt and Preparata [NP82] and by Guibas and Seidel [GS87] in the case where the segments of S b (and S r ) are the edges of a subdivision with convex faces. With the exception of the algorithm of Chazelle et al. [CEGS94], all these algorithms construct the resulting arrangement and therefore have degree 4. The algorithm of Chazelle et al. requires to sort the intersection points of two segments with a vertical line passing through an endpoint. Therefore it is of degree 3. We propose instead an algorithm that computes all the intersections but not the arrangement. This algorithm uses only predicates of degree - 2 and has time We say that a point E i is vertically visible from a segment S b 2 S b if the vertical line segment joining E i with S b does not intersect any other segment in S b (the same notion is applicable to S r ). For two intersecting segments S b 2 S b and S r be a vertical line to the right of A br such that no other segment intersects L between S b and S r (i.e., S b and S r are adjacent). We let T br denote the wedge de-ned by S b and S r in the slab between L and L(I kl ). RR n# 3270 Our algorithm is based on the following observation : contains blue endpoints if and only if it contains a blue endpoint that is vertically visible from S b . Similarly, T br contains red endpoints if and only if it contains a red endpoint vertically visible from S r . Proof : The suOEcient condition is trivial, so we only prove necessity. Assume without loss of generality that S r be the subset of the blue endpoints that belong to T br and CH + (E) their upper convex hull. Clearly, all vertices of CH + (E) are vertically visible from S b . 2 Our algorithm has two phases. The second one is the lazy algorithm of Section 5. The -rst one can be considered as a preprocessing step that will help to eOEciently -nd active pairs of segments. More speci-cally, our objective is to develop a quick test of the emptiness condition based on the previous lemma. The preprocessing phase is aimed at identifying the candidate endpoints for their potential belonging to wedges formed by intersecting adjacent pairs. Referring to S b (and analogously for S r ), we -rst sweep the segments of S b and construct for each blue segment S b the lists b and L b of blue endpoints that are vertically visible from S b and lie respectively below and above S b . The sweep takes time O(n log n) and the constructed lists are sorted by increasing abscissa. Since there is no intersection point, only predicates of degree - 2 are used. The total size of the lists r , and L r is O(n). As mentioned above, the crucial point is to decide whether the wedge T br of a pair of intersecting segments S b and S r adjacent in Y contains or not endpoints of other segments. Without any loss of generality we assume that S r ! S b in Y . If such endpoints exist, then T br contains either a blue vertex of CH red vertex of CH We will show below that, using predicates of degree - 2, the lists can be preprocessed in time O(n log n) and that deciding whether T br contains or not endpoints can be done in time O(log n), using only predicates of degree at most 2. Assuming for the moment that this primitive is available, we can execute the plane sweep algorithm described earlier. Speci-cally, we sweep S b and S r simultaneously, INRIA using the lazy sweep-line algorithm of Section 5 y . Each time we detect a pair of (adjacent) intersecting segments S b and S r , we can decide in time O(log n) whether they are "active" or "not active", using only predicates of degree - 2. We sum up the results of this section in the following theorem : Theorem 11 Given n line segments in the plane belonging to two sets S b and S r , where no two segments in S b (analogously, in S r ) intersect, there exists an algorithm of optimal degree 2 that reports all intersecting pairs in O((n using O(n) storage. We now return to the implementation of the primitive described above. Suppose that, for some segment S i or r) we have constructed the upper convex hull Then we can detect in O(log n) time if an element of a list, say lies above some segment S r . More speci-cally, we -rst identify among the edges of slopes are respectively smaller and greater than the slope of S r . This only requires the evaluation of O(log predicates of degree 2. It then remains to decide whether the common endpoint E of the two reported edges lies above or below the line containing S r . This can be answered by evaluating the orientation predicate orient(E; A r The crucial requirement of the adopted data structure is the ability to eOEciently To this purpose, we propose the following solution. The data structure associated with a list or r) represents the upper convex hull CH . (Similarly, the data structure associated to a list L represents the lower convex hull CH .) This implies that a binary search on the convex hull slopes uniquely identi-es the test vertex. Since the elements of each list are already sorted by increasing x-coordinates, the data structures can be constructed in time proportional to their sizes, therefore in O(n) time in total. It can be easily checked that only orientation predicates (of degree 2) are involved in this process. To guarantee the availability of CH we have to ensure that our data structure can eOEciently handle the deletion of elements. As elements are deleted in order of increasing abscissa, this can be done in amortized O(log per deletion [HS90, HS96]. It follows that preprocessing all lists takes O(n) time, uses O(n) space and only requires the evaluation of predicates of degree - 2. y We can adopt the policy of processing all active pairs before the next endpoint. RR n# 3270 7 An eOEcient implementation of the lazy algorithm under the exact arithmetic model of degree 2 We shall run the lazy algorithm of Section 6.1 under the exact arithmetic model of degree 2, i.e. Predicates 1 and 2 are evaluated exactly but Predicate 3 is implemented with exact arithmetic of degree 2 as explained in Section 4.3. Several intersection points may now be found to have the same abscissa as an endpoint. We re-ne the lazy algorithm in the following way. Let E i be the last processed endpoint and let be the endpoint with an abscissa strictly greater than the abscissa of E i to be processed next. An active pair (S k ; S l ) will be processed if and only if its intersection point is found to lie to the right of E i and not to the right of E i+1 . We claim that this policy leads to eOEcient veri-cation of the emptiness condition. Indeed, the intersections of all prime pairs belong to kl and, by the monotonicity property, I kl will be found to be The crucial observation that drastically reduces the time complexity is the following. A pair of adjacent segments (S k ; S l ) encountered in Y between Y whose intersection point is found to lie in slab active if and only if kl (E kl ). Indeed, since I kl is found to be ! x E i+2 , the monotonicity property implies that I kl ! x E i+2 . Therefore, when checking if a pair is active, it is suOEcient to consider just the next endpoint, not all of them. Theorem 6 therefore applies. If no two endpoints have the same x-coordinate, the algorithm can use the same data structures as the algorithm in Section 6.1 and its time complexity is clearly the same as for the Bentley-Ottmann's algorithm. Otherwise, we construct X on the distinct abscissae of the endpoints and store all endpoints with identical x-coordinates in a secondary search structure with endpoints sorted by y-coordinates. This secondary structure will allow to determine if a pair is active in logarithmic time by binary search. We conclude with the following theorem : Theorem 12 Under the exact arithmetic model of degree 2, the instance of the lazy algorithm described above solves Pb1 in O((n INRIA 8 Conclusion Further pursuing our investigations in the context of the exact-computation para- digm, in this paper we have illustrated that important problems on segment sets (such as intersection report, arrangement, and trapezoidal map), which are viewed as equivalent under the Real-RAM model of computation, are distinct if their arithmetic degree is taken into account. This sheds new light on robustness issues which are intimately connected with the notion of algorithmic degree and illustrates the richness of this new direction of research. For example, we have shown that the well-known plane-sweep algorithm of Bentley- Ottmann uses more machinery than strictly necessary, and can be appropriately modi-ed to report segment intersections with arithmetic capabilities very close to optimal and no sacri-ce in performance. Another result of our work is that exact solutions of some problems can be obtained even if approximate (or even random) evaluations of some predicates are performed. More speci-cally, using less powerful arithmetic than demanded by the application, we have been able to compute the vertices of an arrangement of line segments by constructing an arrangement which may be dioeerent from the actual one (and may not even correspond to any set of straight line segments) but still have the same vertex set. Our work shows that the sweep-line algorithm is more robust than usually believed, proposes practical improvements leading to robust implementations, and provides a better understanding of the sweeping line paradigm. The key to our technique is to relax the horizontal ordering of the sweep. This is one step further after similar attemps aimed though at dioeerent purposes [Mye85, MS88, EG89]. A host of interesting open questions remain. One such question is to devise an output-sensitive algorithm for reporting segment intersections with optimal time complexity and with optimal algorithmic degree (that is, 2). It would also be interesting to examine the plane-sweep paradigm in general. For example, with regard to the construction of Voronoi diagrams in the plane, one should elucidate the reasons for the apparent gap between the algorithmic degrees of Fortune's plane-sweep solution and of the (optimal) divide-and-conquer and incremental algorithms. RR n# 3270 Acknowledgments We are indebted to H. Br#nnimann for having pointed out an error in a previous version of this paper and to O. Devillers for discussions that lead to Lemma 3. S. Pion, M. Teillaud and M. Yvinec are also gratefully acknowledged for their comments on this work. --R An optimal algorithm for Applications of random sampling to on-line algorithms in computational geometry Computing exact geometric predicates using modular arithmetic with single precision. Jochen K On degeneracy in geometric computations. Algorithms for reporting and counting geometric intersections. Introduction to higher algebra. Comments on iAlgorithms for reporting and counting geometric intersectionsj. EOEcient exact evaluation of signs of determinants. An optimal algorithm for intersecting line segments in the plane. Algorithms for bichromatic line segment problems and polyhedral terrains. Triangulating a simple polygon in linear time. Applications of random sampling in computational geometry Topologically sweeping an arran- gement Computational geometry in practice. Computational geometry and software engineering: Towards a geometric computing environment. EOEcient exact arithmetic for computational geometry. What every computer scientist should know about AEoating- point arithmetic Computing convolutions by reciprocal search. Robust set operations on polyhedral solids. The problems of accuracy and robustness in geometric computation. Applications of a semi-dynamic convex hull algorithm Robust proximity queries in implicit Voronoi diagrams. Double precision geometry: a general technique for calculating line and segment intersections using rounded arithmetic. An O(E log E On properties of AEoating point arithmetics: numerical stability and the cost of accurate computations. Robust Algorithms in a Program Library for Geometric Com- putation Robust adaptive AEoating-point geometric predi- cates Towards exact geometric computation. --TR --CTR Olivier Devillers , Alexandra Fronville , Bernard Mourrain , Monique Teillaud, Algebraic methods and arithmetic filtering for exact predicates on circle arcs, Proceedings of the sixteenth annual symposium on Computational geometry, p.139-147, June 12-14, 2000, Clear Water Bay, Kowloon, Hong Kong Ferran Hurtado , Giuseppe Liotta , Henk Meijer, Optimal and suboptimal robust algorithms for proximity graphs, Computational Geometry: Theory and Applications, v.25 n.1-2, p.35-49, May Leonardo Guerreiro Azevedo , Ralf Hartmut Gting , Rafael Brand Rodrigues , Geraldo Zimbro , Jano Moreira de Souza, Filtering with raster signatures, Proceedings of the 14th annual ACM international symposium on Advances in geographic information systems, November 10-11, 2006, Arlington, Virginia, USA Menelaos I. Karavelas , Ioannis Z. Emiris, Root comparison techniques applied to computing the additively weighted Voronoi diagram, Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms, January 12-14, 2003, Baltimore, Maryland Elmar Schmer , Nicola Wolpert, An exact and efficient approach for computing a cell in an arrangement of quadrics, Computational Geometry: Theory and Applications, v.33 n.1-2, p.65-97, January 2006

segment intersection;robust algorithms;plane sweep;computational geometry

352753

Binary Space Partitions for Fat Rectangles.

We consider the practical problem of constructing binary space partitions (BSPs) for a set S of n orthogonal, nonintersecting, two-dimensional rectangles in ${\Bbb R}^3$ such that the aspect ratio of each rectangle in $S$ is at most $\alpha$, for some constant $\alpha \geq 1$. We present an $n2^{O(\sqrt{\log n})}$-time algorithm to build a binary space partition of size $n2^{O(\sqrt{\log n})}$ for $S$. We also show that if $m$ of the $n$ rectangles in $S$ have aspect ratios greater than $\alpha$, we can construct a BSP of size $n\sqrt{m}2^{O(\sqrt{\log n})}$ for $S$ in $n\sqrt{m}2^{O(\sqrt{\log n})}$ time. The constants of proportionality in the big-oh terms are linear in $\log \alpha$. We extend these results to cases in which the input contains nonorthogonal or intersecting objects.

Introduction How to render a set of opaque or partially transparent objects in IR 3 in a visually realistic way is a fundamental problem in computer graphics [12, 22]. A central component of this problem is hidden-surface removal: given a set of ob- jects, a viewpoint, and an image plane, compute the scene visible from the viewpoint on the image plane. Because of its importance, the hidden-surface removal problem has Support was provided by National Science Foundation research grant CCR-93-01259, by Army Research Office MURI grant DAAH04- 96-1-0013, by a Sloan fellowship, by a National Science Foundation NYI award and matching funds from Xerox Corp, and by a grant from the U.S.- Israeli Binational Science Foundation. Address: Box 90129, Department of Computer Science, Duke University, Durham, NC 27708-0129. Email: pankaj@cs.duke.edu y Support was provided by Army Research Office grant DAAH04-93- G-0076. This work was partially done when the author was at Duke Uni- versity. Address: Max-Planck-Institut f?r Informatik, Im Stadtwald, 66 Saarbr-ucken, Germany. Email: eddie@math.uri.edu z This author is affiliated with Brown University. Support was provided in part by National Science Foundation research grant CCR-9522047 and by Army Research Office MURI grant DAAH04-96-1-0013. Address: Box 90129, Department of Computer Science, Duke University, Durham, NC 27708-0129. Email: tmax@cs.duke.edu x Support was provided in part by National Science Foundation re-search grant CCR-9522047, by Army Research Office grant DAAH04- 93-G-0076, and by Army Research Office MURI grant DAAH04-96- 1-0013. Address: Box 90129, Department of Computer Science, Duke University, Durham, NC 27708-0129. Email: jsv@cs.duke.edu been studied extensively in both the computer graphics and the computational geometry communities [11, 12]. One of the conceptually simplest solutions to this problem is the z-buffer algorithm [6, 12]. This algorithm sequentially processes the objects; and for each object it updates the pixels of the image plane covered by the object, based on the distance information stored in the z-buffer. A very fast hidden-surface removal algorithm can be obtained by implementing the z-buffer in hardware. However, the cost of a hardware z-buffer is very high. Only special-purpose and costly graphics engines contain fast z-buffers, and z-buffers implemented in software are generally inefficient. Even when fast hardware z-buffers are present, they are not fast enough to handle the huge models (containing hundreds of millions of polygons) that often have to be displayed in real time. As a result, other methods have to be developed either to "cull away" a large subset of invisible polygons so as to decrease the rendering load on the z-buffer (when models are large; e.g., see [23]) or to completely solve the hidden-surface removal problem (when there are very slow or no z-buffers). One technique to handle both of these problems is the binary space partition (BSP) introduced by Fuchs et al. [14]. They used the BSP to implement the so-called "painter's al- gorithm" for hidden-surface removal, which draws the objects to be displayed on the screen in a back-to-front order (in which no object is occluded by any object earlier in the order). In general, it is not possible to find a back-to-front order from a given viewpoint for an arbitrary set of objects. By fragmenting the objects, the BSP ensures that from any viewpoint a back-to-front order can be determined for the fragments. Informally, a BSP for a set of objects is a tree each of whose nodes is associated with a convex region of space. The regions associated with the leaves of the tree form a convex decomposition of space, and the interior of each region does not intersect any object. The fragments created by the BSP are stored at appropriate nodes of the BSP. Given a viewpoint p, the back-to-front order is determined by a suitable traversal of the BSP. For each node v of the BSP, the objects in one of v's subtrees are separated from those in v's other subtree by a hyperplane. The viewpoint p will lie in one of the regions bounded by the hyperplane at v. The traversal recursively visits first the child of v corresponding to the halfspace not containing p and then the other child of v. The efficiency of the traversal, and thus of the hidden-surface removal algorithm, depends upon the size of the BSP. The BSP has subsequently proven to be a versatile data structure with applications in many other problems that arise in practice-global illumination [5], shadow generation [7, 8, 9], ray tracing [19], visibility problems [3, 23], solid geometry [17, 18, 24], robotics [4], and approximation algorithms for network flows and surface simplification [2, 16]. Although several simple heuristics have been developed for constructing BSPs of reasonable sizes [3, 13, 14, 23, 24], provable bounds were first obtained by Paterson and Yao. They show that a BSP of size O(n 2 ) can be constructed for n disjoint triangles in IR 3 , which is optimal in the worst case [20]. But in graphics-related applications, many common environments like buildings are composed largely of orthogonal rectangles. Moreover, many graphics algorithms approximate non-orthogonal objects by their orthogonal bounding boxes and work with the bounding boxes [12]. In another paper, Paterson and Yao show that a BSP of size O(n n) exists for n non-intersecting, orthogonal rectangles in IR 3 [21]. This bound is optimal in the worst case. In all known lower bound examples of orthogonal rectangles in IR 3 requiring BSPs of n), most of the rectangles are "thin." For example, Paterson and Yao's lower bound proof uses a configuration of \Theta(n) orthogonal rect- angles, arranged in a p n \Theta n \Theta n grid, for which any BSP has n). All rectangles in their construction have aspect n). Such configurations of thin rectangles rarely occur in practice. Many real databases consist mainly of "fat" rectangles, i.e., the aspect ratios of these rectangles are bounded by a constant. An examination of four datasets-the Sitterson Hall, the Orange United Methodist Church Fellowship Hall, and the Sitterson Hall Lobby databases from the University of North Carolina at Chapel Hill and the model of Soda Hall from the University of California at Berkeley-shows that most of the rectangles in these models have aspect ratio less than 30. It is natural to ask whether BSPs of near-linear size can be constructed if most of the rectangles are "fat." We call a rectangle fat if its aspect ratio (the ratio of the longer side to the shorter side) is at most ff, for a fixed constant ff - 1. A rectangle is said to be thin if its aspect ratio is greater than ff. In this paper, we consider the following problem: Given a set S of n non-intersecting, orthogonal, two-dimensional rectangles in IR 3 , of which m are thin and the remaining are fat, construct a BSP for S. We first show how to construct a BSP of size O( log n ) for n fat rectangles in IR 3 (i.e., We then show that if m ? 0, a BSP of size n m2 O( log n ) can be built. This bound comes close to the lower bound of \Omega\Gamma n m) . We finally prove two important extensions to these re- sults. We show that an np2 O( log n )-size BSP exists if p of the n input objects are non-orthogonal. Unlike in the case of orthogonal objects, fatness does not help in reducing the worst-case size of BSPs for non-orthogonal objects. In particular, there exists a set of n fat triangles in IR 3 for which any BSP non-orthogonal objects can be approximated by orthogonal bounding boxes. The resulting bounding boxes might intersect each other. Motivated by this observation, we also consider the problem where n fat rectangles contain k intersecting pairs of rectangles, and we show that we can construct a BSP of size (n + k2 O( log n ). There is a lower bound of on the size of such a BSP. In all cases, the constant of proportionality in the big-oh terms is linear in log ff, where ff is the maximum aspect ratio of the fat rectangles. Our algorithms to construct these BSPs run in time proportional to the size of the BSPs they build, except in the case of non-orthogonal objects, when the running time exceeds the size by a factor of p. Experiments demonstrate that our algorithms work well in practice and construct BSPs of near-linear size when most of the rectangles are fat, and perform better than Paterson and Yao's algorithm for orthogonal rectangles [1]. As far as we are aware, ours is the first work to consider BSPs for the practical and common case of two- dimensional, fat polygons in IR 3 . de Berg considers a weaker model, the case of fat polyhedra in IR 3 (a polyhedron is said to be fat if its volume is at least a constant fraction of the volume of the smallest sphere enclosing it), although his results extend to higher dimensions [10]. One of the main ingredients of our algorithm is an O(n log n)-size BSP for a set of n fat rectangles that are "long" with respect to a box B, i.e., none of the vertices of the rectangles lie in the interior of B. To prove this result, we crucially use the fatness of the rectangles. We can use this procedure to construct a BSP of size O(n 4=3 ) for fat rectangles. The algorithm repeatedly applies cuts that bisect the set of vertices of rectangles in the input set S until all sub-problems have long rectangles and the total size of the sub-problems is O(n 4=3 ), at which point we can invoke the algorithm for long rectangles. We improve the size of the BSP to log n ) by simultaneously simulating the algorithm for long rectangles and partitioning the vertices of rectangles in S in a clever manner. The rest of the paper is organized as follows: Section 2 gives some preliminary definitions. In Section 3, we show how to build an O(n log n)-size BSP for n long rectangles. Then we show how to construct a BSP of size O(n 4=3 ) in Section 4. Sections 5 and 6 present and analyze a better algorithm that constructs a BSP of size O( log n ) . We extend this result in Section 7 to construct BSPs in cases when some objects in the input are (i) thin, (ii) non-orthogonal, or (iii) intersecting. We conclude in Section 8 with some open problems. Due to lack of space, we defer many proofs to the full version of the paper. 2. Geometric preliminaries A binary space partition B for a set S of pairwise- disjoint, (d \Gamma 1)-dimensional, polyhedral objects in IR d is a tree recursively defined as follows: Each node v in B represents a convex region R v and a set of objects that intersect R v . The region associated with the root is IR d itself. If S v is empty, then node v is a leaf of B. Otherwise, we partition v's region R v into two convex regions by a cutting the set of objects in S v that lie in H v . If we let H be the positive halfspace and H \Gamma v the negative halfspace bounded by H v , the regions associated with the left and right children of v are R v and R tively. The left subtree of v is a BSP for the set of objects and the right subtree of v is a BSP for the set of objects S g. The size of B is the number of nodes in B. Suppose v is a node of B. In all our algorithms, the region R v associated with v is a box (rectangular paral- lelepiped). We say that a rectangle r is long with respect to R v if none of the vertices of r lie in the interior of R v . Otherwise, r is said to be short. A long rectangle is said to its edges lie on the boundary of R v ; otherwise it is non-free. A free cut is a cutting plane that divides S into two non-empty sets and does not cross any rectangle in S. Note that the plane containing a free rectangle is a free cut. Free cuts will be very useful in preventing excessive fragmentation of the objects in S. We will often focus on a box B and construct a BSP for the rectangles intersecting it. Given a set of rectangles R, let be the set of rectangles obtained by clipping the rectangles in R within B. For a set of points P , let PB be the subset of P lying in the interior of B. Although a BSP is a tree, we will often discuss just how to partition the region represented by a node into two convex regions. We will not explicitly detail the associated construction of the actual tree itself. z-axis y-axis x-axis Right face Top face Front face Figure 1. Different classes of rectangles. 3. BSPs for long fat rectangles Let S be a set of fat rectangles. Assume that all the rectangles in S are long with respect to a box B. In this section, we show how to build a BSP for SB , the set of rectangles clipped within B. The box B has six faces-top, bottom, front, back, right, and left. We assume, without loss of gen- erality, that the back, bottom, left corner of B is the origin (i.e., the back face of B lies on the yz-plane). See Figure 1. A rectangle s belongs to the top class if two parallel edges of s are contained in the top and bottom faces of B. We similarly define the front and right classes. A long rectangle belongs to at least one of these three classes; a non-free rectangle belongs to a unique class. See Figure 1 for examples of rectangles belonging to different classes. In general, SB can have all three classes of rectangles. We first exploit the fatness of the rectangles to prove that whenever all three classes are present in SB , a small number of cuts can divide B into boxes each of which has only two classes of rectangles. Then we describe an algorithm that constructs a BSP when all the rectangles belong to only two classes. We first state two preliminary lemmas that we will use below and in Section 5. The first lemma characterizes a set of rectangles that are long with respect to a box and belong to one class. The second lemma applies to two classes of long rectangles. The parameter a is real and non-negative. C be a box, P a set of points in the interior of C, and R a set of rectangles long with respect to C . If the rectangles in RC belong to one class, (i) there exists a face g of the box C that contains one of the edges of each rectangle in RC . be the set of vertices of the rectangles in RC that lie in the interior of g. In time, we can find a plane h that partitions C into two boxes C 1 and C 2 such that (jV " C 2. C be a box, P a set of points in the interior of C, and R a set of long rectangles with respect to C such that the rectangles in RC belong to two classes. We can find two parallel that partition C into three boxes either (ii) there is some 1 - i - 3 such that jRC i all rectangles in RC i belong to the same class. 3.1. Reducing three classes to two classes Let B and SB be as defined earlier. Assume, without loss of generality, that the longest edge of B is parallel to the x- axis. The rectangles in SB that belong to the front class can be partitioned into two subsets: the set R of rectangles that are vertical (and parallel to the right face of box B) and the set T of rectangles that are horizontal (and parallel to the top face of box B). See Figure 2(a). Let e be the edge of B that lies on the z-axis. The intersection of each rectangle in R with the back face of B is a segment parallel to the z-axis. r denote the projection of this segment onto the z-axis, and let - r. Let z be the endpoints of intervals in - R that lie in the interior of e but not in the interior of any interval of - R. (Note that be less than 2jRj, as in Figure 2(b), if some of the projected segments overlap.) Similarly, for each rectangle t in the set T , we define ~ t to be the projection of t on the y-axis, and ~ ~ t. Let y be the y-coordinates of the vertices of intervals in ~ T defined in the same way as We divide B into kl boxes by drawing the planes and the planes Figure 2(b). This decomposition of B into kl boxes can easily be constructed in a tree-like fashion by performing cuts. We refer to these cuts as ff-cuts. If any resulting box has a free rectangle (such as t in Figure 2(b)), we divide that box into two boxes by applying the C be the set of boxes into which B is partitioned in this manner. We can prove the following theorem about the decomposition of B into C. This is the only place in the whole algorithm where we use the fatness of the rectangles in S. Lemma 3 The set of boxes C formed by the above process satisfies the following properties: y-axis z-axis c a x-axis s r (a) z 3 z 0 a (b) Figure 2. (a) Rectangles belonging to the sets R and T . (b) The back face of B; dashed lines are intersections of the back face with the ff-cuts. (i) Each box C in C has only two classes of rectangles, (ii) There are at most 26bffc 2 n boxes in C, and be the endpoints of e, the edge of the that lies on the z-axis. Similarly, define y 0 and y l to be the endpoints of the edge of B that lies on the y-axis. C be a box in C. If C does not contain a rectangle from T [ R, the proof is trivial since the rectangles in T and R together constitute the front class. Suppose C contains rectangles from the set R. Rectangles in R belong to the front class and are parallel to the right face of B. We claim that C cannot have any rectangles from the right class. To see this claim, consider an edge of C parallel to the z-axis. The endpoints of this edge have z-coordinates z i and z i+1 , for some contains a rectangle from R, the interval z i z i+1 must be covered by projections of rectangles from R onto the z-axis. Any rectangle from the right class inside C must intersect one of the rectangles whose projections cover z i z i+1 . That cannot happen since the rectangles in S do not intersect each other. A similar proof shows that if C contains rectangles from T , then C is free of rectangles in the top class. We first show that that both k and l are at most 3. Let a (respectively, b; c) denote the length of the edges of B parallel to the z-axis (respectively, y- axis, x-axis). By assumption, a; b - c. Let r be a rectangle from R with dimensions z and x, where z - x. Consider - r, the projection of r onto the z-axis. Suppose that - r lies in the interior of the edge e of B lying on the z-axis. Since r is a rectangle in the front class and is parallel to the right face of B, we know that z - a - the rectangle supporting r in the set S ; has dimensions - z and - x, where z and x - x. (If i is 0 or k \Gamma 1, we cannot claim that z; in these cases, it is possible that z is much less than - z.) See Figure 3. We see that a It follows that the length of - r, and hence the length of z i z i+1 , is at least a=ff. Since every alternate interval for at least one rectangle s in R, k is at most 2bffc + 3. In a similar manner, l is also at most 2bffc + 3. This implies that B is divided into at most kl - (2bffc boxes by the planes and the planes Each such box C can contain at most n rectangles. Hence, at most n free cuts can be made inside C. The free cuts can divide C into at most n boxes. This implies that the set C has at most at most 26bffc 2 n boxes. Each rectangle r in SB is cut into at most kl pieces. The edges of these pieces form an arrangement on r. Each face of the arrangement is one of the at most kl rectangles that r is partitioned into. Only 2(k of the arrangement have an edge on the boundary of r. All other faces can be used as free cuts. Hence, after all possible free cuts are used in the boxes into which B is divided by the kl cuts, only 2(k pieces of each rectangle in SB survive. This proves that for two classes of long rectangles Let C be one of the boxes into which B is partitioned in Section 3.1. We now present an algorithm for constructing r a r z x Figure 3. Projections of - r (the dashed rectan- shaded rectangle), and the right face of B onto the zx-plane. a BSP for the set of clipped rectangles SC , which has only two classes of long rectangles. We recursively apply the following steps to each of the boxes produced by the algorithm until no box contains a rectangle. 1. If SC has a free rectangle, we use the free cut containing that rectangle to split C into two boxes. 2. If SC has two classes of rectangles, we use Lemma 2 (with to split C into at most three boxes using two parallel free cuts. 3. If SC has only one class of rectangles, we split C into two by a plane as suggested by Lemma 1 (with and We first analyze the algorithm for two classes of long rectangles. The BSP produced has the following struc- ture: If Step 3 is executed at a node v, then Step 2 is not invoked at any descendant of v. In view of Lemma 2, repeated execution of Steps 1 or 2 on SC constructs in O(jSC j log jS C of the BSP with O(jSC nodes such that each leaf in TC has only one class of rectangles and the total number of rectangles in all the leaves is at most jS C j. At each leaf v of the tree recursive invocations of Steps 1 and 3 build a BSP of size O(jS v j log jS v j) in O(jS v j log jS v for details). Since where the sum is taken over all leaves v of TC , the total size of the BSP constructed inside C is O(jSC j log jS C j). We now analyze the overall algorithm for long rect- angles. The algorithm first applies the ff-cuts to the rectangles in SB , as described in Section 3.1. Consider the set of boxes C produced by the ff-cuts. Each of the boxes in C contains only two classes of rectangles (by Lemma 3(i)). In view of the above discus- sion, for each box C 2 C, we can construct a BSP for SC of size O(jSC j log jS C Lemma 3(ii) and 3(iii) imply that the total size of the BSP n). The BSP can be built in the same time. We can now state the following theorem. Theorem 1 Let S be a set of n fat rectangles and B a box so that all rectangles in S are long with respect to B. Then an O(n log n)-size BSP for the clipped rectangles SB can be constructed in O(n log n) time. The constants of proportionality in the big-oh terms are linear in ff 2 , where ff is the maximum aspect ratio of the rectangles in S : Remark: In our algorithm for two classes of long rectan- gles, by using in Step 3 above the algorithm of Paterson and Yao for constructing linear-size BSPs for orthogonal segments in the plane [21], rather than their O(n log n) algorithm for arbitrarily-oriented segments in the plane [20], we can improve the size of the BSP to linear. This improvement implies that we can construct linear-size BSPs for long rect- angles. We will not need this improved result below, except in Section 4. 4. BSPs of size O(n 4=3 ) In this section, we present a simple algorithm that constructs a BSP of size O(n 4=3 ) for n fat rectangles. We then use the intuition gained from the O(n 4=3 ) algorithm to develop an improved BSP algorithm in Section 5. We analyze the improved algorithm in Section 6. We need a definition before describing the algorithm. A bisecting cut is an orthogonal cut that divides B into two boxes and bisects the set of vertices of rectangles in S that lie in the interior of B. The algorithm for fat rectangles proceeds in phases. A phase is a sequence of three bisecting cuts, with exactly one cut perpendicular to each of the three orthogonal directions. After each phase, if a box contains a free rectangle, we use the corresponding free cut to further divide the box into two. We begin the first phase with a box enclosing all the rectangles with at most 4n vertices in its interior (since there are n rectangles in S each with four vertices) and continue executing phases of bisecting cuts until each node has no vertex in its interior. At termination, each node contains only long rectangles. We then invoke the algorithm for long rectangles to construct a BSP in each of these nodes. The crux of the analysis of the size of the BSP produced by this algorithm is counting how many pieces one rectangle can split into when subjected to a specified number of phases. To this effect, we use the following result due to Paterson and Yao [21]. Lemma 4 (Paterson-Yao) A rectangle that has been subjected to d phases of cuts (with free cuts used whenever possible) is divided into O(2 d ) rectangles. Theorem 2 A BSP in IR 3 of size O(n 4=3 ) can be constructed for n fat orthogonal rectangles. The constant of proportionality in the big-oh term is linear in ff 2 , where ff is the maximum aspect ratio of the input rectangles. vertices in its interior, one phase of cuts partitions B into boxes each of which has at most k=8 vertices in its interior. Since we start with n rectangles that have at most 4n vertices, the number of phases executed by the above algorithm is at most d(log n)=3 now implies that the total number of rectangles formed once all the phases are executed is O(n \Theta 2 d(log n)=3+2=3e stage, all nodes have only long rectangles. Hence, Theorem 1 and the remark at the end of Section 3 imply that constructing a BSP in each of these nodes increases the total size of the BSP only by a constant factor. This proves the theorem. 2 5. The improved algorithm The algorithm proceeds in rounds. Each round simulates a few steps of the algorithm for long rectangles as well as partitions the vertices of the rectangles in S into a small number of sets of approximately equal size. At the beginning of the ith round, where i ? 0, the algorithm has a top of the BSP for be the set of boxes associated with the leaves of B i containing at least one rectan- gle. The initial tree B 1 consists of one node and Q 1 consists of one box that contains all the input rectangles. Our algorithm maintains the invariant that for each box long rectangles in SB are non-free. If Q i is empty, we are done. Otherwise, in the ith round, for each box we construct a top subtree TB of the BSP for the set SB and attach it to the corresponding leaf of B i . This gives us the new top subtree B i+1 . Thus, it suffices to describe how to build the tree TB on a box B during a round. Let F ' SB be the set of rectangles in SB that are long with respect to B. Set k to be the number of vertices of rectangles in SB that lie in the interior of B (note that each such vertex is a vertex of an original rectangle in the input set S). By assumption, all rectangles in F are non-free. We choose a parameter a, which remains fixed throughout the round. We pick log(f+k) to optimize the size of the BSP that the algorithm creates (see Section 6). We now describe the ith round in detail. rectangles in SB are long, we apply the algorithm described in Section 3 to construct a BSP for SB . Otherwise, we perform a sequence of cuts in two stages that partition B as follows: Separating Stage: We apply the ff-cuts, as described in Section 3.1. We make these cuts with respect to the rectangles in F, i.e., we consider only those rectangles of SB that are long with respect to B. In each box so formed, if there is a free rectangle, we apply the free cut along that rectangle. Let C be the resulting set of boxes. Dividing Stage: We refine each box C in C by applying cuts, similar to the ones made in Section 3.2, as described below. We recursively invoke the dividing stage on each box that C is partitioned into. Let kC denote the number of vertices of rectangles in SC that lie in the interior of C. The set FC is the set of rectangles in F that are clipped within C . 1. If C has any free rectangle, we use the free cut containing that rectangle to split C into two boxes. 2. If jF a, we do nothing. 3. If the rectangles in FC belong to two classes, let PC denote the set of vertices of the rectangles in SC that lie in the interior of C . We apply two with 4. If the rectangles in FC belong to just one class, we apply one cut h using Lemma 1, with and The cuts introduced during the dividing stage can be made in a tree-like fashion. At the end of the dividing stage, we have a set of boxes so that for each box D in this set, SD does not contain any free rectangle and jF a. Notice that as we apply cuts in C and in the resulting boxes, rectangles that are short with respect to C may become long with respect to the new boxes. We ignore these new long rectangles until the next round, except when they induce a free cut. 6. Analysis of the improved algorithm We now analyze the size of the BSP constructed by the algorithm and the time complexity of the algorithm. In a round, the algorithm constructs a top subtree TB on a box B for the set of clipped rectangles SB . Recall that F is the set of rectangles long with respect to B. For a node C in let TC be the subtree of TB rooted at C , OE C the number of long rectangles in FC , and nC the number of long rectangles in SC n FC . For a box D corresponding to a leaf of TB , let f D be the number of long rectangles in SD . Note that f D counts both the "old" long rectangles in FD (pieces of rectangles that were long with respect to B) and the "new" long rectangles in SD nFD (pieces of rectangles that were short with respect to B, but became long with respect to D due to the cuts made during the round) ; f Lemma 5 For a box D associated with a leaf of TB , we have a: We know that nD is at most k (since a rectangle in SD n FD must be a piece of a rectangle short with respect to B, and there are at most k such short rectangles). Hence, Since OE D +akD - (f +ak)=2 a; the lemma follows. 2 For a box C in TB , we use the notation LC to denote the set of leaves in TC . Lemma 6 Let C be a box associated with a node in TB . If all rectangles in FC belong to one class, then a. Lemma 7 Let C be a box associated with a node in TB . If all rectangles in FC belong to two classes, then a. Lemma 8 The tree TB constructed on box B in a round has the following properties: : The bound on since each vertex in the interior of SB lies in the interior of at most one box of LB . Next, we will use Lemmas 6 and 7 to prove a bound on D2LB fD . A similar argument will prove the bound on jL B j. Let C be the set of boxes into which B is partitioned by the separating stage. See Figure 4. Let C be a box in C. Since all rectangles in FC belong to at most two classes, Lemmas 6 and 7 imply that Separating Stage Dividing Stage z - Figure 4. The tree TB constructed in a round. a. The boxes in C correspond to the leaves of a top subtree of . Therefore, the total number of long rectangles in the boxes associated with the leaves of TB is which by equation (1) is By an argument similar to the one used to prove Lemma 3, we have also know that Therefore, we obtain O :We now bound the size of the BSP constructed by the algorithm. Let S(f; denote the maximum size of the BSP produced by the algorithm for a box that contains f long rectangles and k vertices in its interior. If Theorem 1 implies that S(f; log f). For the case k ? 0, by Lemma 8(iii), we construct the subtree on B of size O(f log a + a 3=2 k) in one round, and recursively construct subtrees for each box in the set of leaves LB . Hence, when k ? 0, where and f D a for every box D in LB . The solution to this recurrence is where the constant of proportionality in the big-oh term is linear in log ff. The intuition behind this solution is that each round increases the number of "old" long rectangles by at most a constant factor, while also creating O(a 3=2 "new" long rectangles. The depth of each round is O(log a). Choosing log(f+k) balances the total increase in the number of "old" rectangles (over all the rounds) and the total increase in the number of "new" rectangles. Since all operations at a node can be performed in time linear in the number of rectangles at that node, the same bound can be obtained for the running time of the algorithm. 4n at the beginning of the first round, we obtain the following theorem. Theorem 3 Given a set S of n rectangles in IR 3 such that the aspect ratio of each rectangle in S is bounded by a constant ff - 1, we can construct a BSP of size O( log for S in time log n ) . The constants of proportionality in the big-oh terms are linear in log ff. 7. Extensions In this section, we show how to modify the algorithm of Section 5 to handle the following three cases: (i) some of the rectangles are thin, (ii) some of the rectangles are non- orthogonal, and (iii) the input consists of intersecting fat rectangles. 7.1. Fat and thin rectangles Let us assume that the input rectangles, consisting of m thin rectangles in T and rectangles in F. Given a box B, let f be the number of long rectangles in FB , let k be the number of vertices of rectangles in FB that lie in the interior of B, and let t be the number of rectangles in TB . The algorithm we use now is very similar to the algorithm for fat rectangles. We fix the parameter log(f+k) . 1. If SB contains a free rectangle, we use the corresponding free cut to split B into two boxes. 2. If we use the algorithm for long rectangles to construct a BSP for B. 3. If t - (f + k), we use the algorithm by Paterson and Yao for orthogonal rectangles in IR 3 to construct a BSP 4. If (f t, we perform one round of the algorithm described in Section 5. This algorithm is recursively invoked on all boxes that B is split into. Let S(k; f; t) be the size of the BSP produced by this algorithm for a box with k vertices in its interior, f long rectangles in FB , and t thin rectangles in . Analyzing the algorithm's behavior as in Section 5, we can show that S(k; f; (see [21] for details), and when (f O(f log a The solution to this recurrence is where the constant of proportionality in the first big-oh term is linear in log ff. The following theorem is immediate. Theorem 4 A BSP of size n m2 O( log n ) can be constructed in n m2 O( log n ) time for n rectangles in IR 3 , of which m are thin. The constants of proportionality in the big-oh terms are linear in log ff, where ff is the maximum aspect ratio of the fat rectangles. There exists a set of m thin rectangles and rectangles in IR 3 for which any BSP has m). 7.2. Fat rectangles and non-orthogonal rectangles Suppose p objects in the input are non-orthogonal and the rest are fat rectangles. The algorithm we use is very similar to the algorithm in Section 7.1, except in two places. In Step 1, we check whether we can make free cuts through the non-orthogonal objects too. In Step 3, if the number of non-orthogonal object at a node dominates the number of fat rectangles, we use Paterson and Yao's algorithm for triangles in IR 3 to construct a BSP of size quadratic in the number of objects in cubic time [20]. Theorem 5 A BSP of size np2 O( log n ) can be constructed in np 2 2 O( log n ) time for n objects in IR 3 , of which p are non-orthogonal and the rest are fat rectangles. The constants of proportionality in the big-oh terms are linear in log ff, where ff is the maximum aspect ratio of the fat rectangles. 7.3. Intersecting fat rectangles We now consider the case when the n fat rectangles contain intersecting pairs. For each intersecting pair of rectangles, we break one of the rectangles in the pair into a constant number of smaller pieces such that the pieces do not intersect the other rectangle in the pair. This process creates a total of n rectangles. Some or all of the "new" O(k) rectangles may be thin. We then use the algorithm of Section 7.1 to construct a BSP for the rectangles. The theorem below follows. Theorem 6 A BSP of size (n + k2 O( log n ) can be constructed in (n k2 O( log n ) time for n rectangles in IR 3 , which have k intersecting pairs of rectangles. The constants of proportionality in the big-oh terms are linear in log ff, where ff is the maximum aspect ratio of the fat rectangles. There exists a set of n rectangles in IR 3 , containing k intersecting pairs, for which any BSP has k). 8. Conclusions Since worst-case complexities for BSPs are very high triangles in IR 3 orthogonal rectangles in IR 3 ) and all known examples that achieve the worst case use mainly skinny objects, we have made the natural assumption that objects are fat and have shown that this assumption allows smaller worst-case size of BSPs. We have implemented these algorithms. The practical results are very encouraging and are presented in a companion paper [1]. It seems very probable that BSPs of size smaller than log n ) can be built for n orthogonal rectangles of bounded aspect-ratio in IR 3 . The only lower bound we have is the n) bound. It would be interesting to see if algorithms can be developed to construct BSPs of optimal size. Similar improvements can be envisioned for Theorems 4, 5 and 6. An even more challenging open problem is determining the right assumptions that should be made about the input objects and the graphics display hardware so that provably fast and practically efficient algorithms can be developed for doing hidden-surface elimination of these objects. A preliminary investigation into an improved model for graphics hardware has been made by Grove et al. [15]. Acknowledgments We would like to thank Seth Teller for providing us with the Soda Hall dataset created at the Department of Computer Science, University of California at Berkeley. We would also like to thank the Walk-through Project, Department of Computer Science, University of North Carolina at Chapel Hill for providing us with the datasets for Sitterson Hall, the Orange United Methodist Church Fellowship Hall, and the Sitterson Hall Lobby. --R Practical methods for constructing binary space partitions for orthogonal objects. Surface approximation and geometric partitions. Increasing Update Rates in the Building Walk-through System with Automatic Model-space Subdivision and Potentially Visible Set Calculations Motion planning using binary space partitions. Modeling Global Diffuse Illumination for Image Synthesis. A Subdivision Algorithm for Computer Display of Curved Surfaces. Near real-time object-precision shadow generation using BSP trees-master thesis Near real-time shadow generation using bsp trees Fast object-precision shadow generation for areal light sources using BSP trees Linear size binary space partitions for fat ob- jects A survey of object-space hidden surface re- moval Computer Graphics: Principles and Practice. On visible surface generation by a priori tree structures. The object complexity model for hidden-surface elimination On maximum flows in polyhedral domains. SCULPT an interactive solid modeling tool. Merging BSP trees yields polyhedral set operations. Application of BSP trees to ray-tracing and CSG evaluation Efficient binary space partitions for hidden-surface removal and solid modeling Optimal binary space partitions for orthogonal objects. A characterization of ten hidden surface algorithms. Visibility Computations in Densely Occluded Polyhedral Environments. Set operations on polyhedra using binary space partitioning trees. --TR --CTR John Hershberger , Subhash Suri , Csaba D. Toth, Binary space partitions of orthogonal subdivisions, Proceedings of the twentieth annual symposium on Computational geometry, June 08-11, 2004, Brooklyn, New York, USA Csaba D. Tth, Binary space partitions for line segments with a limited number of directions, Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms, p.465-471, January 06-08, 2002, San Francisco, California Csaba D. T'oth, A note on binary plane partitions, Proceedings of the seventeenth annual symposium on Computational geometry, p.151-156, June 2001, Medford, Massachusetts, United States Mark de Berg , Micha Streppel, Approximate range searching using binary space partitions, Computational Geometry: Theory and Applications, v.33 n.3, p.139-151, February 2006 B. Aronov , A. Efrat , V. Koltun , Micha Sharir, On the union of -round objects, Proceedings of the twentieth annual symposium on Computational geometry, June 08-11, 2004, Brooklyn, New York, USA

solid modelling;aspect ratio;computer graphics;computational geometry;binary space partitions;rectangles

352757

An Optimal Algorithm for Monte Carlo Estimation.

A typical approach to estimate an unknown quantity $\mu$ is to design an experiment that produces a random variable Z, distributed in [0,1] with E[Z]=\mu$, run this experiment independently a number of times, and use the average of the outcomes as the estimate. In this paper, we consider the case when no a priori information about Z is known except that is distributed in [0,1]. We describe an approximation algorithm ${\cal A}{\cal A}$ which, given $\epsilon$ and $\delta$, when running independent experiments with respect to any Z, produces an estimate that is within a factor $1+\epsilon$ of $\mu$ with probability at least $1-\delta$. We prove that the expected number of experiments run by ${\cal A}{\cal A}$ (which depends on Z) is optimal to within a constant factor {for every} Z.

Introduction The choice of experiment, or experimental design, forms an important aspect of statistics. One of the simplest design problems is the problem of deciding when to stop sampling. For example, suppose Z are independently and identically distributed according to Z in the interval [0; 1] with mean -Z . From Bernstein's inequality, we know that if N is fixed proportional to ln(1=ffi)=ffl 2 and with probability at least approximates -Z with absolute error ffl. Often, however, -Z is small and a good absolute error estimate of -Z is typically a poor relative error approximation of -Z . We say ~ -Z is an (ffl; ffi)-approximation of -Z if In engineering and computer science applications we often desire an (ffl; ffi)-approximation of -Z in problems where exact computation of -Z is NP-hard. For example, many researchers have devoted substantial effort to the important and difficult problem of approximating the permanent of valued matrices [1, 4, 5, 9, 10, 13, 14]. Researchers have also used (ffl; ffi)-approximations to tackle many other difficult problems, such as approximating probabilistic inference in Bayesian networks [6], approximating the volume of convex bodies [7], solving the Ising model of statistical mechanics [11], solving for network reliability in planar multiterminal networks [15, 16], approximating the number of solutions to a DNF formula [17] or, more generally, to a GF[2] formula [18], and approximating the number of Eulerian orientations of a graph [19]. and let oe 2 Z denote the variance of Z. Define We first prove a slight generalization of the Zero-One Estimator Theorem [12, 15, 16, 17]. The new theorem, the Generalized Zero-One Estimator Theorem, proves that if Z (1) then S=N is an (ffl; ffi)-approximation of -Z . To apply the Generalized Zero-One Estimator Theorem we require the values of the unknown quantities -Z and oe 2 Z . Researchers circumvent this problem by computing an upper bound - on ae Z =- 2 Z , and using - in place of ae Z =- 2 Z to determine a value for N in Equation (1). An a priori upper bound - on ae Z =- 2 Z that is close to ae Z =- 2 Z is often very difficult to obtain, and a poor bound leads to a prohibitive large bound on N . To avoid the problem encountered with the Generalized Zero-One Estimator Theo- rem, we use the outcomes of previous experiments to decide when to stop iterating. This approach is known as sequential analysis and originated with the work of Wald on statistical decision theory [22]. Related research has applied sequential analysis to specific Monte Carlo approximation problems such as estimating the number of points in a union of sets [17] and estimating the number of self-avoiding walks [20]. In other related work, Dyer et al describe a stopping rule based algorithm that provides an upper bound estimate on -Z [8]. With probability the estimate is at most (1 the estimate can be arbitrarily smaller than - in the challenging case when - is small. We first describe an approximation algorithm based on a simple stopping rule. Using the stopping rule, the approximation algorithm outputs an (ffl; ffi)-approximation of -Z after expected number of experiments proportional to \Upsilon=- Z . The variance of the random variable Z is maximized subject to a fixed mean -Z if Z takes on value 1 with probability -Z and 0 with probability . In this case, oe 2 and the expected number of experiments run by the stopping-rule based algorithm is within a constant factor of optimal. In general, however, oe Z is significantly smaller than -Z , and for small values of oe 2 Z the stopping-rule based algorithm performs 1=ffl times as many experiments as the optimal number. We describe a more powerful algorithm, the AA algorithm, that on inputs ffl, ffi, and independently and identically distributed outcomes Z generated from any random variable Z distributed in [0; 1], outputs an (ffl; ffi)-approximation of -Z after an expected number of experiments proportional to \Upsilon \Delta ae Z =- 2 Z . Unlike the simple, stopping-rule based algorithm, we prove that for all Z, AA runs the optimal number of experiments to within a constant factor. Specifically, we prove that if BB is any algorithm that produces an (ffl; ffi)-approximation of -Z using the inputs ffl, ffi, and Z 1 runs an expected number of experiments proportional to at least \Upsilon \Delta ae Z =- 2 Z . (Canetti, Evan and Goldreich prove the related lower bound \Omega\Gammand (1=ffi)=ffl 2 ) on the number of experiments required to approximate -Z with absolute error ffl with probability at least 1 \Gamma ffi [2].) Thus we show that for any random variable Z, AA runs an expected number of experiments that is within a constant factor of the minimum expected number. The AA algorithm is a general method for optimally using the outcomes of Monte-Carlo experiments for approximation-that is, to within a constant factor, the algorithm uses the minimum possible number of experiments to output an (ffl; ffi)-approximation on each problem instance. Thus, AA provides substantial computational savings in applications that employ a poor upper bound - on ae Z =- 2 Z . For example, the best known a priori bound on - for the problem of approximating the permanent of size n is superpolynomial in n [13]. Yet, for many problem instances of size n, the number of experiments run by AA is significantly smaller than this bound. Other examples exist where the bounds are also extremely loose for many typical problem instances [7, 10, 11]. In all those applica- tions, we expect AA to provide substantial computational savings, and possibly render problems that were intractable, because of the poor upper bounds on ae Z =- 2 Z , amenable to efficient approximation. Approximation Algorithm In Subsection 2.1, we describe a stopping rule algorithm for estimating -Z . This algorithm is used in the first step of the approximation algorithm AA that we describe in Subsection 2.2. 2.1 Stopping Rule Algorithm Let Z be a random variable distributed in the interval [0; 1] with mean -Z . Let Z be independently and identically distributed according to Z. Stopping Rule Algorithm Input Parameters: (ffl; ffi) with Initialize N / 0, S / 0 While Output: ~ Stopping Rule Theorem: Let Z be a random variable distributed in [0; 1] with -Z be the estimate produced and let NZ be the number of experiments that the Stopping Rule Algorithm runs with respect to Z on input ffl and ffi. Then, The proof of this theorem can be found in Section 5. 2.2 Approximation Algorithm AA The (ffl; ffi)-approximation algorithm AA consists of three main steps. The first step uses the stopping rule based algorithm to produce an estimate -Z that is within a constant factor of -Z with probability at least 1 \Gamma ffi. The second step uses the value of - -Z to set the number of experiments to run in order to produce an estimate - ae Z that is within a constant factor of ae with probability at least 1 \Gamma ffi. The third step uses the values of -Z and - ae Z produced in the first two steps to set the number of experiments and runs this number of experiments to produce an (ffl; ffi)-estimate of ~ -Z of -Z . Let Z be a random variable distributed in the interval [0; 1] with mean -Z and variance Z . Let Z two sets of random variables independently and identically distributed according to Z. Approximation Algorithm AA Input Parameters: (ffl; ffi), with 1: Run the Stopping Rule Algorithm using Z parameters fflg and ffi=3. This produces an estimate -Z of -Z . Step 2: Set -Z and initialize S / 0. For ae Z / maxfS=N; ffl -Z g. Step 3: Set ae Z =- 2 Z and initialize S / 0. For ~ -Z / S=N . Output: ~ -Z AA Theorem: Let Z be any random variable distributed in [0; 1], let be the mean of Z, oe 2 Z be the variance of Z, and ae g. Let ~ -Z be the approximation produced by AA and let NZ be the number of experiments run by AA with respect to Z on input parameters ffl and ffi. Then, (2) There is a universal constant c 0 such that Pr[NZ - c 0 \Upsilon (3) There is a universal constant c 0 such that E[NZ Z . We prove the AA Theorem in Section 6. 3 Lower Bound Algorithm AA is able to produce a good estimate of -Z using no a priori information about Z. An interesting question is what is the inherent number of experiments needed to be able to produce an (ffl; ffi)-approximation of -Z . In this section, we state a lower bound on the number of experiments needed by any (ffl; ffi)-approximation algorithm to estimate -Z when there is no a priori information about Z. This lower bound shows that, to within a constant factor, AA runs the minimum number of experiments for every random variable Z. To formalize the lower bound, we introduce the following natural model. Let BB be any algorithm that on input (ffl; ffi) works as follows with respect to Z. Let Z independently and identically distributed according to Z with values in the interval [0; 1]. BB runs an experiment, and on the N th run BB receives the value ZN . The measure of the running time of BB is the number of experments it runs, i.e., the time for all other computations performed by BB is not counted in its running time. BB is allowed to use any criteria it wants to decide when to stop running experiments and produce an estimate, and in particular BB can use the outcome of all previous experiments. The estimate that BB produces when it stops can be any function of the outcomes of the experiments it has run up to that point. The requirement on BB is that is produces an (ffl; ffi)-approximation of -Z for any Z. This model captures the situation where the algorithm can only gather information about -Z through running random experiments, and where the algorithm has no a priori knowledge about the value of -Z before starting. This is a reasonable pair of assumptions for practical situations. It turns out that the assumption about a priori knowledge can be substantially relaxed: the algorithm may know a priori that the outcomes are being generated according to some known random variable Z or to some closely related random variable Z 0 , and still the lower bound on the number of experiments applies. Note that the approximation algorithm AA fits into this model, and thus the average number of experiments it runs with respect to Z is minimal for all Z to within a constant factor among all such approximation algorithms. Lower Bound Theorem: Let BB be any algorithm that works as described above on input (ffl; ffi). Let Z be a random variable distributed in [0; 1], let -Z be the mean of Z, oe 2 Z be the variance of Z, and ae g. Let ~ -Z be the approximation produced by BB and let NZ be the number of experiments run by BB with respect to Z. Suppose that BB has the the following properties: (1) For all Z with -Z ? 0, E[NZ (2) For all Z with -Z ? 0, Then, there is a universal constant c ? 0 such that for all Z, E[NZ Z . We prove this theorem in Section 7. 4 Preliminaries for the Proofs We begin with some notation that is used hereafter. Let - For fixed ff; fi - 0, we define the random variables The main lemma we use to prove the first part of the Stopping Rule Theorem provides bounds on the probabilities that the random variables k are greater than zero. We first form the sequences of random variables e di real valued d. We prove that these sequences are supermartingales when 0 - d - 1 and Z , i.e., for all k ? 0 and similarly, We then use properties of supermartingales to bound the probabilities that the random k are greater than zero. For these and subsequent proofs, we use the following two inequalities: Inequality 4.1 For all ff, e ff Inequality 4.2 Let :72. For all ff with jffj - 1, Lemma 4.3 For jdj - 1, E[e dZ Z . Proof: Observe that E[e dZ But from Inequality 4.2, Taking expectations and applying Inequality 4.1 completes the proof. 2 Lemma 4.4 For 0 - d - 1, and for fi -doe 2 Z , the sequences of random variables supermartingales. Proof: For k - 1, and thus, Similarly, for k - 1 and thus from Lemma 4.3, and Thus, for fi -doe 2 Z , and directly from the properties of conditional expectations and of martingales. Lemma 4.5 If j is a supermartingale then for all This lemma is the key to the proof of the first part of the Stopping Rule Theorem. In addition, from this lemma we easily prove a slightly more general version of the Zero-One Estimator Theorem. Lemma 4.6 For any fixed N ? 0, for any fi - 2-ae Z , \GammaN and \GammaN Proof: Recall the definitions of i N from Equations (3) and (4). Let Then, the left-hand side of Equation (5) is equivalent to Pr[i and the left-hand side of Equation (6) is equivalent to Pr[i \Gamma N and i 0\Gamma N . We now give the remainder of the proof of Equation (5), using i 0+ N , and omit the remainder of the analogous proof of Equation (6) which uses i 0\Gamma N in place of i 0+ N . For any implies that d - 1. Note also that since ae Z - oe 2 Z . Thus, by Lemma 4.4, e di 0+ N is a supermartingale. Thus, by Lemma 4.5 E[e di 0+ \GammaN Since e di 0+ 0 is a constant, E[e di 0+ completing the proof of Equation (5). 2 We use Lemma 4.6 to generalize the Zero-One Estimator Theorem [17] from f0; 1g-valued random variables to random variables in the interval [0; 1]. Generalized Zero-One Estimator Theorem: Let Z variables independent and identically distributed according to Z. If ffl ! 1 and then Pr Proof: The proof follows directly from Lemma (4.6), using noting that ffl- Z - 2-ae Z and that N \Delta (ffl- Z 5 Proof of the Stopping Rule Theorem We next prove the Stopping Rule Theorem. The first part of the proof also follows directly from Lemma 4.6. Recall that Stopping Rule Theorem: Let Z be a random variable distributed in [0; 1] with -Z be the estimate produced and let NZ be the number of experiments that the Stopping Rule Algorithm runs with respect to Z on input ffl and ffi. Then, Proof of Part (1): Recall that ~ It suffices to show that We first show that Pr[NZ ! \Upsilon 1 =(- Z (1+ ffl))] - ffi=2. Let Assuming that -Z (1 the definition of \Upsilon 1 and L implies that Since NZ is an integer, NZ only if NZ - L. But NZ - L if and only if SL - \Upsilon 1 . Thus, Noting that ffl- Z - fi - 2-ae Z , Lemma (4.6) implies that Using inequality (7) and noting that ae Z -Z , it follows that this is at most ffi=2. The proof that Pr[NZ ? \Upsilon 1 =(- Z Proof of Part (2): The random variable NZ is the stopping time such that Using Wald's Equation [22] and E[NZ and thus, :Similar to the proof of the first part of the Stopping Rule Theorem, we can show that and therefore with probability at least 1 \Gamma ffi=2 we require at most (1 experiments to generate an approximation. The following lemma is used in the proof of the AA Theorem in Section 6. Stopping Rule Lemma: Proof of the Stopping Rule Lemma: E[1=~- Z directly from Part (2) of the Stopping Rule Theorem and the definition of NZ . E[1=~- 2 can be easily proved based on the ideas used in the proof of Part (2) of the Stopping Rule Theorem. 2 6 Proof of the AA Theorem AA Theorem: Let Z be any random variable distributed in [0; 1], let -Z be the mean of Z, oe 2 Z be the variance of Z, and ae g. Let ~ -Z be the approximation produced by AA and let NZ be the number of experiments run by AA with respect to Z on input parameters ffl and ffi. Then, (2) There is a universal constant c 0 such that Pr[NZ - c 0 \Upsilon (3) There is a universal constant c 0 such that E[NZ Z . Proof of Part (1): From the Stopping Rule Theorem, after Step (1) of AA, -Z ffl) holds with probability at least 1 \Gamma ffi=3. Let We show next that if -Z ffl), then in Step (2) the choice of \Upsilon 2 guarantees that - ae Z - ae Z =2. Thus, after Steps (1) and (2), \Phi - ae Z =- 2 Z with probability at least 1 \Gamma ffi=3. But by the Generalized Zero-One Estimator Theorem, for ae Z =- 2 ae Z =- 2 Z , Step (3) guarantees that the output ~ -Z of AA satisfies Pr[-Z For all i, let - and observe that, Z . First assume that Z . If oe 2 ffl)ffl- Z then from the Generalized Zero-One Estimator Theorem, after at most (2=(1 \Gamma experiments, ae Z =2 - S=N - 3ae Z =2 with probability at least 1 \Gamma 2ffi=3. Thus - ae Z - ae Z =2. If ffl- Z - oe 2 ffl)ffl- Z then ffl- Z - oe 2 ffl)), and therefore, - ae Z - ffl -Z - ae Z =2. Next, assume that oe 2 . But Steps (1) and (2) guarantee that - ae Z - ffl - ffl), with probability at least 1 \Gamma ffi=3. Proof of Part (2): AA may fail to terminate after O(\Upsilon \Delta ae Z =- 2 experiments either because Step (1) failed with probability at least ffi=2 to produce an estimate - -Z such that -Z (1 \Gamma ffl), or, because in Step (2), for oe 2 ffl)ffl- Z , and, S=N is not O(ffl- Z ) with probability at least 1 \Gamma ffi=2. But Equation 8 guarantees that Step (1) of AA terminates after O(\Upsilon \Delta ae Z =- 2 with probability at least 1 \Gamma ffi=2. In addition, we can show, similarly to Lemma 4.6, that if oe 2 Thus, for N - 2\Upsilon \Delta ffl=- Z , we have that Pr[S=N - 4ffl- Z Proof of Part (3): Observe that from the Stopping Rule Theorem, the expected number of experiments in Step (1) is O(ln(1=ffi)=(ffl- Z )). From the Stopping Rule Lemma, the expected number of experiments in Step (2) is O(ln(1=ffi)=(ffl- Z )). Finally, in Step (3) observe that E[-ae Z =- 2 Z ae Z and - -Z are computed from disjoint sets of independently and identically distributed random variables. From the Stopping Rule Lemma Z ] is O(ln(1=ffi)=- 2 Furthermore, observe that E[-ae Z Z and E[ffl- Z Z and Thus, the expected number of experiments in Step (3) is O(ln(1=ffi) 2. 7 Proof of Lower Bound Theorem Lower Bound Theorem: Let BB be any algorithm that works as described above on input (ffl; ffi). Let Z be a random variable distributed in [0; 1], let -Z be the mean of Z, oe 2 Z be the variance of Z, and ae g. Let ~ -Z be the approximation produced by BB and let NZ be the number of experiments run by BB with respect to Z. Suppose that BB has the the following properties: (1) For all Z with -Z ? 0, E[NZ (2) For all Z with -Z ? 0, Pr[-Z Then, there is a universal constant c ? 0 such that for all Z, E[NZ Z . Let f Z (x) and f Z 0 (x) denote two given distinct probability mass (or in the continuous case, the density) functions. Let Z independent and identically distributed random variables with probability density f(x). Let HZ denote the hypothesis and let H Z 0 denote the hypothesis denote the probability that we reject HZ under f Z and let fi denote the probability that we accept H Z 0 under f Z 0 . The sequential probability ratio test minimizes the number of expected sample size both under HZ and H Z 0 among all tests with the same error probabilities ff and fi. Theorem 7.1 states the result of the sequential probability ratio test. We prove the result for completeness, although similar proofs exist [21]. Theorem 7.1 If T is the stopping time of any test of HZ against H Z 0 with error probabilities ff and fi, and EZ [T ]; ff and ff Proof: For the independent and identically distributed random variables Z 1 k . For stopping time T , we get from Wald's first identity and Next, let\Omega denote the space of all inputs on which the test rejects HZ , and denotes its complement. Thus, by definition, we require that Similarly, we require that Pr Z From the properties of expectations, we can show that Z[\Omega c ]; and we can decompose T j\Omega\Gamma and observe that from Inequality 4.1, EZ But I\Omega ]=Pr where I\Omega denotes the characteristic function for the set\Omega\Gamma Thus, since Y we can show that and finally, ff Similarly, we can show that and, Thus, ff and we prove the first part of the lemma. Similarly, proves the second part of the lemma. 2 Corollary 7.2 If T is the stopping time of any test of HZ against H Z 0 with error probabilities ff and fi such that ff and Proof: If ff ff ff achieves a minimum at Substitution of completes the proof. Lemma 7.5 proves the Lower Bound Theorem for oe 2 We begin with some definitions. Let Z . use Inequality 4.2 to show that Taking expectations completes the proof. 2 Lemma 7.4 2doe 2 Z =/. denotes the derivative of / with respect to d, the proof follows directly from Lemma 7.3. 2 Lemma 7.5 If oe 2 Proof: Let T denote the stopping time of any test of HZ against H Z 0 . Note that Z then, by Lemmas 7.3 -). Thus, to test HZ against H 0 Z , we can use the BB with input ffl such that -Z (1 for ffl we obtain ffl - ffl=(2(2 ffl). But Corollary 7.2 gives a lower bound on the expected number of experiments E[N run by BB with respect the Z. We observe that where the inequality follows from Lemma 7.3. We let Z and substitute to complete the proof. 2 We now prove the Lower Bound Theorem that holds also when oe 2 We define the density Lemma 7.6 If -Z - 1=4 then Proof: Observe that E Z 0 Lemma 7.7 \GammaE Z Proof: Observe that Taking expectations completes the proof. 2 Lemma 7.8 If -Z - 1=4 then E[NZ Proof: Let T denote the stopping time of any test of HZ against H Z 0 . From Lemma 7.6, and since -Z - 1, =4. Thus, to test HZ against H 0 Z , we can use the BB with input ffl such that -Z (1+ ffl Solving for ffl we obtain ffl - ffl=(8 ffl). But Corollary 7.2 gives a lower bound on the expected number of experiments E[N run by BB with respect the Z. Next observe that, by Lemma 7.7, \Gamma! Z in Corollary 7.2 is at most 2ffl- Z . Substitution of completes the proof. 2 Proof of Lower Bound Theorem: Follows from Lemma 7.5 and Lemma 7.8. 2 --R "How hard is it to marry at random? (On the approximation of the permanent)" "Lower bounds for sampling algorithms for estimating the average " "An optimal algorithm for Monte-Carlo estimation (extended abstract)" "Polytopes, permanents and graphs with large factors" "Approximating the Permanent of Graphs with Large Factors," "An optimal approximation algorithm for Bayesian inference" "A random polynomial time algorithm for approximating the volume of convex bodies" "A Mildly Exponential Time Algorithm for Approximating the Number of Solutions to a Multi-dimensional Knapsack Problem" "An analysis of a Monte Carlo algorithm for estimating the permanent" "Conductance and the rapid mixing property for Markov Chains: the approximation of the permanent resolved" "Polynomial-time approximation algorithms for the Ising model" "Random generation of combinatorial structures from a uniform distribution" "A mildly exponential approximation algorithm for the permanent" "A Monte-Carlo Algorithm for Estimating the Permanent" "Monte Carlo algorithms for planar multiterminal network reliability problems" "Monte Carlo algorithms for enumeration and reliability problems" "Monte Carlo approximation algorithms for enumeration problems" "Approximating the Number of Solutions to a GF[2] For- mula," "On the number of Eulerian orientations of a graph" "Testable algorithms for self-avoiding walks" Sequential Analysis Sequential Analysis --TR --CTR Michael Luby , Vivek K. Goyal , Simon Skaria , Gavin B. Horn, Wave and equation based rate control using multicast round trip time, ACM SIGCOMM Computer Communication Review, v.32 n.4, October 2002 Ronald Fagin , Amnon Lotem , Moni Naor, Optimal aggregation algorithms for middleware, Journal of Computer and System Sciences, v.66 n.4, p.614-656, 1 June Fast approximate probabilistically checkable proofs, Information and Computation, v.189 n.2, p.135-159, March 15, 2004

monte carlo estimation;sequential estimation;stochastic approximation;stopping rule;approximation algorithm

352762

Separating Complexity Classes Using Autoreducibility.

A set is autoreducible if it can be reduced to itself by a Turing machine that does not ask its own input to the oracle. We use autoreducibility to separate the polynomial-time hierarchy from exponential space by showing that all Turing complete sets for certain levels of the exponential-time hierarchy are autoreducible but there exists some Turing complete set for doubly exponential space that is not.Although we already knew how to separate these classes using diagonalization, our proofs separate classes solely by showing they have different structural properties, thus applying Post's program to complexity theory. We feel such techniques may prove unknown separations in the future. In particular, if we could settle the question as to whether all Turing complete sets for doubly exponential time are autoreducible, we would separate either polynomial time from polynomial space, and nondeterministic logarithmic space from nondeterministic polynomial time, or else the polynomial-time hierarchy from exponential time.We also look at the autoreducibility of complete sets under nonadaptive, bounded query, probabilistic, and nonuniform reductions. We show how settling some of these autoreducibility questions will also lead to new complexity class separations.

Introduction While complexity theorists have made great strides in understanding the structure of complexity classes, they have not yet found the proper tools to do nontrivial separation of complexity classes such as P and NP. They have developed sophisticated diagonalization, combinatorial and algebraic techniques but none of these ideas have yet proven very useful in the separation task. Back in the early days of computability theory, Post [13] wanted to show that the set of noncomputable computably enumerable sets strictly contains the Turing-complete computably enumerable sets. In what we now call iPost's Programj (see [11, 15]), Post tried to show these classes dioeer by -nding a property that holds for all sets in the -rst class but not for some set in the second. We would like to resurrect Post's Program for separating classes in complexity theory. In particular we will show how some classes dioeer by showing that their complete sets have dioeerent structure. While we do not separate any classes not already separable by known diagonalization techniques, we feel that re-nements to our techniques may yield some new separation results. In this paper we will concentrate on the property known as iautoreducibility.j A set A is autoreducible if we can decide whether an input x belongs to A in polynomial-time by making queries about membership of strings dioeerent from x to A. Trakhtenbrot [16] -rst looked at autoreducibility in both the unbounded and space-bounded models. Ladner [10] showed that there exist Turing-complete computably enumerable sets that are not autoreducible. Ambos-Spies [1] -rst transferred the notion of autoreducibility to the polynomial-time setting. More recently, Yao [19] and Beigel and Feigenbaum [5] have studied a probabilistic variant of autoreducibility known as icoherence.j In this paper, we ask for what complexity classes do all the complete sets have the autoreducibil- ity property. In particular we show: ffl All Turing-complete sets for \Delta EXP are autoreducible for any constant k, where \Delta EXP the sets that are exponential-time Turing-reducible to \Sigma P k . ffl There exists a Turing-complete set for doubly exponential space that is not autoreducible. Since the union of all sets \Delta EXP k coincides with the exponential-time hierarchy, we obtain a separation of the exponential-time hierarchy from doubly exponential space and thus of the polynomial-time hierarchy from exponential space. Although these results also follow from the space hierarchy theorems [9] which we have known for a long time, our proof does not directly use diagonalization, rather separates the classes by showing that they have dioeerent structural properties. Issues of relativization do not apply to this work because of oracle access (see [8]): A polynomial-time autoreduction can not view as much of the oracle as an exponential or doubly exponential computation. To illustrate this point we show that there exists an oracle relative to which some complete set for exponential time is not autoreducible. Note that if we can settle whether the Turing-complete sets for doubly exponential time are all autoreducible one way or the other, we will have a major separation result. If there exists a Turing-complete set for doubly exponential time that is not autoreducible, then we get that the exponential-time hierarchy is strictly contained in doubly exponential time thus that the polynomial-time hierarchy is strictly contained in exponential time. If all of the Turing-complete sets for doubly exponential time are autoreducible, we get that doubly exponential time is strictly contained in doubly exponential space, and thus polynomial time strictly in polynomial space. We will also show that this assumption implies a separation of nondeterministic logarithmic space from nondeterministic polynomial time. Similar implications hold for space bounded classes (see Section 5). Autoreducibil- ity questions about doubly exponential time and exponential space thus remain an exciting line of research. We also study the nonadaptive variant of the problem. Our main results scale down one exponential as follows: ffl All truth-table-complete sets \Delta P are truth-table-autoreducible for any constant k, where \Delta P denotes that sets polynomial-time Turing-reducible to \Sigma P k . ffl There exists a truth-table-complete set for exponential space that is not truth-table-autoreducible. Again, -nding out whether all truth-table-complete sets for intermediate classes, namely polynomial space and exponential time, are truth-table-autoreducible, would have major implications. In contrast to the above results we exhibit the limitations of our approach: For the restricted reducibility where we are only allowed to ask two nonadaptive queries, all complete sets for EXP, EXPSPACE, EEXP, EEXPSPACE, etc., are autoreducible. We also argue that uniformity is crucial for our technique of separating complexity classes, because our nonautoreducibility results fail in the nonuniform setting. Razborov and Rudich [14] show that if strong pseudo-random generators exist, inatural proofsj can not separate certain nonuniform complexity classes. Since this paper relies on uniformity in an essential way, their result does not apply. Regarding the probabilistic variant of autoreducibility mentioned above, we can strengthen our results and construct a Turing-complete set for doubly exponential space that is not even probabilistically autoreducible. We leave the analogue of this theorem in the nonadaptive setting open: Does there exist a truth-table complete set for exponential space that is not probabilistically truth-table autoreducible? We do show that every truth-table complete set for exponential time is probabilistically truth-table autoreducible. So, a positive answer to the open question would establish that exponential time is strictly contained in exponential space. A negative answer, on the other hand, would imply a separation of nondeterministic logarithmic space from nondeterministic polynomial time. Here is the outline of the paper: First, we introduce our notation and state some preliminaries in Section 2. Next, in Section 3 we establish our negative autoreducibility results, for the adaptive as well as the nonadaptive case. Then we prove the positive results in Section 4, where we also brieAEy look at the randomized and nonuniform settings. Section 5 discusses the separations that follow from our results and would follow from improvements on them. Finally, we conclude in Section 6 and mention some possible directions for further research. 1.1 Errata to conference version A previous version of this paper [6] erroneously claimed proofs showing all Turing complete sets for EXPSPACE are autoreducible and all truth-table complete sets for PSPACE are nonadaptively autoreducible. Combined with the additional results in this version, we would have a separation of NL and NP (see Section 5). However the proofs in the earlier version failed to account for the growth of the running time when recursively computing previous players' moves. We use the proof technique in Section 3 though unfortunately we get weaker theorems. The original results claimed in the previous version remain important open questions as resolving them either way will yield new separation results. Notation and Preliminaries Most of our complexity theoretic notation is standard. We refer the reader to the textbooks by Balc#zar, D#az and Gabarr# [4, 3], and by Papadimitriou [12]. We use the binary alphabet 1g. We denote the dioeerence of a set A with a set B, i.e., the subset of elements of A that do not belong to B, by A n B. For any integer k ? 0, a \Sigma k -formula is a Boolean expression of the form where OE is a Boolean formula, Q i denotes 9 if i is odd, and 8 otherwise, and the n i 's are positive integers. We say that (1) has alternations. A \Pi k -formula is just like (1) except starts with a 8-quanti-er. It also has k \Gamma 1 alternations. A QBF k -formula is a \Sigma k -formula (1) or a \Pi k -formula For any integer k ? 0, \Sigma P k denotes the k-th \Sigma-level of the polynomial-time hierarchy. We de-ne these levels recursively by \Sigma P k . The \Delta-levels of the polynomial-time and exponential-time hierarchy are de-ned as \Delta P respectively \Delta EXP k . The polynomial-time hierarchy PH equals the union of all sets \Delta P k , and the exponential-time hierarchy EXPH similarly the union of all sets \Delta EXP k . A reduction of a set A to a set B is a polynomial-time oracle Turing machine M such that A. We say that A reduces to B and write A 6 P Turing). The reduction M is nonadaptive if the oracle queries M makes on any input are independent of the oracle, i.e., the queries do not depend upon the answers to previous queries. In that case we write A 6 P for truth-table). Reductions of functions to sets are de-ned similarly. If the number of queries on an input of length n is bounded by q(n), we write A 6 P bounded by some constant, we write A 6 P We denote the set of queries of M on input x with oracle B by Q M B(x); in case of nonadaptive reductions, we omit the oracle B in the notation. If the reduction asks only one query and answers the answer to that query, we write For any reducibility 6 P r and any complexity class C, a set C is 6 P r -hard for C if we can 6 P r -reduce every set A 2 C to C. If in addition C 2 C, we call C 6 P r -complete for C. For any integer k ? 0, the set TQBF k of all true QBF k -formulae is 6 P m -complete for \Sigma P k . For reduces to the fact that the set SAT of satis-able Boolean formulae is 6 P m -complete for NP. Now we get to the key concept of this paper: De-nition 2.1 A set A is autoreducible if there is a reduction M of A to itself that never queries its own input, i.e., for any input x and any oracle B, x 62 Q M B(x). We call such M an autoreduction of A. We will also discuss randomized and nonuniform variants. A set is probabilistically autoreducible if it has a probabilistic autoreduction with bounded two-sided error. Yao [19] -rst studied this concept under the name icoherencej. A set is nonuniformly autoreducible if it has an autoreduction that uses polynomial advice. For all these notions, we can consider both the adaptive and the nonadaptive case. For randomized autoreducibility, nonadaptiveness means that the queries only depend on the input and the random seed. 3 Nonautoreducibility Results In this section, we show that large complexity classes have complete sets that are not autoreducible. Theorem 3.1 There is a 6 P 2\GammaT -complete set for EEXPSPACE that is not autoreducible. Most natural classes containing EEXPSPACE, e.g., EEEXPTIME and EEEXPSPACE, also have this property. We can even construct the complete set in Theorem 3.1 to defeat every probabilistic autoreduc- tion: Theorem 3.2 There is a 6 P 2\GammaT -complete set for EEXPSPACE that is not probabilistically autore- ducible. In the nonadaptive setting, we obtain: Theorem 3.3 There is a 6 P 3\Gammatt -complete set for EXPSPACE that is not nonadaptively autore- ducible. Unlike the case of Theorem 3.1, our construction does not seem to yield a truth-table complete set that is not probabilistically nonadaptively autoreducible. In fact, as we shall show in Section 4.3, such a result would separate EXP from EXPSPACE. See also Section 5. We will detail in Section 4.3 that our nonautoreducibility results do not hold in the nonuniform setting. 3.1 Adaptive Autoreductions Suppose we want to construct a nonautoreducible Turing-complete set for a complexity class C, i.e., a set A such that: 1. A is not autoreducible. 2. A is Turing-hard for C. 3. A belongs to C. If C has a 6 P m -complete set K, realizing goals 1 and 2 is not too hard: We can encode K in A, and at the same time diagonalize against all autoreductions. A straightforward implementation would be to encode K(y) as A(h0; yi), and stage-wise diagonalize against all 6 P -reductions M by picking for each M an input x not of the form h0; yi that is not queried during previous stages, and setting (x). However, this construction does not seem to achieve the third goal. In particular, deciding the membership of a diagonalization string x to A might require computing inputs y of length jxj c , assuming M runs in time n c . Since we have to do this for all potential autoreductions M , we can only bound the resources (time, space) needed to decide A by a function in t(n !(1) ), where t(n) denotes the amount of resources some deterministic Turing machine accepting K uses. That does not suOEce to keep A inside C. To remedy this problem, we will avoid the need to compute K(y) on large inputs y, say of length at least jxj. Instead, we will make sure we can encode the membership of such strings to any set, not just K, and at the same time diagonalize against M on input x. We will argue that we can do this by considering two possible coding regions at every stage as opposed to a -xed one: the left region L, containing strings of the form h0; yi, and the right region R, similarly containing strings of the form h1; yi. The following states that we can use one of the regions to encode an arbitrary sequence, and set the other region such that the output of M on input x is -xed and indicates the region used for encoding. Either it is the case that for any setting of L there is a setting of R such that M A (x) accepts, or for any setting of R there is a setting of L such that M A (x) rejects. (*) This allows us to achieve goals 1 and 2 from above as follows. In the former case, we will set encode K in L (at that stage); otherwise we will set encode K in R. Since the value of A(x) does not aoeect the behavior of M A on input x, we diagonalize against M in both cases. Also, in any case so deciding K is still easy when given A. Moreover - and crucially - in order to compute A(x), we no longer have to decide K(y) on large inputs y, of length jxj or more. Instead, we have to check whether the former case in (*) holds or not. Although quite complex a task, it only depends on M and on the part of A constructed so far, not on the value of K(y) for any input of length jxj or more: We verify whether we can encode any sequence, not just the characteristic sequence of K for lengths at least jxj, and at the same time diagonalize against M on input x. Provided the complexity class C is suOEciently powerful, we can perform this task in C. There is still a catch, though. Suppose we have found out that the former case in (*) holds. Then we will use the left region L to encode K (at that stage), and we know we can diagonalize against M on input x by setting the bits of the right region R appropriately. However, deciding exactly how to set these bits of the noncoding region requires, in addition to determining which region we should use for coding, the knowledge of K(y) for all y such that jxj 6 jyj 6 jxj c . In order to also circumvent the need to decide K for too large inputs here, we will use a slightly stronger version of (*) obtained by grouping quanti-ers into blocks and rearranging them. We will partition the coding and noncoding regions into intervals. We will make sure that for any given interval, the length of a string in that interval (or any of the previous intervals) is no more than the square of the length of any string in that interval. Then we will block-wise alternately set the bits in the coding region according to K, and the corresponding ones in the noncoding region so as to maintain the diagonalization against M on input x as in (*). This way, in order to compute the bit A(h1; zi) of the noncoding region, we will only have to query K on inputs y with jyj 6 jzj 2 , as opposed to for an arbitrarily large c depending on M as was the case before. This is what happens in the next lemma, which we prove in a more general form, because we will need the generalization later on in Section 5. Lemma 3.4 Fix a set K, and suppose we can decide it simultaneously in time t(n) and space s(n). be a constructible monotone unbounded function, and suppose there is a deterministic Turing machine accepting TQBF that takes time t 0 (n) and space s 0 (n) on QBF- formulae of size 2 n fi(n) with at most 2 log fi(n) alternations. Then there is a set A such that: 1. A is not autoreducible. 2. K 6 P A. 3. We can decide A simultaneously in time O(2 n 2 \Deltat(n 2 )+2 n \Deltat 0 (n)) and space O(2 n 2 Proof (of Lemma 3.4) Fix a function fi satisfying the hypotheses of the Lemma. Let M be a standard enumeration of autoreductions clocked such that M i runs in time n fi(i) on inputs of length n. Our construction starts out with A being the empty set, and then adds strings to A in subsequent stages de-ned by the following sequence: Note that since M i runs in time n fi(i) , M i can not query strings of length n i+1 or more on input 0 n i . Fix an integer i ? 1 and let . For any integer j such that 0 6 j 6 log fi(m), let I j denote the set of all strings with lengths in the interval [m 2 j 1)). Note that log fi(m) forms a partition of the set I of strings with lengths in [m; m fi(m) the property that for any 0 6 k 6 log fi(m), the length of any string in [ k I j is no more than the square of the length of any string in I k . During the i-th stage of the construction, we will encode the restriction Kj I of K to I into use the string 0 m for diagonalizing against M i , applying the next strengthening of (*) to do so: 3.1 For any set A, at least one of the following holds: or Here we use (Q z y ) y2Y as a shorthand for Q z y 1 and all variables are quanti-ed over f0; 1g. Without loss of generality we assume that the range of the pairing function h\Delta; \Deltai is disjoint from 0 . Proof (of Claim 3.1) Fix A. If (2) does not hold, then its negation holds, i.e, Switching the quanti-ers log fi(m) in (4) yields the weaker statement (3). if formula (2) holds then for the lexicographically -rst value satisfying where A endfor A the lexicographically -rst value satisfying where A endfor A endif Figure 1: Stage i of the construction of the set A in Lemma 3.4 Figure 1 describes the i-th stage in the construction of the set A. Note that the lexicographically -rst values in this algorithm always exist, so the construction works -ne. We now argue that the resulting set A satis-es the properties of Lemma 3.4. 1. The construction guarantees that by the end of stage Since M i on input 0 m can not query 0 m (because M i is an autoreduction) nor any of the strings added during subsequent stages (because M i does not even have the time to write down any of these strings), A(0 m holds for the -nal set A. So, M i is not an autoreduction of A. Since this is true of any autoreduction M i , the set A is not autoreducible. 2. During stage i, we encode Kj I in the left region ioe we do not put 0 m into A; otherwise we encode Kj I in the right region. So, for any y 2 I , Therefore, A. 3. First note that A only contains strings of the form 0 m with and strings of the form hb; yi with b 2 f0; 1g and y 2 \Sigma . Assume we have executed the construction of A up to but not including stage i. The additional work to decide the membership to A of a string belonging to the i-th stage, is as follows. case does not hold and (2) is a QBF 2 log fi(m) -formula of size case hb; zi where Then hb; zi 2 A ioe z 2 K, which we can decide in time t(jzj). case hb; zi where Say z 2 I k , 0 6 k 6 log fi(m). In order to compute whether hb; zi 2 A, we run the part of stage i corresponding to the values of j in Figure 1 up to and including k. This involves computing K on [ k I j and deciding O(2 jzj ) QBF 2 log fi(m) -formulae of size O(2 m fi(m) ). The latter takes O(2 jzj \Delta t 0 (m)) time. Since every string in [ k I j is of size no more than , we can do the former in time O(2 jzj 2 A similar analysis also shows that we can perform the stages up to but not including i in time All together, this yields the time bound claimed for A. The analysis of the space complexity is analogous. (Lemma Using the upper bound 2 n fi(n) for s 0 (n), the smallest standard complexity class to which Lemma 3.4 applies, seems to be EEXPSPACE. This results in Theorem 3.1. Proof (of Theorem 3.1) In Lemma 3.4, set K a 6 P m -complete set for EEXPSPACE, and In section 4.2, we will see that 6 P 2\GammaT in the statement of Theorem 3.1 is optimal: Theorem 4.5 shows that Theorem 3.1 fails for 6 P 2\Gammatt . We note that the proof of Theorem 3.1 carries through for 6 EXPSPACE -reductions with polynomially bounded query lengths. This implies the strengthening given by Theorem 3.2. 3.2 Nonadaptive Autoreductions Diagonalizing against nonadaptive autoreductions M is easier. If M runs in time -(n), there can be no more than -(n) coding strings that interfere with the diagonalization, as opposed to 2 -(n) in the adaptive case. This allows us to reduce the complexity of the set constructed in Lemma 3.4 as follows. Lemma 3.5 Fix a set K, and suppose we can decide it simultaneously in time t(n) and space s(n). be a constructible monotone unbounded function, and suppose there is a deterministic Turing machine accepting TQBF that takes time t 0 (n) and space s 0 (n) on QBF- formulae of size n fi(n) with at most 2 log fi(n) alternations. Then there is a set A such that: 1. A is not nonadaptively autoreducible. 2. K 6 P A. 3. We can decide A simultaneously in time O(2 n \Delta (t(n 2 )+ t 0 (n))) and space O(2 n +s(n 2 )+s 0 (n)). Proof (of Lemma 3.5) The construction of the set A is the same as in Lemma 3.4 (see Figure 1) apart from the following dioeerences: now is a standard enumeration of nonadaptive autoreductions clocked such that runs in time n fi(i) on inputs of length n. Note that the set QM (x) of possible queries M makes on input x contains no more than jxj fi(i) elements. ffl During stage i ? 1 of the construction, I denotes the set of all strings y with lengths in denotes the set of strings in I with lengths in [m Note that the only ' y 's and r y 's that aoeect the validity of the predicate iM A 0 formula (2) and the corresponding formulae in Figure 1, are those for which y 2 I . ffl At the end of stage i in Figure 1, we add the following line: A This ensures coding K(y) for strings y with lengths in [n are queried by M i on input 0 m . Although not essential, we choose to encode them in both the left and the right region. Adjusting time and space bounds appropriately, the proof that A satis-es the 3 properties claimed, carries over. There is an additional case in the analysis of the third point, namely the one of an input of the form hb; zi where b 2 I . Then, by construction, which we can decide in time t(jzj). The crucial simpli-cation over the adaptive case lies in the fact that (2) and the similar formulae in Figure 1 now become QBF 2 log fi(n) -formulae of size O(n fi(n) ) as opposed to of size O(2 n fi(n) ) in Lemma 3.4. (Lemma 3.5) As a consequence, we can lower the space complexity in the equivalent of Theorem 3.1 from doubly exponential to singly exponential, yielding Theorem 3.3. In section 4.2, we will show we can not reduce the number of queries from 3 to 2 in Theorem 3.3. If we restrict the number of queries the nonadaptive autoreduction is allowed to make to some -xed polynomial, the proof technique of Theorem 3.3 also applies to EXP. In particular, we obtain: Theorem 3.6 There is a 6 P 3\Gammatt -complete set for EXP that is not 6 P btt -autoreducible. 4 Autoreducibility Results For small complexity classes, all complete sets turn out to be autoreducible. Beigel and Feigenbaum [5] established this property of all levels of the polynomial-time hierarchy as well as of PSPACE, the largest class for which it was known to hold before our work. In this section, we will prove it for the \Delta-levels of the exponential-time hierarchy. As to nonadaptive reductions, the question was even open for all levels of the polynomial-time hierarchy. We will show here that the 6 P tt -complete sets for the \Delta-levels of the polynomial-time hierarchy are nonadaptively autoreducible. For any complexity class containing EXP, we will prove that the 6 P 2\Gammatt -complete sets are 6 P 2\Gammatt -autoreducible. Finally, we will also consider nonuniform and randomized autoreductions. Throughout this section, we will assume without loss of generality an encoding fl of a computation of a given oracle Turing machine M on a given input x with the following properties. fl will be a marked concatenation of successive instantaneous descriptions of M , starting with the initial instantaneous description of M on input x, such that: ffl Given a pointer to a bit in fl, we can -nd out whether that bit represents the answer to an oracle query by probing a constant number of bits of fl. ffl If it is the answer to an oracle query, the corresponding query is a substring of the pre-x of fl up to that point, and we can easily compute a pointer to the beginning of that substring without probing fl any further. ffl If it is not the answer to an oracle query, we can perform a local consistency check for that bit which only depends on a constant number of previous bit positions of fl and the input x. Formally, there exist a function g M and a predicate e M , both polynomial-time computable, and a constant c M such that the following holds: For any input x, any index i to a bit position in fl, and any j, 1 is an index no larger than i, and indicates whether fl passes the local consistency test for its i-th bit fl i . Provided the pre-x of fl up to but not including position i is correct, the local consistency test is passed ioe fl i is correct. We call such an encoding a valid computation of M on input x ioe the local consistency tests (5) for all the bit positions i that do not correspond to oracle answers, are passed, and the other bits equal the oracle's answer to the corresponding query. Any other string we will call a computation. 4.1 Adaptive Autoreductions We will -rst show that every 6 P T -complete set for EXP is autoreducible, and then generalize to all \Delta-levels of the exponential-time hierarchy. Theorem 4.1 Every 6 P T -complete set for EXP is autoreducible. Here is the proof idea: For any of the standard deterministic complexity classes C, we can decide each bit of the computation on a given input x within C. So, if A is a 6 P T -complete set for C that can be decided by a machine M within the con-nes of the class C, then we can 6 P -reduce deciding the i-th bit of the computation of M on input x to A. Now, consider the two (possibly invalid) computations we obtain by applying for every bit position the above reduction, answering all queries except for x according to A, and assuming x 2 A for one computation, and x 62 A for the other. Note that the computation corresponding to the right assumption about A(x), is certainly correct. So, if both computations yield the same answer (which we can eOEciently check using A without querying x), that answer is correct. If not, the other computation contains a mistake. We can not check both computations entirely to see which one is right, but given a pointer to the -rst incorrect bit of the wrong computation, we can eOEciently verify that it is mistaken by checking only a constant number of bits of that computation. The pointer is again computable within C. In case C ' EXP, using a 6 P T -reduction to A and assuming x 2 A or x 62 A as above, we can determine the pointer with oracle A (but without querying x) in polynomial time, since the pointer's length is polynomially bounded. We now -ll out the details. Proof (of Theorem 4.1) Fix a 6 P T -complete set A for EXP. Say A is accepted by a Turing machine M such that the computation of M on an input of size n has length 2 p(n) for some -xed polynomial p. Without loss of generality the last bit of the computation gives the -nal answer. Let g M , e M , and c M be the formalization of the local consistency test for M as described by (5). Let -(hx; ii) denote the i-th bit of the computation of M on input x. We can compute - in EXP, so there is an oracle Turing machine R - 6 P -reducing - to A. Let oe(x) be the -rst such that R Anfxg exists. Again, we can compute oe in EXP, so there is a 6 P T -reduction R oe from oe to A. Consider the algorithm in Figure 2 for deciding A on input x. The algorithm is a polynomial-time oracle Turing machine with oracle A that does not query its own input x. We now argue that it correctly decides A on input x. We distinguish between two cases: case R Anfxg Since at least one of the computations R Anfxg - (hx; \Deltai) or R A[fxg coincides with the actual computation of M on input x, and the last bit of the computation equals the -nal decision, correctness follows. if R Anfxg then accept ioe R A[fxg else i / R A[fxg oe (x) accept ioe e M (x; endif Figure 2: Autoreduction for the set A of Theorem 4.1 on input x case R Anfxg contains a mistake. Variable i gets the correct value of the index of the -rst incorrect bit in this computation, so the local consistency test for R Anfxg being the computation of M on input x fails on the i-th bit, and we accept x. If x 62 A, R Anfxg - (hx; \Deltai) is a valid computation, so no local consistency test fails, and we reject x. (Theorem 4.1) The local checkability property of computations used in the proof of Theorem 4.1 does not relativize, because the oracle computation steps depend on the entire query, i.e., on a number of bits that is only limited by the resource bounds of the base machine, in this case exponentially many. We next show that Theorem 4.1 itself also does not relativize. Theorem 4.2 Relative to some oracle, EXP has a 6 P 2\GammaT -complete set that is not autoreducible. Proof Note that EXP has the following property: Property 4.1 There is an oracle Turing machine N running in EXP such that for any oracle B, the set accepted by N B is 6 P m -complete for EXP B . Without loss of generality, we assume that N runs in time 2 n . Let K B denote the set accepted by We will construct an oracle B and a set A such that A is 6 P 2\GammaT -complete for EXP B and is not The construction of A is the same as in Lemma 3.4 (see Figure 1) with for that the reductions M i now also have access to the oracle B. We will encode in B information about the construction of A that reduces the complexity of A relative to B, but do it high enough so as not to destroy the 6 P 2\GammaT -completeness of A for EXP B nor the diagonalizations against 6 P B -autoreductions. We construct B in stages along with A. We start with B empty. Using the notation of Lemma 3.4, at the beginning of stage i, we add to B ioe property (2) does not hold, and at the end of sub-stage j, we union B with (2) holds at stage i Note that this does not aoeect the value of K B (y) for , nor the computations of M i on inputs of size at most m (for suOEciently large i such that m log It follows from the analysis in the proof of Lemma 3.4 that the set A is 6 P 2\GammaT -hard for EXP B and not 6 P B Regarding the complexity of deciding A relative to B, note that the encoding in the oracle B allows us to eliminate the need for evaluating QBF log fi(n) -formulae of size 2 n fi(n) . Instead, we just query B on easily constructed inputs of size O(2 n 2 Therefore, we can drop the terms corresponding to the QBF log fi(n) -formulae of size 2 n fi(n) in the complexity of A. Consequently, A 2 EXP B . (Theorem 4.2) Theorem 4.2 applies to any complexity class containing EXP that has Property 4.1, e.g., EXPSPACE, EEXP, EEXPSPACE, etc. Sometimes, the structure of the oracle allows to get around the lack of local checkability of oracle queries. This is the case for oracles from the polynomial-time hierarchy, and leads to the following extension of Theorem 4.1: Theorem 4.3 For any integer k ? 0, every 6 P T -complete set for \Delta EXP k+1 is autoreducible. The proof idea is as follows: Let A be a 6 P T -complete set accepted by the deterministic oracle Turing machine M with oracle TQBF k . First note that there is a polynomial-time Turing machine N such that a query q belongs to the oracle TQBF k ioe where the y ' 's are of size polynomial in jqj. We consider the two purported computations of M on input x constructed in the proof of Theorem 4.1. One of them belongs to a party assuming x 2 A, the other one to a party assuming x 62 A. The computation corresponding to the right assumption is correct; the other one might not be. Now, suppose the computations dioeer, and we are given a pointer to the -rst bit position where they disagree, which turns out to be the answer to an oracle query q. Then we can have the two parties play the k-round game underlying (6): The party claiming q 2 TQBF k plays the existentially quanti-ed y ' 's, the other one the universally quanti-ed y ` 's. The players' strategies will consist of computing the game history so far, determining their optimal next move, 6 P -reducing this computation to A, and -nally producing the result of this reduction under their respective assumption about A(x). This will guarantee that the party with the correct assumption plays optimally. Since this is also the one claiming the correct answer to the oracle query q, he will win the game, i.e., N(q; y answer bit. The only thing the autoreduction for A has to do, is determine the value of N(q; y in polynomial time using A as an oracle but without querying x. It can do that along the lines of the base case algorithm given in Figure 2. If during this process, the local consistency test for 's computation requires the knowledge of bits from the y ` 's, we compute these via the reduction de-ning the strategy of the corresponding player. The bits from q we need, we can retrieve from the M-computations, since both computations are correct up to the point where they -nished generating q. Once we know N(q; y easily decide the correct assumption about A(x). The construction hinges on the hypothesis that we can 6 P -reduce determining the player's moves to A. Computing these moves can become quite complex, though, because we have to recursively reconstruct the game history so far. The number of rounds k being constant, seems crucial for keeping the complexity under control. The conference version of this paper [6] erroneously claimed the proof works for EXPSPACE, which can be thought of as alternating exponential time with an exponential number of alternations. Establishing Theorem 4.3 for EXPSPACE would actually separate NL from NP, as we will see in Section 5. Proof (of Theorem 4.3) Let A be a 6 P T -complete set for \Delta EXP k accepted by the exponential-time oracle Turing machine M with oracle TQBF k . Let g M , e M , and c M be the formalization of the local consistency test for M as described by (5). Without loss of generality there is a polynomial p and a polynomial-time Turing machine N such that on inputs of size n, M makes exactly 2 p(n) oracle queries, all of the form where q has length 2 p 2 (n) . Moreover, the computations of N in (7) each have length 2 p 3 (n) , and their last bit represents the answer; the same holds for the computations of M on inputs of length n. Let g N , e N , and c N be the formalization of the local consistency test for N . We -rst de-ne a bunch of functions computable in \Delta EXP k+1 . For each of them, say -, we -x an oracle Turing machine R - that 6 P -reduces - to A, and which the -nal autoreduction for A will use. The proofs that we can compute these functions in \Delta EXP are straightforward. Let -(hx; ii) denote the i-th bit of the computation of M TQBF k on input x, and oe(x) the -rst i (if any) such that R Anfxg ii). The roles of - and oe are the same as in the proof of Theorem 4.1: We will use R - to -gure out whether both possible answers for the oracle query lead to the same -nal answer, and if not, use R oe to -nd a pointer i to the -rst incorrect bit (in any) of the simulated computation getting the negative oracle answer x 62 A. If i turns out not to point to an oracle query, we can proceed as in the proof of Theorem 4.1. Otherwise, we will make use of the following functions and associated reductions to A. We de-ne the functions j ' and y ' inductively for At each level ' we -rst de-ne j ' , which induces a reduction R j ' , and then de-ne y ' based on R j ' . All of these functions take an input x such that the i-th bit of R Anfxg - (hx; \Deltai) is the answer to an oracle query (7), where oe (x). We de-ne j ' (x) as the lexicographically least y such that if this value does not exist, we set j ' . Note that the right-hand side of (8) is 1 ioe y ' is existentially quanti-ed in (7). R A[fxg R Anfxg The condition on the right-hand side of (9) means that we use the hypothesis x 2 A to compute y ' (x) from R j ' in case: ffl either y ' is existentially quanti-ed in (7) and the player assuming x 2 A claims (7) holds, ffl or else y ' is universally quanti-ed and the player assuming x 2 A claims (7) fails. Otherwise we use the hypothesis x 62 A. In case i points to the answer to an oracle query (7), the functions j ' and the reductions R j ' incorporate the moves during the successive rounds of the game underlying (7). The reduction R j ' together with the player's assumption about membership of x to A, determines the actual move during the '-th round, namely R A[fxg if the '-th round is played by the opponent assuming otherwise. The condition on the right-hand side of (9) guarantees that the existentially quanti-ed variables are determined by the opponent claiming the query (7) is a true formula, and the universally quanti-ed ones by the other opponent. In particular, (9) ensures that the opponent with the correct claim about (7) has a wining strategy. Provided it exists, the function de-nes a winning move during the '-th round of the game for the opponent playing that round, given the way the previous rounds were actually played (as described by the y(x)'s). For odd ', i.e., y ' is existentially quanti-ed, it tries to set y ' such that the remainder of (7) holds; otherwise it tries to set y ' such that the remainder of (7) fails. The actual move may dioeer from the one given by j ' in case the player's assumption about x 2 A is incorrect. The opponent with the correct assumption plays according to j ' . Since that opponent also makes the correct claim about (7), he will win the game. In any case, N(q; y hold ioe (7) holds. Finally, we de-ne the functions - and - , which have a similar job as the functions - respectively oe, but for the computation of N(q; y instead of the computation of M TQBF k (x). More precisely, -(hx; ri) equals the r-th bit of the computation of N(q; y 1 where the are de-ned by (9), and the bit with index oe (x) in the computation R Anfxg the answer to the oracle query (7). We de-ne -(x) to be the -rst r (if any) for which R Anfxg R A[fxg ri), provided the bit with index oe (x) in the computation R Anfxg - (hx; \Deltai) is the answer to an oracle query. Now we have these functions and corresponding reductions, we can describe an autoreduction for A. On input x, it works as described in Figure 3. We next argue that the algorithm correctly decides A on input x. Checking the other properties required of an autoreduction for A is straightforward. We only consider the cases where R Anfxg points to the answer to an oracle query in R Anfxg \Deltai). We refer to the analysis in the proof of Theorem 4.1 for the remaining cases. case R Anfxg points to the -rst incorrect bit of R Anfxg - (hx; \Deltai), which turns out to be the answer to an oracle query, say (7). yields the correct oracle answer to (7), R Anfxg and we accept x. If x 62 A, both R Anfxg give the correct answer to the oracle query i points to in the computation R Anfxg \Deltai). So, they are equal, and we reject x. if R Anfxg then accept ioe R A[fxg else i / R A[fxg oe (x) if the i-th bit of R Anfxg \Deltai) is not the answer to an oracle query then accept ioe e M (x; else if R Anfxg then accept ioe R Anfxg else r / R A[fxg accept ioe e N (q; y R Anfxg where q denotes the query described in R Anfxg to which the i-th bit in this computation is the answer and R A[fxg R Anfxg endif endif endif Figure 3: Autoreduction for the set A of Theorem 4.3 on input x case R Anfxg Then, as described in Figure 3, we will use the local consistency test for R Anfxg being the computation of N(q; y 1 (x)). Apart from bits in the purported computation R Anfxg - (hx; \Deltai), this test may also need bits from q and from the y ' (x)'s. The y ` (x)'s can be computed straightforwardly using their de-nition (9). The bits from q we might need, can be retrieved from R Anfxg \Deltai). This is because our encoding scheme for computations has the property that the query q is a substring of the pre-x of the computation up to the position indexed by i. Since either R Anfxg - (hx; \Deltai) is correct everywhere, or else i is the -rst position where is is incorrect, the description of q in R Anfxg - (hx; \Deltai) is correct in any case. Moreover, we can easily compute a pointer to the beginning of the substring q of R Anfxg - (hx; \Deltai) from i. - (hx; \Deltai) has an error as a computation of gets assigned the index of the -rst incorrect bit in this computation, so the local consistency check fails, and we accept x. If x 62 A, R Anfxg - (hx; \Deltai) is a valid computation of N(q; y 1 so every local consistency test is passed, and we reject x. (Theorem 4.3) 4.2 Nonadaptive Autoreductions So far, we constructed autoreductions for 6 P T -complete sets A. On input x, we looked at the two candidate computations obtained by reducing to A, answering all oracle queries except for x according to A, and answering query x positively for one candidate, and negatively for the other. If the candidates disagreed, we tried to -nd out the right one, which always existed. We managed to get the idea to work for quite powerful sets A, e.g., EXP-complete sets, by exploiting the local checkability of computations. That allowed us to -gure out the wrong computation without going through the entire computation ourselves: With help from A, we -rst computed a pointer to the -rst mistake in the wrong computation, and then veri-ed it locally. We can not use this adaptive approach for constructing nonadaptive autoreductions. It seems like -guring out the wrong computation in a nonadaptive way, requires the autoreduction to perform the computation of the base machine itself, so the base machine has to run in polynomial time. Then checking the computation essentially boils down to verifying the oracle answers. Using the game characterization of the polynomial-time hierarchy along the same lines as in Theorem 4.3, we can do this for oracles from the polynomial-time hierarchy. Theorem 4.4 For any integer k ? 0, every 6 P tt -complete set for \Delta P k+1 is nonadaptively autore- ducible. Parallel to the adaptive case, an earlier version of this paper [6] stated Theorem 4.4 for unbounded k, i.e., for PSPACE. However, we only get the proof to work for constant k. In Section 5, we will see that proving Theorem 4.4 for PSPACE would separate NL from NP. The only additional diOEculty in the proof is that in the nonadaptive setting, we do not know which player has to perform the even rounds, and which one the odd rounds in the k-round game underlying a query like (6). But we can just have them play both scenarios, and afterwards -gure out the relevant run. Proof (of Theorem 4.4) Let A be a 6 P tt -complete set for \Delta P k accepted by the polynomial-time oracle Turing machine M with oracle TQBF k . Without loss of generality there is a polynomial p and a polynomial-time Turing machine N such that on inputs of size n, M makes exactly p(n) oracle queries q, all of the where q has length p 2 (n). Let q(x; i) denote the i-th oracle query of M TQBF k on input x. Note that k . g. The set Q belongs to \Delta P , so there is a 6 P -reduction RQ from Q to A. If for a given input x, R A[fxg Q and R Anfxg Q agree on hx; ji for every 1 6 j 6 p(jxj), we are home: We can simulate the base machine M using R A[fxg as the answer to the j-th oracle query. Otherwise, we will make use of the following functions computable in \Delta P corresponding oracle Turing machines R j 1 tt -reductions to A, and functions . As in the proof of Theorem 4.3, we de-ne j ' and y ' inductively k. They are de-ned for inputs x such that there is a smallest 1 6 i 6 p(jxj) for which R Anfxg ii). The value of j ' (x) equals the lexicographically least we set j ' string does not exist. The right-hand side of (11) is 1 ioe y ' is existentially quanti-ed in (10). R A[fxg R Anfxg The condition on the right-hand side of (12) means that we use the hypothesis x 2 A to compute y ' (x) from R j ' in case: ffl either y ' is existentially quanti-ed in (10) and the assumption x 2 A leads to claiming that ffl or else y ' is universally quanti-ed and the assumption x 2 A leads to claiming that (10) fails. The intuitive meaning of the functions j ' and the reductions R j ' is similar to in the proof of Theorem 4.3: They capture the moves during the '-th round of the game underlying (10) for i). The function j ' encapsulates an optimal move during round ' if it exists, and the reduction R j ' under the player's assumption regarding membership of x to A, produces the actual move in that round. The condition on the right-hand side of (12) guarantees the correct alternation of rounds. We refer to the proof of Theorem 4.3 for more intuition. Consider the algorithm in Figure 4. Note that the only queries to A the algorithm in Figure 4 if R Anfxg then accept ioe M accepts x when the j-th oracle query is answered R A[fxg else i / -rst j such that R Anfxg accept ioe N(q; y where q denotes the i-th query of M on input x when the answer to the j-th oracle query is given by R A[fxg and R A[fxg R Anfxg endif Figure 4: Nonadaptive autoreduction for the set A of Theorem 4.4 on input x needs to make, are the queries of RQ dioeerent from x on inputs hx; ji for 1 6 j 6 p(jxj), and the queries of R j ' dioeerent from x on input x for 1 6 ' 6 k. Since RQ and the R j ' 's are nonadaptive, it follows that Figure 4 describes a 6 P tt -reduction to A that does not query its own input. A similar but simpli-ed argument as in the proof of Theorem 4.3 shows that it accepts A. So, A is nonadaptively autoreducible. (Theorem 4.4) Next, we consider more restricted reductions. Using a dioeerent technique, we show: Theorem 4.5 For any complexity class C, every 6 P 2\Gammatt -complete set for C is 6 P 2\Gammatt -autoreducible, provided C is closed under exponential-time reductions that only ask one query which is smaller in length. In particular, Theorem 4.5 applies to In view of Theorems 3.1 and 3.3, this implies that Theorems 3.1, 3.3, and 4.5 are optimal. The proof exploits the ability of EXP to simulate all polynomial-time reductions to construct an auxiliary set D within C such that any 6 P 2\Gammatt -reductions of D to some -xed complete set A has a property that induces an autoreduction on A. Proof (of Theorem 4.5) be a standard enumeration of 6 P 2\Gammatt -reductions such that M i runs in time n i on inputs of size n. Let A be a 6 P 2\Gammatt -complete set for C. Consider the set D that only contains strings of the form h0 decided by the algorithm of Figure 5 on such an input. Except for deciding A(x), the algorithm runs case truth-table of M i on input h0 i ; xi with the truth-value of query x set to A(x) constant: rejects of the form iy 62 Aj: accept ioe x 62 A otherwise: accept ioe x 2 A endcase Figure 5: Algorithm for the set D of Theorem 4.5 on input h0 in exponential time. Therefore, under the given conditions on C, there is a 6 P -reduction M j from D to A. The construction of D diagonalizes against every 6 P -reduction M i of D to A whose truth-table on input h0 would become constant once we -lled in the membership bit for x. Therefore, for every input x, one of the following cases holds for the truth-table of M j on input h0 ffl The reduced truth-table is of the form iy 2 Aj with y 6= x. ffl The reduced truth-table is of the form iy 62 Aj with y 6= x. ffl The truth-table depends on the membership to A of 2 strings dioeerent from x. Then M A j does not query x on input h0 j ; xi, and accepts ioe x 2 A. The above analysis shows that the algorithm of Figure 6 describes a 6 P 2\Gammatt -reduction of A. then accept ioe M A accepts else y / unique element of QM j accept ioe y 2 A endif Figure Autoreduction constructed in the proof of Theorem 4.5 (Theorem 4.5) 4.3 Probabilistic and Nonuniform Autoreductions The previous results in this section trivially imply that the 6 P T -complete sets for the \Delta-levels of the exponential-time hierarchy are probabilistically autoreducible, and the 6 P tt -complete sets for the \Delta-levels of the polynomial-time hierarchy are probabilistically nonadaptively autoreducible. Randomness allows us the prove more in the nonadaptive case. First, we can establish Theorem 4.4 for EXP: Theorem 4.6 Let f be a constructible function. Every 6 P f(n)\Gammatt -complete set for EXP is probabilistically O(f(n))\Gammatt -autoreducible. In particular, every 6 P tt -complete set for EXP is probabilistically nonadaptively autoreducible. Proof (of Theorem 4.6) Let A be a 6 P f(n)\Gammatt -complete set for EXP. We will apply the PCP Theorem for EXP [2] to A. Lemma 4.7 ([2]) There is a constant k such that for any set A 2 EXP, there is a polynomial-time Turing machine V and a polynomial p such that for any input x: ffl If x 2 A, then there exists a proof oracle - such that Pr ffl If x 62 A, then for any proof oracle - Pr Moreover, V never makes more than k proof oracle queries, and there is a proof oracle ~ - 2 EXP independent of x such that (13) holds for - in case x 2 A. Translating Lemma 4.7 into our terminology, we obtain: Lemma 4.8 There is a constant k such that for any set A 2 EXP, there is a probabilistic 6 P reduction N , and a set B 2 EXP such that for any input x: ffl If x 2 A, then N B (x) always accepts. ffl If x 62 A, then for any oracle C, N C (x) accepts with probability at most 1 3 . Let R be a 6 P f(n)\Gammatt -reduction of B to A, and consider the probabilistic reduction M A that on input runs N on input x with oracle R A[fxg . M A is a probabilistic 6 P k \Deltaf (n)\Gammatt -reduction to A that never queries its own input. The following shows it de-nes a reduction from A: accepts. ffl If x 62 A, then for accepts with probability at most 1(Theorem 4.6) Note that Theorem 4.6 makes it plausible why we did not manage to scale down Theorem 3.2 by one exponent to EXPSPACE in the nonadaptive setting, as we were able to do for our other results in Section 3 when going from the adaptive to the nonadaptive case: This would separate EXP from EXPSPACE. We suggest the extension of Theorem 4.6 to the \Delta-levels of the exponential-time hierarchy as an interesting problem for further research. Second, Theorem 4.4 also holds for NP: Theorem 4.9 All 6 P tt -complete sets for NP are probabilistically nonadaptively autoreducible. Proof (of Theorem 4.9) Fix a 6 P tt -complete set A for NP. Let RA denote a length nondecreasing 6 P m -reduction of A to SAT. De-ne the set is a Boolean formula with, say m variables and 9 a there is a 6 P -reduction RW from W to A. We will use the following probabilistic algorithm by Valiant and Vazirani [18]: Lemma 4.10 ([18]) There exists a polynomial-time probabilistic Turing machine N that on input a Boolean formula ' with n variables, outputs another quanti-er such ffl If ' is satis-able, then with probability at least 1 4n , OE has a unique satisfying assignment. ffl If ' is not satis-able, then OE is never satis-able. Now consider the following algorithm for A: On input x, run N on input RA (x), yielding a Boolean formula OE with, say m variables, and it accepts ioe OE(R A[fxg evaluates to true. Note that this algorithm describes a probabilistic 6 P tt -reduction to A that never queries its own input. Moreover: ffl If x 2 A, then with probability at least 1 , the Valiant-Vazirani algorithm N produces a Boolean formula OE with a unique satisfying assignment ~ a OE . In that case, the assignment we use (R A[fxg we accept x. ffl If x 62 A, any Boolean formula OE which N produces has no satisfying assignment, so we always reject x. Executing \Theta(n) independent runs of this algorithm, and accepting ioe any of them accepts, yields a probabilistic nonadaptive autoreduction for A. (Theorem 4.9) So, for probabilistic autoreductions, we get similar results as for deterministic ones: Low end complexity classes turn out to have the property that their complete sets are autoreducible, whereas high end complexity classes do not. As we will see in more detail in the next section, this structural dioeerence yields separations. If we allow nonuniformity, the situation changes dramatically. Since probabilistic autoreducibil- ity implies nonuniform autoreducibility [5], all our positive results for small complexity classes carry over to the nonuniform setting. But, as we will see next, the negative results do not, because also the complete sets for large complexity classes become autoreducible, both in the adaptive and in the nonadaptive case. So, uniformity is crucial for separating complexity classes using autoreducibility, and the Razborov-Rudich result [14] does not apply. Feigenbaum and Fortnow [7] de-ne the following concept of #P-robustness, of which we also consider the nonadaptive variant. De-nition 4.1 A set A is #P-robust if #P A ' FP A ; A is nonadaptively #P-robust if #P A FP A tt . Nonadaptive #P-robustness implies #P-robustness. For the usual deterministic and nondeterministic complexity classes containing PSPACE, all 6 P T -complete sets are #P-robust. For the deterministic classes containing PSPACE, it is also true that the 6 P tt -complete sets are nonadaptively #P-robust. The following connection with nonuniform autoreducibility holds: Theorem 4.11 All #P-robust sets are nonuniformly autoreducible. All nonadaptively #P-robust sets are nonuniformly nonadaptively autoreducible. Proof Feigenbaum and Fortnow [7] show that every #P-robust language is random-self-reducible. Beigel and Feigenbaum [5] prove that every random-self-reducible set is nonuniformly autoreducible (or iweakly coherentj as they call it). Their proofs carry over to the nonadaptive setting. It follows that the 6 P tt -complete sets for the usual deterministic complexity classes containing PSPACE are all nonuniformly nonadaptively autoreducible. The same holds for adaptive reductions, in which case the property is also true of nondeterministic complexity classes containing PSPACE. In particular, we get the following: Corollary 4.12 All 6 P T -complete sets for NEXP, EXPSPACE, EEXP, NEEXP, EEXPSPACE, . are nonuniformly autoreducible. All 6 P tt -complete sets for PSPACE, EXP, EXPSPACE, . are nonuniformly nonadaptively autoreducible. 5 Separation Results In this section, we will see how we can use the structural property of all complete sets being autoreducible to separate complexity classes. Based on the results of Sections 3 and 4, we only get separations that were already known: EXPH 6= EEXPSPACE (by Theorems 4.3 and 3.1), Theorems 4.6 and 3.2), and PH 6= EXPSPACE (by Theorems 4.4 and 3.3, and also by scaling down EXPH 6= EEXPSPACE). However, settling the question for certain other classes, would yield impressive new separations. We summarize the implications in Figure 7. Theorem 5.1 In Figure 7, a positive answer to a question from the -rst column, implies the separation in the second column, and a negative answer, the separation in the third column. question yes no Are all 6 P T -complete sets for EXPSPACE autoreducible? NL 6= NP PH 6= PSPACE Are all 6 P T -complete sets for EEXP autoreducible? Are all 6 P tt -complete sets for PSPACE 6 P Are all 6 P tt -complete sets for EXP 6 P Are all 6 P tt -complete sets for EXPSPACE probabilistically 6 P Figure 7: Separation results using autoreducibility Most of the entries in Figure 7 follow directly from the results of the previous sections. In order to -nish the table, we use the next lemma: Lemma 5.2 If NL, we can decide the validity of QBF-formulae of size t and with ff alternations on a deterministic Turing machine M 1 in time t O(c ff ) and on a nondeterministic Turing machine M 2 in space O(c ff log t), for some constant c. Proof (of Lemma 5.2) by Cook's Theorem we can transform in polynomial time a \Pi 1 -formula with free variables into an equivalent \Sigma 1 -formula with the same free variables, and vice versa. Since we can decide the validity of \Sigma 1 -formulae in polynomial-time. Say both the transformation algorithm T and the satis-ability algorithm S run in time n c for some constant c. Let OE be a QBF-formula of size t with ff alternations. Consider the following algorithm for deciding OE: Repeatedly apply the transformation T to the largest suOEx that constitutes a \Sigma 1 - or until the whole formula becomes \Sigma 1 , and then run S on it. This algorithm correctly decides the truth of OE. Since the number of alternations decreases by one during every iteration, it makes at most ff calls to T , each time at most raising the length of the formula to the power c. It follows that the algorithm runs in time t O(c ff ) . Moreover, since padding argument shows that DTIME[- time constructible function - . Therefore the result holds. (Lemma 5.2) This allows us to improve Theorems 3.2 and 3.3 as follows under the hypothesis Theorem 5.3 If there is a 6 P 2\GammaT -complete set for EXPSPACE that is not probabilistically autoreducible. The same holds for EEXP instead of EXPSPACE. Proof Combine Lemma 5.2 with the probabilistic extension of Lemma 3.4 used in the proof of Theorem 3.2. Theorem 5.4 If there is a 6 P 3\Gammatt -complete set for PSPACE that is not nonadaptively autoreducible. The same holds for EXP instead of PSPACE. Proof Combine Lemma 5.2 with Lemma 3.5. Now, we have all ingredients for establishing Figure 7: Proof (of Theorem 5.1) The NL 6= NP implications in the iyesj-column of Figure 7 immediately follow from Theorems 5.3 and 5.4 by contraposition. By Theorem 3.1, a positive answer to the 2nd question in Figure 7 would yield EEXP 6= EEXPSPACE, and by Theorem 3.3, a positive answer to the 4th question would imply EXP 6= EXPSPACE. By padding, both translate down to P 6= PSPACE. Similarly, by Theorem 4.3, a negative answer to the 2nd question would imply EXPH 6= EEXP, which pads down to PH 6= EXP. A negative answer to the 4th question would yield PH 6= EXP directly by Theorem 4.4. By the same token, a negative answer to the 1st question results in EXPH 6= EXPSPACE and PH 6= PSPACE, and a negative answer to the 3rd question in PSPACE. By Theorem 4.6, a negative answer to the last question implies EXP 6= EXPSPACE and P 6= PSPACE. We note that we can tighten all of the separations in Figure 7 a bit, because we can apply Lemmata 3.4 and 3.5 to smaller classes than in Theorems 3.1 respectively 3.3. One improvement along these lines that might warrant attention is that we can replace iNL 6= NPj in Figure 7 by This is because that condition suOEces for Theorems 5.3 and 5.4, since we can strengthen Lemma 5.2 as follows: Lemma 5.5 If coNP ' NP " NSPACE[log O(1) n], we can decide the validity of QBF-formulae of size t and with ff alternations on a deterministic Turing machine M 1 in time t O(c ff ) and on a nondeterministic Turing machine M 2 in space O(d ff log d t), for some constants c and d. 6 Conclusion We have studied the question whether all complete sets are autoreducible for various complexity classes and various reducibilities. We obtained a positive answer for lower complexity classes in Section 4, and a negative one for higher complexity classes in Section 3. This way, we separated these lower complexity classes from these higher ones by highlighting a structural dioeerence. The resulting separations were not new, but we argued in Section 5 that settling the very same question for intermediate complexity classes, would provide major new separations. We believe that re-nements to our techniques may lead to them, and would like to end with a few words about some thoughts in that direction. One does not have to look at complete sets only. Let C 1 ' C 2 . Suppose we know that all complete sets for C 2 are autoreducible. Then it suOEces to construct, e.g., along the lines of Lemma 3.4, a hard set for C 1 that is not autoreducible, in order to separate C 1 from C 2 . As we mentioned at the end of Section 5, we can improve Theorem 3.1 a bit by applying Lemma 3.4 to smaller space-bounded classes than EEXPSPACE. We can not hope to gain much, though, since the coding in the proof of Lemma 3.4 seems to be DSPACE[2 n fi(n) ]-complete because of the log fi(n) -formulae of size 2 n fi(n) involved for inputs of size n. The same holds for Theorem 3.3 and Lemma 3.5. Generalizations of autoreducibility may allow us to push things further. For example, one could look at k(n)-autoreducibility where k(n) bits of the set remain unknown to the querying machine. Theorem 4.3 goes through for k(n) 2 O(log n). Perhaps one can exploit this leeway in the coding of Lemma 3.4 and narrow the gap between the positive and negative results. As discussed in Section 5, that would yield interesting separations. Finally, one may want to look at other properties than autoreducibility to realize Post's Program in complexity theory. Perhaps another concept from computability theory or a more arti-cial property can be used to separate complexity classes. Acknowledgments We would like to thank Manindra Agrawal and Ashish Naik for very helpful discussions. We are also grateful to Carsten Lund and Muli Safra for answering questions regarding the PCP Theorem. We thank the anonymous referees for their nice suggestions on how to present our results. --R Proof veri-cation and hardness of approximation problems On being incoherent without being very hard. Using autoreducibility to separate complexity classes. On the random-self-reducibility of complete sets The role of relativization in complexity theory. On the computational complexity of algorithms. Mitotic recursively enumerable sets. Classical Recursion Theory Computational Complexity. Recursively enumerable sets of positive integers and their decision problems. Natural proofs. Recursively Enumerable Sets and Degrees. On autoreducibility. On autoreducibility. NP is as easy as detecting unique solutions. Coherent functions and program checkers. --TR --CTR Christian Glaer , Mitsunori Ogihara , A. Pavan , Alan L. Selman , Liyu Zhang, Autoreducibility, mitoticity, and immunity, Journal of Computer and System Sciences, v.73 n.5, p.735-754, August, 2007 Luca Trevisan , Salil Vadhan, Pseudorandomness and Average-Case Complexity Via Uniform Reductions, Computational Complexity, v.16 n.4, p.331-364, December 2007

complexity classes;autoreducibility;completeness;coherence

352775

Self-Testing without the Generator Bottleneck.

Suppose P is a program designed to compute a function f defined on a group G. The task of self-testing P, that is, testing if P computes f correctly on most inputs, usually involves testing explicitly if P computes f correctly on every generator of G. In the case of multivariate functions, the number of generators, and hence the number of such tests, becomes prohibitively large. We refer to this problem as the generator bottleneck. We develop a technique that can be used to overcome the generator bottleneck for functions that have a certain nice structure, specifically if the relationship between the values of the function on the set of generators is easily checkable. Using our technique, we build the first efficient self-testers for many linear, multilinear, and some nonlinear functions. This includes the FFT, and various polynomial functions. All of the self-testers we present make only O(1) calls to the program that is being tested. As a consequence of our techniques, we also obtain efficient program result-checkers for all these problems.

Introduction . The notions of program result-checking, self-testing, and self-correcting as introduced in [4, 17, 5] are powerful tools for attacking the problem of program correctness. These methods offer both realistic and efficient tools for software verification. Various useful mathematical functions have been shown to have self-testers and self-correctors; some examples can be found in [5, 3, 17, 9, 14, 18, 1, 19, 21, 6]. The theoretical developments in this area are at the heart of the recent breakthrough results on probabilistically checkable proofs and the subsequent results that show non-approximability of hard combinatorial problems. Suppose we are given a program P designed to compute a function f . Informally, a self-tester for f distinguishes the case where P computes f correctly always from the case where P errs frequently. A result-checker for a function f takes as input a program P and an input q to P , and outputs PASS when P correctly computes f always and outputs FAIL if P (q) 6= f(q). Given a program P that computes f correctly on most inputs, a self-corrector for f is a program P sc that uses P as an oracle and computes f correctly on every input with high probability. 1.1. Definitions and Basics. Before we discuss our results, we present the basic definitions of testers, checkers, etc., and state some desirable properties of these programs. Let f be a function on a domain D and let P be a program that purports to compute f . The testers, correctors, and checkers we define are probabilistic programs that take P as an oracle, and in addition, take one or more of the following parameters as input: an accuracy parameter ffl that specifies the conditions that P is expected to This paper unifies the preliminary versions which appeared in the 27th Annual Symposium on Theory of Computing [10] and in the 15th Annual Foundations of Software Technology and Theoretical Computer Science [16]. y Department of Computer Science, Cornell University, Ithaca, NY 14853-7501 (ergun@cs.cornell.edu). This work is partially supported by ONR Young Investigator Award N00014-93-1-0590, the Alfred P. Sloan Research Award, and NSF grant DMI-91157199. z Department of Computer Science, Cornell University, Ithaca, NY 14853-7501 (ravi@cs.cornell.edu). This work is partially supported by ONR Young Investigator Award N00014-93-1-0590, the Alfred P. Sloan Research Award, and NSF grant DMI-91157199. x Department of Computer Science, University of Houston, Houston, Most of this work performed while the author was at SUNY/Buffalo, supported in part by K. Regan's NSF grant CCR-9409104. UN, S. R. KUMAR, AND D. SIVAKUMAR meet, and a confidence parameter ae that is an upper bound on the probability that the tester/corrector/checker fails to do its job. The following definitions formalize the notions of self-tester [5], self-corrector [5, 17], and result-checker [4]. Definition 1.1 (Self-Tester). An ffl-self-tester for f is a probabilistic oracle program T that, given ae ? 0, satisfies the following conditions: ffl Pr x2D [P Definition 1.2 (Self-Corrector). An ffl-self-corrector for f is a probabilistic oracle program P sc that, given any input y, and ae ? 0, satisfies the following condition: ffl Pr x2D [P Definition 1.3 (Result-Checker). A checker (or result-checker) for f is a probabilistic oracle program C that, given an input y and ae ? 0, satisfies the following conditions: ffl Pr x2D [P outputs PASS, and We now list three important properties that are required of self-testers, self- correctors, and result-checkers. For definiteness, we state these for the case of self- testers. First, the self-tester T should be computationally different from and more efficient than any program that computes f [4]. This restriction ensures that T does not implement the obvious algorithm to compute f (and hence could harbor the same set of bugs, or be computationally inefficient). Furthermore, this ensures that the running time of T is asymptotically better than the running time of the best known algorithm for f . The second important property required of T is that it should not require the knowledge of too many correct values of f . In particular, this rules out the possibility that T merely keeps a large table of the correct values of f for all inputs. The third important property required of a self-tester is efficiency : an efficient self-tester should only make O(1=ffl; lg(1=ae)) calls to P . For constant ffl and ae, an efficient self-tester makes only O(1) calls to the program. (In the rest of the paper, we often write O(1) as a shorthand for O(1=ffl; lg(1=ae)), particularly when discussing the dependence on other parameters of interest.) The following well-known lemma summarizes some relationships between the notions of self-testers, self-correctors, and result-checkers. For the reader's convenience, we sketch the idea of the proof of this lemma, suppressing the details of the accuracy and confidence parameters. Lemma 1.4 ([5]). (a) If f has a self-tester and a self-corrector that make O(1) calls to the program, then f has a result-checker that makes O(1) calls to the program. (b) If f has a result-checker, then it has a self-tester. Proof. (Sketch) For part (a), suppose that f has a self-tester and a self-corrector. Given an input y and oracle access to a program P , first self-test P to ensure that it doesn't err too often. If the self-tester finds P to be too erroneous, output FAIL. Otherwise, compute f(y) by using the self-corrector for P sc and the program P , and output PASS iff P Clearly a perfect program always passes. Suppose P (y) 6= f(y). Then one of the following two cases must occur. The program is too erroneous, in which case the self-tester, and hence the checker, outputs FAIL. The program is not too erroneous, in which case the self-corrector computes f(y) correctly with high probability, so the checker detects that P (y) 6= P sc (y) and outputs FAIL. For part (b), suppose that f has a result-checker. By using the result-checker to test if P randomly chosen inputs x, the fraction of inputs x for which P (x) 6= f(x) can be estimated. Output PASS iff this fraction is less than ffl. A useful tool in constructing self-correctors is the notion of random self-reducibility . The fine details of this notion are beyond the scope of this paper, and we refer the reader to the papers [3, 17] (see also the survey paper [11]). Informally, a function f is randomly self-reducible if evaluation of f on an input can be reduced efficiently to the evaluation of f on one or more random inputs. For a quick example, note that linear functions are randomly self-reducible: to compute f(x), it suffices to pick a random r compute f(x + r) and f(r), and finally obtain All functions that we consider in this paper are efficiently randomly self-reducible; therefore, whenever required, we will always assume that efficient self-correction is possible. 1.2. Building Self-Testers using Properties. The process of self-testing whether a program P computes a function f correctly on most inputs is usually a two-step strategy. First perform some tests to verify that P agrees on most inputs with a function g that belongs to a certain class F of functions that contains f . Then perform some additional tests to verify that the function g is, in fact, the intended function f . The standard way to test whether P agrees with some function in a class F of functions is based on the notion of a robust property . Informally, property I is said to be a robust characterization of a function family F if the following two conditions hold: (1) every f 2 F satisfies I, and (2) if P is a function (program) that satisfies I for most inputs, then P must agree with some g 2 F on most inputs. For example, Blum, Luby, and Rubinfeld [5] establish that the property of linearity serves as a robust property for the class of all linear functions, and use this to build self-testers for linear functions. This generic technique was first formalized in [19]. (Robust Property). A property is a predicate I f property I f (~x) is (ffl; ffi)-robust for a class of functions F over a domain D, if it satisfies the following conditions: \Theta ffl If a function (program) P satisfies Pr ~x2D k [I P there is a function g such that that is, (9g 2 F) such that P agrees with g on all but ffi fraction of inputs. We now outline the process of building self-testers using robust properties (cf. [5]). Let D be a (finite) group with generators e some class of functions from D into some range R. Further assume that the functions in F possess the property of random self-reducibility, and can hence be self-corrected efficiently. Suppose P is a program that purports to compute a specific function f 2 F . Let I f (~x) be a robust property that characterizes F . As mentioned earlier, the process of building self-testers is a two-step process. In the first step, we will ensure that that the program P agrees with some function 2 F on most inputs. To do this, we will use the fact that I f is a robust property that characterizes F . Specifically, the self-tester will estimate the fraction of k-tuples holds. If this fraction is at least 1 \Gamma ffl, then by the robustness of I f , it follows that there is some g 2 F that agrees with P on all but ffi fraction of D. The required estimation can be carried out by random sampling of ~x and testing the property I f . UN, S. R. KUMAR, AND D. SIVAKUMAR The next step is to verify that the function g is the same as the function f that P purports to compute. This is achieved by testing that g(e i generator of the group D. If this is true, then by an easy induction it would follow that g j f . An important point to be mentioned here is that the self-tester has access only to P and not to g; the function g is only guaranteed to exist. Nevertheless, the required values of g may be obtained by using a self-corrected version P sc of P . Another point worth mentioning is that to carry out this step, the self-tester needs to know the values of f on every generator of D. 1.3. The Generator Bottleneck. An immediate application of the basic method outlined above to functions whose domains are vector spaces of large dimension suffers from a major efficiency drawback. For example, if the inputs to the function f are n-dimensional vectors (or n \Theta n matrices), then the number of generators of the domain is n (resp. n 2 ). The straightforward approach of exhaustively testing if P sc agrees with f on each generator by making n (resp. furthermore, the self-tester built this approach requires the knowledge of the correct value of f on n (resp. is large, this makes the overhead in the self-testing process too high. This issue is called the generator bottleneck problem. In this paper, we address the generator bottleneck problem, and solve it for a fairly large class of functions that satisfy some nice structural properties. The self- testers that we build are not only useful in themselves, but are also useful in building efficient result-checkers, which are important for practical applications. 1.4. Our Results. We present a fairly general method of overcoming the generator bottleneck and testing multivariate functions by making only O(1) calls to the program being tested. First we investigate the problem of multivariate linear functions (i.e., the functions f satisfying We show a general technique that can be applied in a natural vector space setting. The main idea is to obtain an easy and uniform way of "generating" all generators from a single generator. Using this idea, we give a simple and powerful condition for a linear function f to be efficiently self-testable on a large vector space. We then apply this scheme to obtain very efficient self-testers for many functions. This includes polynomial differentiation (of arbitrary order), polynomial integration, polynomial "mod" function, etc. We also obtain the first efficient self-tester for Fourier transforms. We then extend this method to the case of multilinear functions (i.e., functions f that are linear in each variable when the other variables are fixed). We build an efficient tester for polynomial multiplication as a consequence. Another application we give is for large finite fields: we show that multilinear functions over finite field extensions of dimension n can be efficiently self-tested with O(1) calls, independent of the dimension n. We also provide a new efficient self-tester for matrix multiplication. We next extend the result to some nonlinear functions. We give self-testers for exponentiation functions that avoid the generator bottleneck. For example, consider the function that computes the square of a polynomial over a finite field: Here we do not have the linearity property that is crucial in the proof for the linear functions. Instead, we use the fact that the Lagrange interpolation identity (cf. Fact 4.1) for polynomials gives a robust characterization. We exhibit a self-tester for the function that makes O(d) calls to the program being tested. Extending the technique when f is a constant degree exponentiation to the case when f is a constant degree polynomial (eg., is a polynomial over a finite field) is much harder. First we show a reduction from multiplication to the computation of low-degree polynomials. Using this reduction and the notion of a result-checker, we construct a self-tester for degree d polynomials over finite field extensions of dimension n that make O(2 d ) calls to the program being tested. 1.5. Related Work. One method that has been used to get around the generator bottleneck has been to exploit the property of downward self-reducibility [5]. The self-testers that use this property, however, have to make\Omega\Gamma383 n) calls to the program depending on the way the problem decomposes into smaller problems. For instance, a tester for the permanent function of n \Theta n matrices makes O(n) calls to the program, whereas a tester for polynomial multiplication that uses similar principles makes O(log n) calls. In [5] a bootstrap tester for polynomial multiplication that makes O(log n) calls to the program being tested is given. It is already known that matrix multiplication can be tested (without any calls to the program) using a result- checker due to Freivalds [13]. The idea of Freivalds' matrix multiplication checker can also be adapted to build testers for polynomial multiplication that make no calls to the program being tested. This approach, however, requires the underlying field to be large (have at least (2 is the degree of the polynomials being multiplied, and fl is a positive constant). Moreover, this scheme requires the tester to perform polynomial evaluations, whereas ours does not. For Fourier transforms, a different result-checker that uses preprocessing has been given independently in [6]. A Useful Fact. The following fact, a variant of the well-known Chernoff-Hoeffding bounds, is often very useful in obtaining error-bounds in sampling 0/1 random variables [15]: Fact 1.6. Let Y independently and identically distributed 0=1 random variables with means -. Let ' - 2. If N - (1=-)(4 e Organization of the Paper. Section 2 discusses the scheme for linear functions over vector spaces; x3 extends the scheme for multilinear functions; x4 outlines the approach for non-linear functions. 2. Linear Functions over Vector Spaces. In this section, we address the problem of self-testing linear functions on a vector space without the generator bot- tleneck. We demonstrate a general technique to self-test without the generator bottleneck and provide several interesting applications of our technique. Definitions. Let V be a vector space of finite dimension n over a field K , and let f be a function from V into a ring R. We are interested in building a self-tester for the case where f(\Delta) is a linear function, that is, f(cff and c 2 K . For the unit vector that has a 1 in the i-th position and 0's in the other positions. The vectors e 1 a collection of basis vectors that span V . Viewed as an Abelian group under vector addition, V is generated by e We assume that the field K is finite, since it is not clear how to choose a random element from an infinite field. The property of linearity I f (ff; fi) was shown to be robust in [5]. Using this and the generic construction of self-testers from robust properties, one obtains the following self-tester for the function f : 6 F. ERG - UN, S. R. KUMAR, AND D. SIVAKUMAR Property Test: Repeat O( 1 ae ) times Pick ff; fi 2R V Verify Reject if the test fails Generator Tests: For Verify P sc If P passes the Property Test then we are guaranteed the existence of a linear function g that is close to P . There are, however, two problems with the Generator Tests: one is that the self-tester is inefficient-if the inputs are vectors of size n, the self-tester makes O(n) calls to the program, which is not desirable. Secondly, the self-tester needs to know the correct value of f on n different points, which is also undesirable. Our primary interest is to avoid this generator bottleneck and solve both of the problems mentioned. The key idea is to find an easy and uniform way that "converts" one generator into the next generator. We illustrate this idea through the following example. Example. Let Pn denote the additive group of all degree n polynomials over a field K . The elements multiplying any generator x k by x gives the next generator x k+1 . For a polynomial q 2 Pn and a scalar c 2 K , let denote the function that evaluates q(c). Clearly E c is linear and satisfies the simple relation E c Suppose P is a program that purports to compute and assume that P has passed Property Test given above. Then we know by robustness of linearity that there is a linear function g that agrees with P on most inputs. Note that g can be computed correctly with high probability via the self- corrector (which are easy to construct for linear functions [5]). Now, rather than verify that g(x k of Pn , we may instead verify that g satisfies the property By an easy induction, this implies that g agrees with E c at all the generators. By linearity of g, it follows that g agrees with E c on all inputs. We are now faced with the task of verifying is too expensive to be tried explicitly and exhaustively. Instead, we prove that it suffices to check with O(1) tests that almost everywhere that we look at. That is, pick many random q 2 Pn , ask the program P sc to compute the values of g(q) and g(xq) and cross-check that holds. In other words, we prove that the property J g (q) j robust (in a restricted sense), under the assumption that g is linear. (In its most general interpretation, robustness guarantees the existence of h that satisfies and that agrees with g on a large fraction of inputs. We actually show that h j g, hence the "restricted sense.") Notice that the number of points on which the self-tester needs to know the value of f is just one, in contrast to n as in the original approach of [5]. Generalization via the Basis Rotation Function '. We note that this idea has a natural generalization to vector spaces. Let ' denote the basis rotation function, i.e., the linear operator on a vector space V that "rotates" the coordinate axes that span . ', which can be viewed as a matrix, defines a one-to-one correspondence from the set of basis vectors to itself: for every i, '(e i . The computational payoff is achieved when there is a simple relation between f(ff) and f('(ff)) for all vectors specifically, we show that the generator bottleneck can be avoided if there is an easily computable function h ';f such that for all ff 2 V . (For instance, for polynomial evaluation E c , From here on, when obvious, we drop the suffix f and simply denote h ';f as h ' . If the function f is linear, the linearity of ' implies that h ' is linear in its second argument in the following sense: h ' (ff What is more important is that h ' be easy to compute, given just ff and f(ff). Using this scheme, we show that many natural functions f have a suitable candidate for h ' . The Generator Tests of [5] can now be replaced by: Basis Test: Verify P sc Inductive Test: Repeat O( 1 ae ) times Pick ff 2R V Verify P sc Reject if the test fails The following theorem proves that this replacement is valid: Theorem 2.1. Suppose f is a linear function from the vector space V into a ring R, and suppose P is a program for f . (a) Let ffl ! 1=2, and suppose P satisfies the following condition: Then the function g defined by is a linear function on V , and g agrees with P on at least 1 \Gamma 2ffl fraction of the inputs. (b) Furthermore, suppose h ' (ff; satisfies the following conditions: (3) Pr ff2V [g('(ff)) 6= h ' (ff; g(ff))] - ffl, where ff is such that '(ff) is defined. Remarks. The above theorem merely lists a set of properties. The fact that this set yields a self-tester is presented in Theorem 2.2. Note that hypotheses (1), (2), and (3) above are conditions on P and g, not tests performed by a self-tester. Proof. The proof that the function g is linear and P sc computes g (with high probability) is due to [5]. For the rest of this proof, we will assume that g is linear and that it satisfies conditions (2) and (3) above. We first argue that it suffices to prove that if the conditions hold, then for every agrees with f on the first basis vector. For i ? 1, the basis vector e i can be obtained by '(e would follow that g computes f correctly on all the basis vectors. Finally, since g is linear, it computes f correctly on all of V , since the vectors in V are just linear combinations of the basis vectors. Now we show that condition (3) implies that 8ff 2 V , an arbitrary element ff 2 V . We will show that the probability over a random UN, S. R. KUMAR, AND D. SIVAKUMAR that positive. Since the equality is independent of fi and holds with nonzero probability, it must be true with probability 1. Now Pr \Theta The first equality in the above is just rewriting. The second equality follows from the linearity of '. The third equality follows from the fact that g is linear. If the random variable fi is distributed uniformly in V , the random variables fi and ff \Gamma fi are distributed identically and uniformly in V . Therefore, by the assumption that g satisfies condition (3), the fourth equality fails with probability at most 2ffl. The fifth equality uses the fact that h ' is linear, and the last equality uses the fact that g is linear. The foregoing theorem shows that if P (and g) satisfy certain conditions, then g, which can be computed using P , is identically equal to the function f . The self-tester comprises the following tests: Linearity Test, Basis Test, and Inductive Test. Theorem 2.2. For any ae ! 1 and ffl ! 1=2, the above three tests comprise a 2ffl-self-tester for f . That is, if a program P computes f correctly on all inputs, then the self-tester outputs PASS with probability 1, and if P computes f incorrectly on more than 2ffl fraction of the inputs, then the self-tester outputs FAIL with probability at least 1 \Gamma ae. Proof. In performing the three tests, the self-tester is essentially estimating the probabilities listed in conditions (1), (2), and (3) of the hypothesis of Theorem 2.1. Note that condition (2) does not involve any probability; rather, the self-tester uses P sc to compute g(e 1 ). By choosing O((1=ffl) log(1=ae)) samples in Linearity Test and Inductive Test and by using the self-corrector with confidence parameter ae=3 in Basis Test the self-tester ensures that its confidence in checking each condition is at least 1 \Gamma (ae=3). correctly, the tester always outputs PASS. Con- versely, suppose the tester outputs PASS. Then with probability ae, the hypotheses of Theorem 2.1 are true. By the conclusion of Theorem 2.1, it follows that a function g that is identical to f exists and that g equals P on at least 1 \Gamma 2ffl fraction of the inputs. 2.1. Applications. We present some applications of Theorem 2.2. We remind the reader that a linear function f on a vector space V is efficiently self-testable without the generator bottleneck if there is a (linear) function h ' that is easily computable and that satisfies In each of our applications f , we show that a suitable function exists that satisfies the above condi- tions. Recall the example of the polynomial evaluation function E c the identity E c holds; in the applications below, we will only establish similar relationships. Also, for the sake of simplicity, we do not give all the technical parameters required; these can be computed by routine calculations following the proofs of the theorems in the last section. Our applications concern linear functions of polynomials. We obtain self-testers for polynomial evaluation, Fourier transforms, polynomial differentiation, polynomial integration, and the mod function of polynomials. Moreover, the vector space setting lets us state some of these results in terms of the matrices that compute linear transforms of vector spaces. Let Pn ' K [x] denote the group of polynomials in x of degree - n over a field K . The group Pn forms a vector space under usual polynomial addition and scalar multiplication by elements from K . The polynomials polynomial has the vector representation (q basis rotation function ' in this case is just multiplication by x, thus that multiplying q by x results in a polynomial of degree n+ 1. To handle this minor detail, we will assume that the program works over the domain Pn+1 and we conclude correctness over Pn . Polynomial Evaluation. For any c 2 K , let E c (q) denote, as described before, the function that returns the value q(c). This function is linear. Moreover, the relation between E c (xq) and E c (q) is simple and linear: E c To self-test a program P that claims to compute E c , the Inductive Test is simply to choose many random q's, and verify that P sc holds. Operators and the Discrete Fourier Transform. If are distinct elements of K , then one may wish to evaluate a polynomial q 2 Pn simultaneously on all points. The ideas for E c extend easily to this case, for any u 2 K , and these relations hold simultaneously. Let ! be a principal (n 1)-st root of unity in K . The operation of converting a polynomial from its coefficient representation to pointwise evaluation at the powers of ! is known as the Discrete Fourier Transform (DFT). DFT has many fundamental applications that include fast multiplication of integers and polynomials. With our nota- tion, the DFT of a polynomial q 2 Pn is simply F The DFT F is linear, and F that here the function h is really n coordinate functions h n. The self-tester will simply choose q's randomly, request the program to compute F (q) and F (xq), and verify for each holds. This suggests the following generalization (for the case of arbitrary vector spaces). Simultaneous evaluation of a polynomial at d corresponds to multiplying the vector p by a . The ideas used to test simultaneous evaluation of polynomials and the DFT extend to give a self-tester for any linear transform that is represented by a Vandermonde matrix. The matrix for the DFT can be written as a Vandermonde matrix F , where . The inverse of the DFT, that is, converting a polynomial from pointwise representation to coefficient form, also has a Vandermonde matrix whose entries are given by e \Gammaij . It follows that the inverse Fourier Transform can be self-tested efficiently. Another point worth mentioning here is that in carrying out the Inductive Test the self-tester does not have to compute det F . All it needs to do is verify that for many randomly chosen q's, the identity ( e F (q))[i] holds. Operators in Elementary Jordan Canonical Form. A linear operator M is said to be in elementary Jordan canonical form if all the diagonal entries of M are c for some all the elements to the left of the main diagonal (the first non-principal diagonal in the lower triangle of M) are 1's. It is easy to verify that where M 0 is a matrix that has a \Gamma1 in the top left corner and a 1 in the bottom right UN, S. R. KUMAR, AND D. SIVAKUMAR corner and zeroes elsewhere. Therefore, for every in the vector space, This gives an easy way to implement the Inductive Test in the self-tester. An attempt to extend this to matrices in Jordan canonical form, or even to diagonal matrices, seems not to work. If, however, a diagonal or shifted diagonal matrix has a special structure, then we can obtain self-testers that avoid the generator problem. For example, the matrix corresponding to the differentiation of polynomials has a special structure: it contains the entries n; on the diagonal above the main diagonal. Differentiation and Integration of Polynomials. Differentiation of polynomials is a linear function We have the explicit form for Integration of polynomials is a linear function I : Pn ! Pn+1 . The explicit form for h is Even though this does not readily fit into our framework (since it is not of the form the proof of Theorem 2.1 can be easily modified to handle this case using the linearity of I . For completeness, we spell out the details for the robustness of the Inductive Test which is the only change required. Lemma 2.3. If g : Pn ! Pn+1 is a linear function that satisfies Pr q2Pn [g(xq) 6= Proof. Pr r2Pn \Theta Since the event holds with positive probability and is independent of r, it holds with probability one. Thus we can avoid the generator bottleneck for these functions. This can be considered as a special case of the previous application. Higher Order Differentiation of Polynomials. Let D k denote the k-th differential operator. It is easy to write a recurrence-like identity for D k in terms of D This gives us a self-tester only in the library setting described in [5, 20], where one assumes that there are programs to compute all these differential operators. If we wish to self-test a program that only computes D k and have no library of lower-order differentials, this assumption is not valid. To remedy this, we will use the following lemma, which is proved in the Appendix. Lemma 2.4. If q is a polynomial in x of degree - k, then Using this identity, the self-tester can perform an Inductive Test. The robustness of the Inductive Test can be established as in the proof of Theorem 2.1. For completeness, we outline the key step here. Let c i denote the coefficient of the term in the sum in Lemma 2.4. Thus c (\Gammax) k\Gammai . Lemma 2.5. If g is a linear function that satisfies Pr q2Pn [ Proof. Pr r2Pn \Theta X Here the first equality is rewriting, the second equality holds with probability 1\Gamma2ffl by the assumption that Pr q [ the event holds with positive probability and is independent of r, it holds with probability one. Thus, testing if g satisfies this identity for most q suffices to ensure that g satisfies this identity everywhere. If g does satisfy this identity, then we know the following: To conclude that g j D k by induction, we need to modify Basis Test to test k base cases: if Mod Function. Let ff 2 K [x] be a monic irreducible polynomial. Let M ff (q) denote the mod function with respect to ff, that is, M ff ff. This is a linear function when the addition is interpreted as mod ff addition. Since ff is monic, the degree of M ff (q) is always less than deg ff. If c 2 K is the coefficient of the highest degree term in M ff (q), we have ae xM ff (q) if deg xM ff (q) ! deg ff As before, in testing if a program P computes the function M ff , step (3) of the self-tester is to choose many q's at random, compute P sc (q) and P sc (xq), and verify that one of the identities P sc holds (depending on the degree of q). 3. Multilinear Functions. In this section we extend the ideas in x2 to multilinear functions. A k-variate function f is called k-linear if it is linear in each of its variables when the other variables are fixed, i.e., f(ff Our main motivating example for a multilinear function is polynomial multiplication which is bilinear. Note that the domain of f is generated by n 2 generators of the form (i.e., pairs of generators of suppose we wish to test P that purports to compute f . The naive approach would require doing the Generator Tests at these n 2 generators. This requires O(n 2 ) calls to P , rendering the self-tester highly inefficient. Blum, Luby, and Rubinfeld [5] give a more efficient bootstrap self-tester that makes O(log O(k) n) calls to P . It can be seen that for general k-linear functions, their method can be extended to yield a tester that makes O(log O(k) n) calls to P . (In our context, it is allowable to think of k as a constant since changing k results in an entirely different function f .) We are interested in reducing the number of calls to P with respect to UN, S. R. KUMAR, AND D. SIVAKUMAR the problem size n for a specific function f . The complexity of the tester we present here is independent of n, and the self-tester is required to know the correct value of f at only one point. As in the previous section, our result applies to many general multilinear functions over large vector spaces. As before, we define a set of properties depending on f , that, if satisfied by P , would necessarily imply that P must be the same as the particular multilinear function f . For simplicity, we present the following theorem for f that is bilinear. This is an analog of Theorem 2.1 for multilinear functions. Theorem 3.1. Suppose f is a bilinear function from V 2 into a ring R, and suppose P is a program for f . (a) Let ffl ! 1=4, and suppose P satisfies the following condition: Then the function g defined by g(ff is a bilinear function on V 2 , and g agrees with P on at least 1 \Gamma 2ffl fraction of the inputs. (b) Furthermore, suppose g satisfies the following conditions: Proof. A simple extension of the proof in [5] shows that g is bilinear. (Better bounds on ffl via a different test can be obtained by appealing to [2].) As in the proof of Theorem 2.1, it suffices to show that given the three conditions, a stronger version of condition (3) holds: g('(ff 1 h (2) . With the addition of this last property, it can be shown that g j f . Taking condition (2) that g(e 1 as the base case and inducting by obtaining (e via an application of ' to either generator for all 1 be shown that g(e i bases elements This, combined with the bilinearity property of g, implies the correctness of g on every input. Now we proceed to show the required intermediate result that given conditions (1) and (2), g satisfies the stronger version of condition (3) that we require above: Pr \Theta h (2) The first equality is a rewriting of terms. Multilinearity of g implies the second and third equalities. If the probability Pr[g(ff the fourth equality fails with probability less than 4ffl. The rest of the equality follows from the multilinearity of h (2) ' and g. If ffl ! 1=4, this probability is nonzero. Since the first and last terms are independent of are equal with nonzero probability, the result follows. A similar approach works for h (1) ' as well. Multilinearity Test: Repeat O( 1 ae ) times Pick Verify Reject if the test fails Basis Test: Verify P sc Inductive Test: Repeat O( 1 ae ) times Pick Verify Verify Reject if the test fails Note that in the latter two tests we use a self-corrected version P sc of P . The notion of self-correctors for multilinear functions over vector spaces is implied by random self-reducibility. It is easy to see that Theorem 3.1 extends to an arbitrary k-linear function so long as ffl ! 1=2 k . Thus, we obtain the following theorem whose proof mirrors that of Theorem 2.2. Theorem 3.2. If f is a k-variate linear function, then for any ae ! 1 and , the above three tests comprise a 2 k ffl-self-tester for f that succeeds with probability at least 1 \Gamma ae. 3.1. Applications. Let q 1 ; q 2 denote polynomials in x. The function M(q that multiplies two polynomials is symmetric and linear in each variable. Moreover, since polynomial multiplication has an efficient self-tester. An interesting application of polynomial multiplication, together with the mod function described in x2.1, is the following. It is well-known that a degree n (finite) extension K of a finite field F is isomorphic to the field F[x]=(ff), where ff is an irreducible polynomial of degree n over F. Under this isomorphism, each element of K is viewed as a polynomial of degree - n over F, addition of two elements is just their sum as polynomials, and multiplication of q by q 1 q 2 mod ff. It follows that field arithmetic (addition and multiplication) in finite extensions of a finite field can be self-tested without the generator bottleneck, that is, the number of calls made to the program being tested is independent of the degree of the field extension. 3.2. Matrix Multiplication. Let Mn denote the algebra of n \Theta n matrices over Mn !Mn denote matrix multiplication. Matrix multiplication 14 F. ERG - UN, S. R. KUMAR, AND D. SIVAKUMAR is a bilinear function; however, since it is a matrix operation rather than a vector oper- ation, it requires a slightly different treatment from the general multilinear functions. Mn , viewed as an additive group, has n 2 generators; one possible set of generators is where each generator E i;j is a matrix that has a 1 in position We note that any generator E i;j can be converted into any other generator E k;' via a sequence of horizontal and vertical rotations obtained by multiplications by the special permutation matrix \Pi: The rotation operations correspond to the ' operator of the model for multilinear func- tions. There are, however, two different kinds of rotations-horizontal and vertical- due to the two-dimensional nature of the input, and the function h defining the behavior of the function with respect to these rotations is not always easily computable short of actually performing a matrix multiplication. We therefore exploit some additional properties of the problem to come up with a set of conditions that are sufficient for P to be computing matrix multiplication f . Let M 0 n denote the subgroup of Mn that contains only matrices with columns all-zero. Theorem 3.3. Let P be a program for f and ffl ! 1=8. (a) Suppose P satisfies the following: ffl. Then the function g defined by g(X; Y bilinear function on M 2 n , and g agrees with P on at least 1 \Gamma 2ffl fraction of the inputs. (b) Furthermore, suppose g satisfies the following conditions: To be able to prove this theorem, we first need to show that the conditions recounted have stronger implications than their statements. Then we will show that these strengthened versions of the conditions imply Theorem 3.3. First, we show that condition (2) implies a stronger version of itself: Lemma 3.4. If condition (2) in Theorem 3.3 holds then g(E Proof. We in fact show something stronger. We show that g(X; Y all Pr \Theta g(X; Y The second equality holds with probability by condition (2). All the rest hold from linearity of g and f . The result follows since ffl ! 1=4. The lemma follows since n . An immediate adaptation of the proof of Lemma 3.4 can be used to extend condition (3) to hold for all inputs. Next, we show that the linearity of g makes it possible to conclude from hypothesis (4) that g is associative. Lemma 3.5. If condition (4) in Theorem 3.3 holds then g is always associative. Proof. Pr \Theta The first equality holds from the linearity of g after expanding X;Y; Z as respectively. The second one is true by the condition 8ffl. The last one is a recombination of terms using linearity. We now have the tools to prove Theorem 3.3. Proof. The bilinearity of g follows from the proof of Theorem 3.1. From Lemmas 3.4 and 3.5, we have that condition (2) can be extended such that conditions (3) and (4) on hold for all inputs. To show that these properties are sufficient to identify g as matrix multiplication, note that from the multilinearity of g we can write, 1-i;j-n 1-k;'-n 1-i;j;k;'-n If implies that g is the same as f . Now, using our assumptions, we proceed to show the former holds: The first equality is just a rewriting of the two generators in terms of other generators. The second one follows from the strengthening of condition (3) that g computes f whenever one of its arguments is equal to a power of \Pi. The third one follows from associativity of g, and the fourth one holds because g is the same as f when the first input is a power of \Pi and because of rewriting of E k+j \Gamma2;' . The fifth equality is true because g computes f correctly when its first argument is E i;1 (as the consequence of condition (2), see Lemma 3.4). The last one is a rewriting of the previous equality, using the associativity of multiplication. Therefore, g is the same function as f . We now present the test for associativity: UN, S. R. KUMAR, AND D. SIVAKUMAR Associativity Test: Repeat O( 1 ae ) times Pick X;Y; Z 2R Mn Verify P sc (X; P sc (Y; Reject if the test fails A self-tester can be built by testing conditions (1), (2), (3), and (4), which correspond to the Property Test the Basis Test, the Inductive Test and the Associativity Test respectively. Note that testing conditions (2) and (3) involve knowing the value of f at random inputs. These inputs, however, come from a restricted subspace which makes it possible to compute f both easily and efficiently. The following theorem is immediate. Theorem 3.6. For any ae ! 1 and ffl ! 1=8, there is an ffl-self-tester for matrix multiplication that succeeds with probability at least 1 \Gamma ae. 4. Nonlinear Functions. In this section, we consider nonlinear functions. Specifically, we deal with exponentiation and constant degree polynomials in the ring of polynomials over the finite fields Z p . It is obvious that exponentiation and constant degree polynomials are clearly defined over this ring. 4.1. Constant Degree Exponentiation. We first consider the function q d for some d (that is, raising a polynomial to the d-th power). Suppose a program P claims to perform this exponentiation for all degree n polynomials q 2 Pn ' K [x]. Using the low-degree test of Rubinfeld and Sudan [19] (see also [14]) we can first test if the function computed by P is close to some degree d polynomial g. As before, using the self-corrected version P sc of P , we can also verify that g(e 1 The induction identity applies, and one can test whether P satisfies this property on most inputs. Now it remains to show that this implies We follow a strategy similar to the case of linear functions, this time using the Lagrange interpolation formula as the robust property that identifies a degree d polynomial. We note that this idea is similar to the use of the interpolation formula by Gemmell, et al. [14], which extends the [5] result from linear functions to low-degree polynomials. Before proceeding with the proof, we state the following fact concerning the Lagrange interpolation identity. Fact 4.1. Let g be a degree d polynomial. For any q 2 Pn , if are distinct elements of Pn , Y ; and also Y The self-tester for comprises the following tests: Degree Test: Verify P is close to a degree d polynomial g (low-degree test) Reject if the test fails Basis Test: Verify P sc Inductive Test: Repeat O( 1 ae ) times Pick ff 2R V Verify P sc Reject if the test fails Let fi denote the probability that the d random choices from the domain produce distinct elements. We will assume that the domain is large enough so that fi is close to 1. Assume that P passes the Degree Test and P sc passes the Basis Test that is, agrees with some degree-d polynomial g on most inputs. We note that the low-degree of test of [19] makes O((1=ffl) log(1=ae)) calls to render a decision with confidence only O((1=ffl) log(1=ae)) calls to compute g correctly with probability (ae=3). Below we sketch the proof that if ffl and P sc passes the Inductive Test then g satisfies that so the time taken by the tester is only \Theta(d) (when ae is a constant). Pr a a a Here a The first equality is Fact 4.1, and applies since g has been verified to be a degree d polynomial. Since the q i 's are uniformly and identically distributed, by Inductive Test the second equality fails with probability 1)ffl. The third equality is just rewriting, and the fourth equality is due to Fact 4.1 (the interpolation identity), which can be applied so long as the q i 's are distinct, an event that occurs with probability fi. Since the equality holds independent of q i 's, if ffl holds with probability 1. Theorem 4.2. The function has an O(1=d)-self-tester that makes O(d) queries. 4.2. Constant Degree Polynomials. Next we consider extending the result of x4.1 to arbitrary degree d polynomials f : Pn ! P nd . Clearly the low-degree test and the basis test work as before. The interpolation identity is valid, too. The missing ingredient is the availability of an identity like "f which, as we have shown above, is a robust property that can be efficiently tested. We show how to get UN, S. R. KUMAR, AND D. SIVAKUMAR around this difficulty; this idea is based on a suggestion due to R. Rubinfeld [private communication]. nd is a degree d polynomial (eg., suppose a program P purports to compute f . Our strategy is to design a self-corrector R for P and to then estimate the fraction of inputs q such that P (q) 6= R(q). The difficulty in implementing this idea by directly using the random self-reducibility of f is that the usefulness of the self-corrector (to compute f correctly on every input) depends critically on our ability to certify that P is correct on most inputs. Since checking whether P is correct on most inputs is precisely the task of self-testing, we seem to be going in cycles. To circumvent this problem, we will design an intermediate multiplication program Q that uses P as an oracle. To design the program Q, we prove the following technical lemma that helps us express d-ary multiplication in terms of f-that is, we establish a reduction from the multilinear function to the nonlinear function f . This reduction is a generalization to degree d of the elementary polarization identity slightly stronger in that it works for arbitrary polynomials of degree d, not just degree-d exponentiation. Lemma 4.3. For x 2 f0; 1g d , let x i denote the i-th bit of x. For any polynomial of degree d, x2f0;1g d Y G d Y Using the reduction given by Lemma 4.3, we will show how to construct an ffl-self- tester T for f (for following the outline sketched. (1) First we build a program Q that performs c-ary multiplication for any c - d d, we can simply multiply by extra 1's). The program Q is then self-tested efficiently (without the generator bottleneck) by using a (1=2 d+1 )-self-tester for the d-variate multilinear multiplication function from x3. The number of queries made to P in this process is O(1), where the constant depends on d (the degree of f) but not on n (the dimensionality of the domain of f ). Thus if Q passes this self-testing step, then it computes multiplication correctly on all but 1=2 d+1 fraction of the inputs. If Q fails the self-testing process, the self-tester T rejects. (2) Next we build a reliable program Q sc that self-corrects Q using the random self-reducibility of the multilinear multiplication function. That is, Q sc can be used to compute c-ary multiplication (for any c - d) correctly for every input with probability at least 1 \Gamma ae for any constant ae ? 0. In particular, by making O(2 d log d) calls to (and hence O(2 2d log d) calls to P ), Q sc can be used to compute multiplication correctly for every input with probability at least 1 \Gamma (1=10d). (3) Next, we use Q sc to build the program R that computes f(q) in the straight-forward way by using Q sc to compute the d required multiplications. If Q sc computes multiplication correctly with probability correctly for any input with probability at least 0:9. Finally, T randomly picks O((1=ffl) log(1=ae)) many samples q and checks if outputs PASS iff P all the chosen values of q. It is easy to see that if P computes f correctly on all inputs, the self-tester T will output PASS with probability one. For the converse, suppose that outputs PASS. We will upper bound the probability of this event by ae. Since Q passes the self-testing step (Step (1)), it computes multiplication correctly on all but 1=2 d+1 fraction of the inputs, and therefore, the the use of the self-corrector Q sc as described in Step (2) is justified. This, in turn, implies the guarantee made of R in Step (3): for every input q, with probability at least 0:9. In Step (4), the probability that P (q) 6= R(q) for a single random q is at least (0:9)ffi. The probability that P for every random input q chosen in Step (4) is therefore at most (1 \Gamma (0:9)ffi) Thus the probability that T outputs PASS, given that Pr q [P (q) 6= f(q)] ? ffl, is at most ae. The following theorem is proven, modulo the proof of Lemma 4.3. Theorem 4.4. The function f(q), where f is a polynomial in q 2 Pn of degree d, has an O(1=2 d )-self-tester that makes O(2 d ) queries. Even though our self-tester makes O(2 d ) queries to test degree-d exponentiation, the number of queries is independent of n, the dimensionality of the domain. Thus, our self-tester is attractive if n is large and d is small. In particular, in conjunction with the testers for finite field arithmetic described in x3.1, the self-testers described here help us to efficiently self-test constant degree polynomials on finite field extensions of large dimension. It remains to prove Lemma 4.3. This lemma is a direct corollary of the following lemma, which illustrates a method to express d-ary multiplication in terms of f . The proof of the next lemma is given in the Appendix. Lemma 4.5. Let d be distinct variables. For x 2 f0; 1g d , let x i denote the i-th bit of x. Then x2f0;1g d Y Appendix . Proof of Lemma 2.4 and Lemma 4.5. Lemma 2.4. If q is a polynomial in x of degree - k, then Proof. By induction on k. The base case obviously true. Let k ? 0, we have and since differentiation is linear, we have Since the first term (i = 0) in the first sum vanishes, and since i \Delta , the first sum evaluates to which 1)!q by the inductive hypothesis. Hence it suffices to show that the second sum evaluates to 0. This summation can be written as tity, this sum can be split into the terms xD(q)). The second term can be seen UN, S. R. KUMAR, AND D. SIVAKUMAR to be (\Gammax)k!D(q) using the induction hypothesis. The first and third terms can be combined to obtain k!(xD(q)) again using the induction hypothesis. Thus, the entire Lemma 4.5. Let d be distinct variables. For x 2 f0; 1g d , let x i denote the i-th bit of x. Then x2f0;1g d Y Proof. The proof uses the Fourier Transform on the boolean cube f0; 1g d (us- ing the standard isomorphism between f0; 1g d and Z d denote the space of functions from Z d into C . F is a (finite) vector space of functions of dimension 2 d . Define the inner product between functions f; x f(x)g(x). For define the function - ff : Z d x i and ff i denote the i-th bits, respectively, of x and ff. It is easy to check that whence every - ff is a character of Z d . Furthermore, it is easy to check that h- ff Therefore, the characters - ff form an orthonormal basis of F , and every function f : Z d unique expansion in this basis as ff f ff - ff . This is called the Fourier transform of f , and the coefficients b f ff are called the Fourier coefficients of f ; by the orthonormality of the basis, b h- ff ; fi. An easy property of the Fourier transform is that for ff 6= 0 d , (in fact, this is true for any non-trivial character of any group). For the proof of the lemma, we note that it suffices to prove the lemma for all complex numbers p i . Fix a list of complex numbers define the function . Then the left hand side of the statement of the lemma is just 2 d b Thus: x2Z d/ d Y f ff x x 0-n1 ;:::;n d -c Y 0-n1 ;:::;n d -c Y x d Y 0-n1 ;:::;n d -c Y x d Y The innermost sum is zero if ff every i, that is, n d, this is impossible, since d, then the only way this can happen is if otherwise, for some i, n i ? 1, and since i, it is easy to see that we have 2 d b Acknowledgments . We are very grateful to Ronitt Rubinfeld for her valuable suggestions and guidance. We thank Manuel Blum and Mandar Mitra for useful discussions. We thank Dexter Kozen for his comments. We are grateful to the two anonymous referees for valuable comments that resulted in many improvements to our exposition. The idea of describing the proof of Lemma 4.5 using Fourier transforms is also due to one of the referees. --R Checking approximate computations over the reals Hiding instances in multioracle queries Designing programs that check their work A theory of testing meets a test of theory Reflections on the Pentium division bug Functional Equations and Modeling in Science and Engineer- ing A note on self-testing/correcting methods for trigonometric functions Locally random reductions in interactive complexity theory Approximating clique is almost NP-complete Fast probabilistic algorithms On self-testing without the generator bottleneck New directions in testing Testing polynomial functions efficiently and over rational do- mains Robust characterizations of polynomials with applications to program testing A Mathematical Theory of Self-Checking Robust functional equations with applications to self-testing/ correcting On the role of algebra in the efficient verification of proofs --TR --CTR M. Kiwi, Algebraic testing and weight distributions of codes, Theoretical Computer Science, v.299 n.1-3, p.81-106, Marcos Kiwi , Frdric Magniez , Miklos Santha, Exact and approximate testing/correcting of algebraic functions: a survey, Theoretical aspects of computer science: advanced lectures, Springer-Verlag New York, Inc., New York, NY, 2002

program correctness;generator bottleneck;self-testing

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An approach to safe object sharing.

It is essential for security to be able to isolate mistrusting programs from one another, and to protect the host platform from programs. Isolation is difficult in object-oriented systems because objects can easily become aliased. Aliases that cross program boundaries can allow programs to exchange information without using a system provided interface that could control information exchange. In Java, mistrusting programs are placed in distinct loader spaces but uncontrolled sharing of system classes can still lead to aliases between programs. This paper presents the object spaces protection model for an object-oriented system. The model decomposes an application into a set of spaces, and each object is assigned to one space. All method calls between objects in different spaces are mediated by a security policy. An implementation of the model in Java is presented.

Introduction In the age of Internet programming, the importance of sound security mechanisms for systems was never greater. A host can execute programs from unknown network sources, so it needs to be able to run each program in a distinct protection domain. A program running in a protection domain is prevented from accessing code and data in another domain, or can only do so under the control of a security policy. Domains protect mistrusting programs from each other, and also protect the host environment. There are several aspects to protection domains: access control, resource allocation and control, and safe termination. This paper concentrates on access control. There are several ways to implement protection domains for programs. Operating systems traditionally implement domains using hardware-enforced address spaces. The trend towards portable programs and mobile code has led nonetheless to virtual machines that enforce protection in software. One example in the object-oriented context is a guarded object [10]. In this approach, a guard object maintains a reference to a guarded object; a request to gain access to the guarded object is mediated upon by the guard. Another example of software enforced protection is found in Java [2], where each protection domain possesses its own name space (set of classes and objects) [17]; domains only share basic system classes. Isolation between Java protection domains is enforced by run-time controls: each assignment of an object reference to a variable in another domain is signaled as a type error (ClassCastException). The difficulty in implementing protection domains in an object-oriented context is the ease with which object aliases can be created [12]. An object is aliased if there is at least two other objects that hold a reference to it. Aliasing is difficult to detect, and unexpected aliasing across domains can constitute a storage channel since information that was not intended for external access can be leaked or modified [15]. Aliasing between domains can be avoided by making their object sets disjoint. Any data that needs to be shared between domains is exchanged by value instead of by reference. Partitioning protection domains into disjoint object graphs is cumbersome when an object needs to be accessible to several domains simultaneously. This is especially true if the object is mutable e.g., application environment objects, as this necessitates continuous copies of the object being made and transmitted. A more serious problem is that some system objects must be directly shared by domains, e.g., system-provided communication objects, and even this limited sharing can be enough to create aliases that lead to storage channels. For instance, there is nothing to prevent an error in the code of a guard object from leaking a reference to the guarded object to the outside world. Techniques that control object aliasing are often cited as a means to enforce security by controlling the spread of object references in programs [12]. These techniques are fundamentally software engineering techniques; their goal is to enforce stronger object encapsulation. Security requires more than this for two main reasons. First, aliasing control techniques are often class-based: they aim to prevent all objects of a class from being referenced, though cannot protect selected instances of that class. Second, security constraints are dynamic in nature and aliasing constraints are not. One example is server containment [7], where the goal is to allow a server to process a client request, but after this request, the server must "forget" the references that it holds for objects transmitted by the client. The goal of server containment is to reduce the server's ability to act as a covert channel [15]. The crux of the problem is that once a reference is ob- tained, it can be used to name an object and to invoke methods of that object. We believe that naming and invocation must be separated, thus introducing access control into the language. Least privilege [23] is one example of a system security property that requires access control for its imple- mentation. Least privilege means that a program should be assigned only the minimum rights needed to accomplish its task. Using an example of file system security, least privilege insists that a directory object not be able to gain access to the files it stores [7] in order to minimize the effects of erroneous directory objects. Thus, a directory can name file objects, but can neither modify nor extract their contents. This paper introduces the Object Spaces model for an object-oriented system; Java [2] is chosen as the implementation platform. A space is a lightweight protection domain that houses a set of objects. All method calls between objects of different spaces are mediated by a security policy, though no attempt is made to control the propagation of object references between spaces. The model allows for safe and efficient object sharing. Its efficiency stems from the fact that copy-by-value of object parameters between domains is no longer needed. The model is safe in the sense that if ever an object reference is leaked to a program in another space, invocation of a method using that reference is always mediated by a security policy. In addition, access between objects of different spaces is prohibited by default; a space must be explicitly granted an access right for a space to invoke an object in it, and the owner of this space may at any time revoke that right. The implementation of the model is made over Java and thus no modifications to the Java VM or language are needed. Each application object is implicitly assigned a space object. An object can create an object in another space, though receives an indirect reference to this object: A bridge object is returned that references the new object. Our implementation assures the basic property that access to an object in another space always goes through a bridge object. Bridges contain a security policy that mediate each cross-space method call. A space is a lightweight protection domain as it only models the access control element of a domain; thread management and resource control issues are not treated since these require modifications to the Java virtual machine. The remainder of this paper is organized as follows. Section 2 outlines the object space model and explains its design choices. Section 3 presents a Java API for the model and examples of its use. Section 4 describes the implementation of the model over Java 2 and gives performance results. Section 5 reviews related work and Section 6 concludes. 2 The Object Space Model The basis of the object space model is to separate the ability to name an object from the ability to invoke methods of that object. This is done by partitioning an application's set of objects into several object spaces. A space contains a set of objects, and possibly some children spaces. Every object of an application is inside of exactly one space. An object may invoke any method of any object that resides in the same space. Method invocation between objects of different spaces is mediated by an application-provided security policy. We start in Section 2.1 with an overview of the object space model. A formal definition of the model is given in Section 2.2, and we present some examples of how the model well-known protection problems in Section 2.3. 2.1 Model Overview The set of spaces of an application is created dynami- cally. On application startup, the initial objects occupy a RootSpace. Objects of this space may then create further spaces; these new spaces are owned by the RootSpace. These children spaces may in turn create further spaces. For each space created by an object, the enclosing space of the creator object becomes the owner space of the new space. The space graph is thus a tree under the ownership relation. (a) (b) (c) Figure 1 Ownership and authorization relations on spaces. An object in a space may invoke methods of an object in another space if the second space is owned by the calling object's space. When the calling object is not in the owner space of the called object, then the calling space must have been explicitly granted the right to invoke methods of objects in the second space by the owner of the second space. The set of spaces is organized hierarchically because this models well the control structure of many applications [22]. Typically, a system separates programs into a set of protection domains since they must be protected from each other. A program's components may also need to be isolated from one another, since for example it may use code from different libraries. In the object space model, a program in a space can map its components to distinct spaces. We decided not to include space destruction within the space model, because of the difficultly of implementing this safely and efficiently. A space, like an object, can be removed by the system when other spaces no longer possess a reference to it (or to the objects inside of it). An object may create other objects in its space without any prohibition. A space may also create objects in its children spaces - this is how a space's initial objects are created. A space s1 may not create objects in another space s2 if s1 is not the parent of space s2 . The goal of this restriction is to prevent s1 from inserting a Trojan Horse object into space s2 that tricks s2 into granting s1 a right for s2 . Figure 1 gives an example of how the space graph of an application develops, and how access rights are introduced. Ownership is represented by arrows and access rights are represented by dashed line arrows. When the system starts, RootSpace (represented by space s0 in the figure) is created. In this example, space s0 creates two children spaces, s1 and s2 , and permits objects of space s1 to invoke objects of space s2 . Objects of space s0 can invoke objects of spaces s1 and s2 by default since space s0 is the owning space of s1 and s2 . In Figure 1b, space s1 creates a child space s3 , and grants it a copy of its access right for s2 . Only space s1 possesses a right on S3 though. In Figure 1c, s3 has created a child space s4 , and granted space s2 an access right for s4 . This means that objects of spaces s2 and s3 can call objects in space s4 . The access control model could be seen as introducing programming complexity because an object that possesses a reference is no longer assured that a method call on the referenced object will succeed. This is also the case for applets in Java where calls issued by applets to system objects are mediated by SecurityManager objects that can reject the call. 2.2 Formal Definition The state of the object space protection system is defined by the triple [S, O, R] where S is the set of spaces, O is the ownership relation and R represents the space access rights. Let N denote the set from which space names are generated. S is a subset of names of newly created spaces are taken from NnS. O is a relation on spaces (N \Theta (N [ fnullg)); we use s1 O s2 to mean that space s1 "is owned by" space s2 . The expression s1 O s2 evaluates to true if (s1 , s2 The null value in the definition of O denotes the owner of RootSpace. Finally, R is a relation of type (N \Theta N ); s1 R s2 means that space s1 possesses the right to invoke methods in space s2 . A space always has the right to invoke (objects in) owned spaces: s1 O s2 ) s2 R s1 . Further, an object can always invoke methods on objects in the same space as itself: 8s:S. s R s. We now define the semantics of the object space opera- tions. The grant(s0 , s1 , s2) operation allows (an object of) space s0 to give (objects of) space s1 the right to invoke objects of space s2 . For this operation to succeed, s0 must be the owning space of s2 or, s0 must already have an access right for s2 and be the owner of s1 . The logic behind this is that a parent space decides who can have access to its chil- dren, and a space may always copy a right that it possesses to its children spaces. if (s2 O s0 ) (s0 R s2 " s1 O s0) then [S, O, (R [ f(s1 , s2)g)] else [S, O, R] Access rights between spaces can also be revoked. A space s can revoke the right from any space that possesses a right for a space owned by s. When a space loses a right, then all of its descendant spaces in the space hierarchy implicitly lose the right also. This is because a space might have acquired a right, granted a copy of that right to a child space, and have the child execute code on its behalf that exploits that access right. The operation revoke(s0 , s1 , s2) is used by (an object of) space s0 to remove the right of (objects of) space s1 to access objects of space s2 . The operation is the reverse of grant. D(s) denotes the set of descendant spaces of s in the space tree; as said, these spaces also have their right revoked. D(s"))g. This operation does not allow an owner to lose its right to access a child space. if (s2 O s0) (s1 O s0 " s1 R s2)) then [S, O, (R n f(s1 , s2 else [S, O, R] A space s may create a new space for which it becomes the owner. The new space is given a fresh name s' (s' 62 S). create(s)[S, O, R] " On system startup, the RootSpace s0 is created. The initial system state is thus [fs0 g, f(s0 , null)g, f(s0 , s0)g]. Finally, each time a method call is effected in the system, an access control decision is made using the checkAccess operation to determine if space s0 may invoke methods on objects in space s1 : In Figure 1a, S is fs0 , s1 , s2g, and O is f(s1 , s0 ), (s2 , s0 ), (s0 , null)g. R contains f(s0 , s1 ), (s0 , s2 ), (s1 , s2)g as well as the pairs (s i , s i ) for each s i in S. In Figure 1b, the sets fs3g, f(s3 , s1)g and f(s3 , s2 ), (s1 , s3g are included in the three elements of the system state. In Figure 1c, s3 has created a child space s4 , and granted space s2 an access right for s4 . Thus, s3 R s4 , s2 R s4 , and s4 O s3 . 2.3 Examples We give some brief examples of how the model can be ex- ploited. More detail is added in Section 3 after a Java API for the model has been presented. 2.3.1 Program Isolation Today's computer users cannot realistically trust that the programs they run are bug or virus free. It is crucial then that the host be able to run a non-trusted program in isolation from its services. This means that client programs not be able to communicate with the services, or that they can only do so under the control of a security policy that decides whether each method call from a program to the servers is permitted. The basis to achieving isolation using the object space model is shown in Figure 2a. The Root space creates a space (Server) for the host service objects, and a client space for each of the user programs. The host's security policy is placed in the Root space, and controls whether the user programs may access the services using the grant and revoke operations. The code of this example is given in Section 3.2. In comparison, the ability to isolate programs in this fashion is awkward in Java using loader spaces. In Java, each program is allocated its own class loader [17], which is responsible for loading versions of the classes for the pro- gram. An object instantiated from a class loaded by one loader is considered as possessing a distinct type to objects of the same class loaded by another loader. This means that the assignment of an object reference in one domain to a variable in another domain constitutes a type error (ClassCastException). This model is inconvenient for client-server communication, since parameter objects must be serialized (transferred by value). (b) (a) Root Root Guard G-Obj (c) user Root Client Server client1 client2 Server user packet Figure Examples with spaces. 2.3.2 Guarded Objects A common example of a mechanism for controlled sharing is guarded objects. We consider two different versions of the guarded object notion here: a Java 2 version [10] and a second more traditional version [23]. In Java, a guard object contains a guarded object. On application startup, only the guard object possesses a reference to the guarded object. This object contains a method which executes a checkAccess method that encapsulates a security policy, and returns a reference to the guarded object if checkAccess permits. This mechanism is useful in contexts where a client must authenticate itself to a server before gaining access to server objects, e.g., a file server that authenticates an access (using checkAccess) before returning a reference to a file. An implementation of the more traditional guarded object notion would never return a reference to the guarded object. Rather, each method call to the guarded object would be mediated by the guard, which would then transfer the call if checkAccess permits, and then transfer back the result object. Both of these approaches have weaknesses however. In the Java version, there is no way to revoke a reference to the guarded object once this reference has been copied outside of the guard object. Revocation is needed in practice to confine the spread of access rights in a system. The problem with the traditional notion of guarded object is that a method of the guarded object may return an object that itself contains a reference to the guarded object. This clearly undermines the role of the guard. Figure 2b illustrates how guarded objects are implemented in the object space model. A guard object is placed in its own space (Guard), and the guard creates a child space (G-Obj) in which it instantiates the guarded object. Space Guard controls what other spaces can access G-Obj. To implement the traditional version of the guarded object paradigm, the guard would never grant access to G-Obj to other spaces. To implement the Java version, getObject() of the guard grants the Client space access to G-Obj in the event of checkAccess succeeding. The guard can at any time revoke access, which is something that cannot be done in traditional implementations, so even if a reference is leaked, a grant operation must also be effected for access to the guarded object to be possible. We give the code of this example in Section 3.2. Guards are required in all systems for stronger encapsula- tion. The goal of encapsulation is be able to make an object public - accessible to other programs - without making its component objects directly accessible. This is often a requirement for kernel interface objects, since a serious error could occur if a user program gained hold of a reference to an internal object. An example of this is the security bug that allowed an applet to gain a reference to its list of code signers in JDK1.1.1, which the applet could then modify [4]. By adding signer objects to this list, the applet could inherit the privileges associated with that signer. private Vector /* of object */ signers; public Vector getSigners()- return signers; The JDK actually used an array to represent the signers [4]; arrays require special treatment in the object space model, as will be seen in Section 4. This example also shows that declaring a variable as private is not enough to control access to the object bound to that variable. In the object space model, stronger encapsulation of internal objects (e.g., signers) is achieved by having these objects instantiated in a space (G-Obj) owned by the kernel interface objects' space (Guard). 2.3.3 Server Containment Servers are shared by several client programs. In an environment where mistrusting programs execute, a server should not be allowed to act as a covert channel by holding onto references to objects passed as parameters in a service request and then subvertly passing these references to a third party. Security requires that a server be contained [7] - the server can no longer gain access to any object after the request has been serviced. A schema for this using spaces is shown in Figure 2c. The packet space is for objects that are being passed as parameter. The server is granted access to these objects for the duration of the service call. This access is revoked following the call. Server containment requires the ability to isolate programs from one another, and the ability to revoke rights on spaces. As such, it uses features also present in the preceding two examples. 3 The Object Space API This section describes the classes of the object space system API, and then presents an example of its use. 3.1 Basic Java Classes There are only three classes that an application requires to use the object space model: IOSObject, Space and RemoteSpace. We briefly describe the role of each, before presenting an implementation over Java 2 in Section 4. The class IOSObject 1 describes an object that possesses an attribute Space. This attribute denotes the space that houses the object. Not all objects of an application need inherit from IOSObject; the only requirement is that the first object instantiated within each space be a subclass of IOSObject since in this way, there is at least one object from which others may obtain a pointer to their enclosing Space object. The API of IOSObject is the following: public class IOSObject protected Space mySpace; public IOSObject(); public final Space getSpace(); The getSpace method enables an object to get a handle on its enclosing space from an IOSObject. The Space class represents an object's handle on its enclosing space. Handles on other spaces are instances of the RemoteSpace class. SpaceObject is an empty interface implemented by both Space and RemoteSpace. public final class Space implements SpaceObject public static RemoteSpace createRootSpace(IOSObject iosObj); protected public RemoteSpace createChildSpace(); public void grant( SpaceObject sourceSpace, SpaceObject targetSpace ); public void revoke( SpaceObject sourceSpace, SpaceObject targetSpace ); public Object newInstance( String className, RemoteSpace target ); public RemoteSpace getParent(); protected void setParent(Space parent); static boolean checkAccess( Space protectedObjSpace, "IOS" comes from "Internet Operating System", which is the name of the project in which the space model was developed. Space callerSpace ) Recall that spaces are organized in a hierarchy: the root of the hierarchy is created with the static method createRootSpace. This method returns a RemoteSpace ob- ject, and the system ensures that this method is called only once. The createChildSpace method creates a child space of the invoking object's space. The grant and revoke methods implement the access control commands of the model (see Section 2.2). The space of the object that invokes either operation is the grantor or revoker space of the operation. The newInstance method creates an object within the specified space. This is how objects are initially created inside of a space. The implementation verifies that the class specified extends IOSObject, so that subsequently created objects have a means to obtain a reference to their Space. Further, only a parent space may execute this method. The goal is to prevent spaces injecting malicious code into a space in the aim of forcing that space to execute a grant that would allow the malicious object space gain an access right to the attacked space. The setParent method is executed by the system when initiating a space; the checkAccess method that consults the security policy. These two methods are only used by the object space model implementation. The third of the classes in the object space API is RemoteSpace: public final class RemoteSpace implements SpaceObject public RemoteSpace(Space sp); This represents a handle on another space. The only user-visible (public) operation is the constructor that allows an object to generate a remote space pointer from the pointer to its enclosing space. This enables a space to transfer a pointer to itself to other spaces and thus allow other spaces to grant it access rights. It is important to note that an object can only possess a Space reference to its enclosing space, and never to other spaces. In this way, the system assures that an object in one space does not force another space to grant it an access right since the grant and revoke operations are only defined in Space, meaning that the system can always identify the space of the invoking object and thus authorize the call. Note also that the Space and RemoteSpace classes are final, meaning that a malicious program cannot introduce Trojan Horse versions of these classes into the system. 3.2 Example code extracts The first example continues the program isolation discussion of Section 2.3.1, and is taken from a newspaper system for the production and distribution of articles [19]. Here we concentrate on a program that compiles an article. For security reasons, we wish to isolate this program from the rest of the system - in particularly from the Storage and graphical Editor objects. This requires being able to meditate all method calls between the client program and the services. These security requirements are summarized in Figure 3. The is a typical example of the need to isolate user programs from the rest of the system. Section 4 gives a performance comparison of an implementation of this example using Java loader spaces with copy-by-value semantics, and the object spaces implementation. Root Storage Article client Client Space Service Space Editor Figure 3 The article packager example. In the code below, Root is the application start-up program that creates two object spaces, and instantiates the objects in each domain. This is the only class of the application that uses the object space model API methods. The Editor class uses several Swing components to offer a front-end user this exchanges request messages and events with the client program. public class Root extends IOSObject- public void start()- // Create the client and server Spaces RemoteSpace RemoteSpace // Allocate access rights; // Create the services . newInstance("GUI.Editor", child1); Storage newInstance("Kernel.Storage", child1); // . and create the client client newInstance("Kernel.client", child2); // Start things running Editor ed = E.init(c); A.init();c.init(ed, A); public static void main( String[] args )- RemoteSpace In this example, the application main starts an instance of Root. This creates two child spaces, child1 and child2, grants a right to each space to invoke object methods in the other. An Editor object and a Storage object are created in space child1 and the client program is installed in child2. The editor is given a reference to the client object (so that it can forward events from the GUI interface) and the client is given a reference to the two service objects. The Root class here is almost identical to that used in an implementation of the guarded object model of Figure 2b. A Guard object and client are created in distinct spaces. In the extract of this example below, the guard has a string (of class IOSString) as guarded object. The class IOSString is our own version of String; the motivation for this class is given in Section 4. import InternetOS.*; import InternetOS.lang.*; public class Guard extends IOSObject- IOSString guardedObject; RemoteSpace guardedSpace; public void init() - newInstance("InternetOS.lang.IOSString", guardedObject. set(new IOSObject("The secret text.")); public Object getObject(IOSString password, RemoteSpace caller)- // if checkAccess(password) -. mySpace.grant(caller, guardedSpace); return guardedObject; The Guard object has an init method that is called by the Root. This method creates a child space (guardedSpace), instantiates the guarded object in this space, and initializes it using its set method (defined in IOSString). The role of the guard is to mediate access requests on getObject. A client must furnish a password string and the guard verifies the password using the guard object's checkAccess method. If the check succeeds, the guard grant's access to the client space and returns the object reference. public class client extends IOSObject- Guard G; public void init(Guard G)- G; IOSString new IOSString("This is my password"); IOSString (IOSString)G.getObject(password, new RemoteSpace(mySpace)); System.out.println("String is "); The client is a program that requests access from the guard by supplying the password and a pointer to its space to getObject. 4 The Object Space Implementation In this section we describe the implementation of our model. We first describe the notion of bridge, which is the mechanism that separates spaces at the implementation level. For portability and prototyping reasons, the current implementation is made over the Java 2 platform, so no modifications to the virtual machine or language were made. A future implementation could integrate the model into the JVM; in this way other aspects of protection domains such as resource control and safe termination can also be treated. We begin in Section 4.1 by describing the basic role of bridge objects. Section 4.2 describes how they are interposed on method calls between spaces, and Section 4.3 explains how bridge classes are generated. Section 4.4 describes in more detail how the object space model interacts and in some cases conflicts with features of the Java language. Section 4.5 presents a performance evaluation of the implementation. 4.1 Bridge Objects So far, we have seen that objects belong to spaces and that they interact either locally inside the same space or issue method calls across space boundaries. Interactions between objects of different spaces are allowed only if the security policy permits. To implement the object space model, a bridge object is interposed between a caller and a callee object when these are located in different spaces. If the caller has the authorization to issue the call, then the bridge forwards the call to the callee, otherwise a security violation exception is raised. We call the callee the protected object, since it is protected from external accesses by the bridge, and we use the term possessor to refer to the caller. This is illustrated in Figure 4, where real references are denoted by arrows; the dashed line arrow denotes a protected reference whose use is meditated by a security policy. The security policy is represented by an access matrix accessible to all bridges; this encodes the authorization relation R defined in Section 2.2. Access Matrix bridge protected object possessor Space 1 Figure 4 A bridge object interposed between object spaces. Bridges are hidden from application programmers. They are purely an implementation technique and do not appear in the API. Assuming security permits, a program behaves as if it has a direct reference to objects in remote spaces. The main exception to this rule are array references which always refer to local copies of arrays, even if the entries in an array can refer to remote objects. We return to the question of arrays in Section 4.4. Consider class Root in Section 3.2; its start method contains the call c.init(ed, A) to transfer references for the editor and storage objects to the client program. The three objects are all in a different space to the Root object; the references used are in fact references to bridge objects even though the programmer of the Root class does not see this. Bridges are implemented using instances of Java Bridge classes, where Bridge is an interface that we provide. Each class C has a bridged class BC constructed for it. The interface of BC is the same as that of C. Further, BC is defined as a subclass of C, which makes it possible to use instances of BC (i.e., the bridges) anywhere that instance of C are expected. The role of a bridge is three-fold: 1. It verifies that the caller can issue a call to the protected object. To be more precise, this results in verifying that the space of the caller can access the space of the callee according to the security policy, consulted using the checkAccess method of class Space. This method is shown in Figure 7. 2. It forwards the request from the possessor to the protected object, if the possessor has the right to access the protected object. 3. It ensures that objects exchanged as parameters between the possessor and the protected object do not become directly accessible from outside their spaces. The protection model is broken if an object obtains a direct reference to an object in another space (a reference is direct if no bridge is interposed between the objects). This can happen during a call if the arguments are directly passed to the callee. Therefore, a bridge can be interposed between the callee and the arguments it re- ceives. Similarly, this wrapping can occur on the object returned from the method call on the protected object. To avoid reference leaks exploiting the Java exception mechanism, bridges are also responsible for catching exceptions raised during the execution of the protected object's method, and for throwing bridged versions of the exceptions to the caller. 4.2 Interposition of Bridges Bridges are introduced into the system when an application object creates an object in a child space using the newInstance method. This method is furnished by the system (in Space) and cannot be redefined by users since it is defined in a final class. In addition to creating the required object and assigning it to the space, newInstance creates a bridge for the new object. A reference to this bridge is returned to the object that initiates the object creation, making the new object accessible to its creator only through the bridge. For instance, in the example 3.2, references E, A, c point to a bridge instead of pointing directly to an Editor, a Storage or a client object since these objects are created using newInstance. The other way that bridge objects appear in the system is during cross-domain calls where the need for protection for arguments and returned objects arise. By default, when a reference to an object is passed through a bridge, a bridge object for the referenced object is generated in the destination space. Nevertheless, if a bridge object for the protected object already exists in the destination space, then a reference to this bridge is returned instead of having a new bridge object generated. This is implemented using a map that maps protected object and space pairs to the bridge used by objects in that space to refer to the protected object. An advantage of this solution is that the same bridge is shared among objects of the same space referring to the same protected object. However, if objects reside in different spaces, they cannot share the same bridge. A second advantage of this is that the time needed to consult the bridge cache is inferior to the time needed to generate a new bridge object. A final advantage concerns equality semantics: the == operator applied to two bridge references to the same protected object always evaluates to true. However, there are cases where bridge interposition is not necessary. For instance, if an object creates another object which resides in the same space as its creator, then a direct reference to the new object is allowed. This is the case when the Java new operator is used, i.e., an object created with new belongs to the same space as its creator and direct invocations are allowed. Further, if a bridge receives a bridge reference as an argument to a call and observes that the protected object of that bridge is actually in the destination space, then the protected object reference is returned in place of the bridge. Space 1 O3 Space 3 Space 1 O3 O2 O2 O4 Figure 5 The creation of bridges between spaces. An example of the interposition of bridge objects is shown in Figure 5. Space 1 possesses an object O1 that creates an object O2 in Space 2 using newInstance; a bridge B1 is created for this reference. O1 then passes a reference for itself to O2; B1 detects that this reference is remote and creates a bridge B2. O2 creates an object O3 locally in its space using new, and obtains a direct reference to it. O3 then creates an object O4 in Space 3, and a bridge B3 is created. Finally, O2 passes a reference for O1 to O4 (via O3); a new bridge B4 is created for this that notes that the reference is from Space 3 to Space 1. The only exception to the above is the exchange of RemoteSpace objects. Objects of this type can be freely passed to foreign spaces without bridge intervention. Con- sequently, direct access to these objects is allowed. These objects are used by other spaces for granting or revoking accesses to their children. Allowing direct sharing of RemoteSpace objects does not lead to reference leaks, since no methods or instance fields of RemoteSpace are accessible to user programs, as can be seen from its API. A Space object transmitted through a bridge is converted to a RemoteSpace. This is needed to ensure the invariant that a Space object can only be referenced by an object enclosed by that space. 4.3 Generating Bridge Classes This subsection describes the generation by our system of the bridge class BC which mediates accesses to instances of class C. Bridge generation always starts from a call to the getBridge method of the BridgeFactory class. This method expects three arguments, the object the bridge has to guard (the protected object), the space of the protected object and the space of the possessor. The method getBridge returns a bridge whose class is a subclass of the protected object's class. If the class file of the bridge does not exist at the time the method is called, construction of the class file is started and instantiation of a new object follows. This method is also responsible for the management of the map that caches bridges interposed between a given space and a protected object and returns a cached bridge if another object in the given space refers to the protected object. An outline of the code of class BridgeFactory is shown in Figure 10. Bridge classes are placed in the same package as the object space implementation. Their constructors are protected, which prevents user code from directly creating instances. The main task behind the generation of a bridge is to pro- duce, for each method m defined in class C as well as in its superclasses, a corresponding method mB in BC that implements the expected functionality of the bridge as described in Section 4.1. The structure of each mB method generated for m is uni- form. First, a piece of code is inserted at the beginning of the method to consult the security policy. If the access is granted, a bridge, instead of the argument itself, is passed to the protected object when forwarding the call. This has to be done for each argument (except if the argument is primitive or of class RemoteSpace, in which case the value of the argument is copied) and this ensures that the protected object cannot possess a direct access to arguments. Once the arguments are converted, the method forwards the call to the protected object. If the call returns a non-primitive value, then as for the arguments, a bridge instead of the returned value is returned to the caller object. Figure 9 presents the bridge generated from the user class FileUpdater shown in Figure 8. To avoid exceptions leaking out internal objects, bridges catch exceptions thrown in the protected object and generate a bridge that encapsulates the exception, before throwing this exception back to the caller. The code that checks whether access to the protected object is allowed is performed in the static method checkAccess defined in class Space. This method takes the space of a protected object as well as the space of its caller as input and consults the access matrix stored in the two dimensional array called authorizations for deciding whether access can be granted or not. The code that interposes a bridge between the arguments of the call and the protected object is present in method getBridgeForArg, whereas the code that interposes a bridge between the returned value and the possessor of the bridge is located inside method getBridgeForReturn. Both methods are defined inside class BridgeFactory as shown in Figure 10. Notice that these methods handle several cases; either the argument (respectively the returned object) is a bridge, a RemoteSpace, a Space or an instance of a user class. If it is a bridge, then the object protected by the bridge is extracted and an appropriate bridge is interposed between the protected object and the callee (respectively the caller). public class BridgeInternal Object protectedObj; Space protectedObjSpace; Space callerSpace; //initialize fields. initialize( Object go, Space goSpace, Space pSpace) -. Figure 6 Class BridgeInternal class Space- // the access matrix static boolean[][] authorizations; static boolean checkAccess( Space protectedObjSpace, Space callerSpace )- return Figure 7 Method checkAccess controls access. Each bridge contains a BridgeInternal object. The role of this object is to store the information related to the state of the bridge, i.e, its protected object, the latter's space, and the space of the possessor. The class BridgeInternal is shown in Figure 6. It is not possible to reserve a field for this information inside a user bridge class because the generic methods getBridgeForArg and getBridgeForReturn need to access this information when they receive any bridge as argument or returned object. Soot is the framework for manipulating Java bytecode [24] that we used for generating the bridge classes. public class FileUpdater public File concatFiles(File file1 , File file2) throws FileNotFoundException throw new FileNotFoundException("File Not Found!"); return file1; Figure 8 A user class example 4.4 Caveats for Java This section looks in more detail at the implications of the object space implementation for Java programs. In partic- ular, several features of the language, such as final classes and methods, are incompatible with the implementation ap- proach. Dealing with these issues means imposing restrictions on the classes of objects that can be referenced across space boundaries. Final and private clauses are important software engineering notions for controlling the visibility of classes in an application. For the object space implementation to work, each class C of which an object is transfered through a bridge has a class BC generated that subclasses C. Final classes thus cannot have bridges generated. In the current implementation, the bridge generator complains if an object is passed whose class contains final clauses, though public class FileUpdaterBridge extends FileUpdater implements Bridge - BridgeInternal new BridgeInternal(); FileUpdaterBridge()- FileUpdaterBridge( Object obj, Space protectedObjSpace, Space callerSpace bi.initialize( obj , protectedObjSpace , callerSpace BridgeInternal getBridgeInternal()-return bi; public File concatFiles(File arg1 , File arg2) throws FileNotFoundException - bi.callerSpace try - File File File returnedObj=((FileUpdater)bi.protectedObj). concatFiles( arg1Bridge , arg2Bridge ); return (File)BridgeFactory. catch (FileNotFoundException e) - throw (FileNotFoundException)BridgeFactory. getBridgeForReturn(e , bi); catch (Throwable e) - throw (RuntimeException)BridgeFactory. getBridgeForReturn(e , bi); else throw new AccessException("Unauthorized Call"); Figure 9 Example bridge class generated. this restriction does not apply to java.lang.Object (see be- low). In order to handle final methods and classes, the object space system loader could remove final modifiers from class BridgeFactory - // maps a pair (objectToProtect , callerSpace) to // to the bridge interposed between them Map objAndCallerSpaceToBridge; static Bridge getBridge( Object object , Space protectedObjSpace, Space callerSpace // This method first checks if the map already // contains a bridge interposed // between the object and callerSpace. // If so, it returns the bridge. // If not, it checks whether the bridge's class // file exists. // If the class file does not exist, this method asks Soot to build one. // Finally, it creates and returns a new instance // of the bridge. static Object getBridgeForArg(Object arg, BridgeInternal currentBI) - BridgeInternal // If the call argument is a bridge on a object // located in the same space as the callee, // return a direct reference to the object. argBI.protectedObjSpace == currentBI.protectedObjSpace return argBI.protectedObj; // The call argument is located in another space. // Get a handle on it. return BridgeFactory. argBI.protectedObjSpace , currentBI.protectedObjSpace else if( arg instanceof RemoteSpace required around RemetoSpace return arg; else if( arg instanceof Space // Do not allow transfer of space objects. return new RemoteSpace((Space)arg); else // The call argument lives in the caller's space. // Get a handle on it. return BridgeFactory. currentBI.callerSpace, currentBI.protectedObjSpace // Same idea as getBridgeForArg but this time // the returned object is protected from the caller static Bridge getBridgeForReturn( Object returnedObj, BridgeInternal currentBI )-. Figure class files before linking. To prevent illegal subclassing, the loader must record the final modifiers in each class already loaded, and verify that further classes loaded do not violate final constraints. The loader must also remove private modifiers from classes BC . This rewriting approach was used by loaders in the JavaSeal [6] system to remove catchs of ThreadDeath exceptions, since catching these exceptions would allow an applet to ignore terminate signals from its parent. The re-writing approach does not work for system classes, as these are loaded and linked by the basic system loader. System classes These classes include the java.lang, java.util and java.io classes. The problem with these classes is that they can be final, e.g., java.lang.String, or they contain final methods that cannot be overridden. The class java.lang.Object must be permitted since every class sub-classes it. The only final methods of this class are notifyAll, notify, wait, and getClass. These methods cannot be overridden, and so invocation of these methods on objects cannot be controlled. The former three methods are used for thread synchronization. However, locking is out of the scope of our access control model; it is an issue for a fully-fledged protection domain model but this requires modification to the virtual machine in any case. Concerning the method getClass, a program that calls getClass on a bridge class gets a class object for the bridged object. How- ever, the constructor of a bridge class is protected, so the program can do nothing with the object. Special treatment is also given to system classes like String and Integer which are final classes in Java 2. Our implementation provides tailored versions of these classes to represent strings and integers exchanged across boundaries. The class IOSString for instance is simply a wrapper around a String object, and can be exchanged between spaces. The reader may have noticed the class IOSString in the paper's examples. The object space implementation also provides a bridged class for IOSString. This class contains a copy of the wrapped String object in IOSString, and is used to transfer the value of the string across spaces in the set method. The API of IOSString is given below. The second constructor takes a String object; this allows a space to create an IOSString from a String locally and to transmit that string value to another domain. public class IOSString implements Serializable- protected String myString; public IOSString()-; public IOSString(String s); public IOSString getString(); public void set(IOSString s); public void print(); Lastly, since String is final and cannot have a bridge defined for it, bridge classes define the toString method to return null in order to avoid direct references to Strings in remote domains. In cases where strings need to be exchanged for convenience, like in exceptions for instance, the user class should define a getMessage method that returns an IOSString. Field accesses Access to fields is also a form of inter-object communication and must be controlled for security. The current implementation however does not yet cater for this. A solution would be for the loader to instrument the bytecode with instructions that consult the access matrix before each field access, or for field accesses to be converted into method calls. The former approach is applied to Java in [21]. In the current implementation of bridges, access to fields in remote objects become access to fields in bridges. These fields do not reflect the corresponding fields in the protected object. Static fields and methods Static methods pose two problems. First, they cannot be redefined in subclasses. Sec- ond, objects referenced by static variables could be shared between protection domains without an access control check taking place. In a fully fledged implementation of protection domains, classes should not be shared between domains [6] to avoid undetected sharing between domains. In the object space implementation, the bridge generator signals an error when an it receives an object of a user class that contains static methods. The problem of static variables is looked at in [8]. This proposal strengthens isolation between loader spaces by keeping a different copy of objects referenced by static variables for each copy of the class used by a loader. Unfor- tunately, we cannot use this solution for the object space implementation since classes are shared across domains. Remote Array Figure 11 Treatment of arrays in object space implementation. Arrays Arrays in Java are objects in the sense that methods defined in Object can be executed on array objects. Unfortunately, an array is not an object in the sense that element selection using "[ ]" is not a method call, and this requires special treatment. Our approach is outlined in Figure 11. Whenever a reference to an array object is copied across a space boundary (i.e., through a bridge), the array is copied locally. The copy is even made if the array contains primitive types like int or char. The implication of this approach is that method calls on array objects do not traverse space boundaries and that array selection is done locally. In effect, copy by value is being used for array ob- jects; an array entry can be modified in one space without a corresponding change in another domain. Entries in copied arrays for objects become bridges if not already so. This means that non-array objects are always named using the same bridge object within a space, even though array objects may be copied. If sharing of arrays is required, then the programmer must furnish an array class that has entry selectors as methods. This solution has an interesting repercussion regarding the example of the bug in Java cited in Section 2.3.2. The signers object was in fact represented by an array "Identity signers[]". If the object space model were used to implement this, then a copy of the signer array would be returned to the caller, whose modifications to the array would remain innocuous. Synchronization on objects is intricately influenced by the interposition of bridges between objects. Two objects located in different spaces and willing to synchronize on the same protected object would experience undesired behavior since they are implicitly performing their operations on two different bridges protecting the protected object instead of acting on the protected object itself. This problem arises if synchronized statements are used instead of relying on solutions that exploit synchronized methods. The latter is perfectly valid since bridges forward calls to protected objects and consequently, locking occurs on the protected objects themselves. Native methods can also lead to security flaws since they could be used to leak object references between spaces and there is no way to control this code. Our current implementation for Java does not allow bridges for classes that possess native methods, except for Object's methods, e.g., hashCode. 4.5 Performance evaluation Efficiency is one of the goals of the object space model. In particular, the cost of mediation of inter-space calls by bridge should be generally inferior to the cost of copy-by-value (of which "serialization" is an example) and the exchange of the byte array over a communication channel. We conducted performance measures for the implementation running over SunOS 5.6 on a 333 MHz Ultra-Sparc-IIi processor using Sun's VM for JDK1.2.1. All measures were obtained after averaging over a large number of iterations. One of the basic measures taken was to compare the cost of a method call between protection domains using the space (bridge) model and the loader (Java serialization) model. We also made comparisons with J-Kernel [25]. The latter allows domains to exchange parameters either by using the Java serialization mechanism, or by using a faster serialization tool developed for J-Kernel or by passing capabilities. A J-Kernel capability is an object that denotes an object in a remote domain; this is J-Kernel's equivalent of a bridged object. The table below shows comparisons for: A) a method call with no parameters, B) for a method call with a string as parameter, and C) for a method call with an article object as a parameter. The Article class is used in the application of Section 3.2, and consists of a hash-table of files representing the article contents, as well as strings for the article attributes. Times are shown in micro-seconds. For A, we estimated that a basic method call without arguments (and serialization) was around 5 nano-seconds. A cross-domain call without arguments is faster in our approach than with J-Kernel. For such a basic call, J-Kernel's overhead can mainly be explained by the thread context switch that has to be performed when crossing domains. In our implementation, the only overhead resides in the security policy check required during cross-domain calls. This cost is quite low since this check reduces to a lookup in an access matrix implemented as a static two dimensional array stored in class Space. Even though accessing the matrix is fast, the trade-off is that space required is O(N 2 ) in the number of Spaces present in the system. Mechanisms that use copy for passing parameters are as expected slower that their counterpart that do not (J-Kernel with capability and our object space model). Further, they do not scale well with the size of arguments. The cost of parameter passing with the object space model is approximately two times slower than passing parameters with capabilities in J-Kernel. The overhead comes from the dynamic creation and lookup of bridges in our model. How- ever, this cost comes with a benefit. Our model has stricter controls on access to objects. In J-Kernel, once a capability is released into the environment, it is not possible to control its spread among domains whereas in our model, we can selectively grant or revoke access to certain domains. Space Serial. J-Kernel Serial. fast copy capability C 2.5 363.2 1004.2 587.2 1.4 The figures give an estimate of the basic mechanism. To get a more general overview, we implemented the article packager example of Section 3.2 using both the object space model and the Java loader model. The space implementation was described earlier. In the loader implementation, a class loader object is created to load the client and article classes. The service classes (Storage & Editor) are loaded with the system loader. A communication channel object is installed between the client and service objects for the exchange of serialized messages. The application is highly interactive, so a direct comparison is not obvious; we therefore compared two types of com- munication: the cost of saving an article stored within a program on disk, and the cost of sending an event message from the GUI Editor to the client. In the loader version, the time to save a small article is approximately 5617 micro-seconds; this cost includes the time to serialize the article. In the space version, the figure was slightly less (5520 micro seconds). This also has an article serialization since the article must be serialized to be saved on disk. The figure includes a creation of a bridge object for the article being passed to the Storage object. The time to send a message from the GUI to the program is about 143 micro-seconds in the space implementation. In the loader implementation, this figure is around 1511 micro-seconds due to serialization. The cost of serialization can be reduced by making the classes of the objects exchanged sharable (have them loaded by the system loader). How- ever, the result of this is to weaken isolation because there is greater scope for aliasing between domains. Regarding space usage, a bridge requires 4 words: a reference to a BridgeInternal object which contains 3 words (reference to guarded object, and references to guarded and possessor spaces). A Space object requires 3 words (an internal Integer identifier, a reference for the parent space, a reference to a hash-map object containing the children spaces); the pointer to the access matrix is static, so is shared by all Space objects. If there are N spaces active in a system, then the overhead of a space is N 2 access matrix entries and NM entries in the hash-map maintained by BridgeFactory that maps object and space pairs to bridges. M denotes the number of objects in the space referenced by objects in other spaces. If all spaces contains M objects, then the maximum number of bridges in the system is this represents the case where all objects in all spaces are referenced by objects in all other spaces. The measures were taken for installed bridge classes. In our implementation, a bridge class is generated on the fly if the class cannot be found on disk. This is a costly process. For instance, a bridge for the Editor class takes around 3.67 seconds to generate (due to parsing of the class file). On startup of the article packager application, the root, service and client spaces are created; this necessitates the creation of 10 bridges, which takes around 6.24 seconds. 5 Related Work This section compares our object space model with related work; it is divided into Java related work, and more general work on program security. 5.1 Java Security Java has an advanced security model that includes protection domains, whose design goal was to isolate applets from each other. The basic mechanism used is the class loader. Each applet in Java is assigned its own class loader which loads a distinct and private version of a class for its protection domain [17]. Java possesses the property that a class of one loader has a distinct type to the same class loaded by another loader. Typing is therefore the basis for isolation since creating a reference from one loader space to another is signaled as a type error. A problem with this model is that dynamic typing can violate the property that spaces do not reference each other. This is because all classes loaded by the basic system loader are shared by all other loader spaces - they are never reloaded. The system loader loads all basic classes (e.g., java.lang.*) so sharing between loader spaces is en- demic. This sharing is enough to lead to aliasing between loader spaces. Consider a class Password which is loaded by two loader spaces i and j; the resulting class versions are Password i and Password j . This class implements the interface PasswordID with methods init and value. Suppose that the interface PasswordID is loaded by the system loader. In this case, the following program allows one password to read the value of the other, that is, the password object of loader space (UID 2) can directly invoke the password object in the other space. public final class Password implements PasswordID- private int UID; private PasswordID sister; private String password; public static void main(String[] args) throws Exception- // Create two loader spaces MyLoader new MyLoader(); MyLoader new MyLoader(); // Root leaks references to each space child1.init(1, "hth3tgh3", child2); child2.init(2, "tr54ybb", child1); public void init(int i, String s, PasswordID R)- public String value()- return password; This program starts by creating two loaders of class MyLoader. This loader reads files from a fixed directory. It delegates loading of all basic Java classes and of the PasswordID interface to the parent (system) loader. The program then creates an instance of Password in each loader space (by asking each loader to load and instantiate an instance of the class). The program grants each password a reference for each other. Despite the fact that each domain has a distinct loader, the call on value by the second password on the first succeeds. Loader spaces are used to implement protection domains in several Java-based systems, e.g., [3, 14, 6, 25]. Isolation is obtained only if the shared classes do not make leaks such as that in the above example. In the object space approach, the model at least guarantees that if ever a reference to an object escapes or is leaked to another space, use of that reference is nevertheless mediated by a security policy. The security policy prohibits calls between spaces by default: an access right for a space must be explicitly granted, and this grant can be undone by a subsequent revocation. One advantage of the loader space model over the object space model is that the former allows a program to control the classes that are loaded into its protection domain. This is important for preventing code injection attacks, where malicious code is inserted into a domain in an attempt to steal or corrupt data. In the object space model, only a parent can force a child space to execute code not foreseen by the program through the newInstance method. However, there is no way to control the classes used by a particular space. Section 4 compared the implementation of the object space model with J-Kernel [25]. Recall that protection domains in J-Kernel are made up of selected shared system classes, user classes loaded by a domain loader, as well as instances of these classes. A capability object is used to reference an object in a remote domain. A call on a capability object transfers control to the called domain; parameters in the call are copied by value unless they are capability objects, which are copied directly. In comparison to the object space model, J-Kernel uses copy-by-value by default, whereas the object space model uses copy-by-reference. J-Kernel must explicitly create a capability for an object to transfer it by reference; the object space model must explicitly serialize an object to copy it by value. The latter approach is a more natural object-oriented choice. Access control is based on capabilities in J- Kernel. A problem with capabilities is that their propagation cannot be controlled: once a domain exports a capability for one of its objects, it can no longer control what other domain receives a copy of the capability. Revocation exists but this entails revoking all copies of a capability, meaning that the distribution of access rights for an object must start again from scratch. In the object space model, an owner can grant and revoke rights for spaces selectively to other spaces. Another difference between the two systems is the presence of the hierarchy in the object space model and the absence of multiple class loaders and class instances. The goal of the JavaSeal kernel [6] is to isolate mobile agents from each other and from the host platform. A protection domain in JavaSeal is known as a seal, and is also implemented using the Java loader mechanism. The set of seals is organized into a hierarchy. A message exchanged between two seals is routed via their common parent seal, which can suppress the message for security reasons. It was in fact our experience with programming a newspaper application [19] over JavaSeal that first motivated the object space model. Many objects such as environment variables and article objects needed to be distributed to several seals. This meant copy-by-value semantics, which we found to be cumbersome for mutable objects like key certificates and article files. We wanted a safe form of object sharing to simplify programming. Interesting similarities exist between the object space model and memory management in real-time Java [5]. The latter has ScopedMemory objects that act as memory heaps for temporary objects. A newly created real-time thread can be assigned a ScopedMemory; alternatively, threads can enter the context of a ScopedMemory by executing its enter() method. The memory object contains a reference counter that is incremented each time that a thread enters it. An object created by a thread in a ScopedMemory is allocated in that memory object. An object (in a ScopedMemory) may create other ScopedMemory objects, thus introducing a hi- erarchy. The goal of the scoped memory model is to avoid use of a (slow) garbage collector to remove objects. When the reference counter of a ScopedMemory object becomes 0, the objects it contains can be removed. To prevent dangling references, an object cannot hold a reference to an object in a sibling ScopedMemory; the JVM dynamically checks all reference assignments to verify this constraint. Compared with the object space model, both approaches use a hierarchy with accesses between spaces being dynamically checked. However, the access restrictions in the object space model can be dynamically modified, and accesses between non-related spaces are possible. On the other hand, the object space model does not deal with resource termination 5.2 Program Security Mechanisms There has been much work on integrating access control into programs. Some approaches annotate programs with calls to a security policy checker [21, 9]. In Java for instance [9], a system class contains a method call to a SecurityManager object that checks whether the calling thread has the right to pursue the call. Another approach to program security uses programming language support. For instance, the languages [13, 18] include the notion of access rights; programs can possess rights for objects and access by a program to an object can only progress if it possesses the access right. Language designers today tend to equate security to type correctness. In this way, security is just another "good be- havior" property of a program, that can be verified using static analysis or dynamic checking [20, 16] Leroy and Rouaix use typing to enforce security in environments running applets [16]. Security in this context means that an applet cannot gain access to certain objects (such as those private to an environment function), and that objects which are accessible can only be assigned a specified set of permitted values. Each system type has special versions t that each define a set of permitted values. For instance, may be String, and t be CLASSPATH with possible values being /applet/public and /bin/java. Each conversion from to t on an object entails verifying that the object respects the permitted values. Environment functions available to applets are bridge-like in the sense that each in-coming reference of type is cast to t. This is similar to the object space model implementation in Java since for each class , a bridge class t is constructed that contains code to verify the system's access control policy. Access permissions are specified by an access matrix in the object space model, rather than by permitted object values. The goal of JFlow [20] is information flow security ; this deals with controlling an attacker's ability to infer information from an object rather than with controlling access to the object's methods. JFlow extends the Java language by associating security labels with variables. A security label denotes the sensitivity of an object's information. JFlow has a static analyzer that ensures that an object does not transfer a reference or data to an object with an inferior security label, as this would constitute a leak. The complexity of the mechanism comes from ensuring that information about objects used in an conditional expression evaluation is not implicitly leaked to objects modified in the scope of the conditional expression. In comparison to these works, the object space implementation relies on typing to ensure that each object access is made using a secure version of the class (i.e., one that includes access control checks). Annotation of classes with checks could be included to check field accesses between objects [21], as we mentioned in Section 4. A related topic to access control is aliasing control, e.g., Confined types [4], Balloons [1] or Islands [11]. Confined types is a recent effort to control the visibility of kernel objects by controlling the visibility of class names. A confined type is a class whose objects are invisible to specific user programs. The advantage of this approach is that the confinement of a type is verified by the compiler. On the other hand, classes cannot be confined and non-confined at the same time. It is important that one can designate some objects of a class as protected, and other objects as public. For instance, the visibility of Strings that represent passwords must be confined, though the class String is a general class that should be accessible to all programs. Another problem with aliasing control is that object visibility can vary during the object's lifetime. For instance, an object given to a server must remain accessible to that server during the server's work-time. Once the server has completed its task, the object should be removed from the server's visibility; this is the server containment property [7]. The object space model controls access on an object basis, and the visibility constraints can be dynamically altered. 6 Conclusions This paper has presented an access control model for an object-oriented environment. The model API is strongly influenced by Java and its loader spaces programming model, though aims to overcome weaknesses in Java access control caused by aliasing. We evaluated our proposition for an implementation over Java 2. Though the implementation has the advantage of portability, it means that we cannot address resource control and domain termination issues. These issues must be treated if object spaces are to become fully-fledged protection domains. Virtual machine support could also be useful to overcome the limitations of the model in Java, e.g., the prohibition of field accesses and the work around of final modifiers. Our results show that interposition of access control programs between objects of different domains can be more efficient than a simple copy-by-value of data between loader spaces. We believe that the object space model is a more natural object programming style than copy-by-value, especially for objects that need to be accessed by many programs and whose value can change often, e.g., environment vari- ables. The model also has the advantage that any leak of a reference between spaces is innocuous if the receiving space has not been explicitly granted the right to use the space of the referenced object. And even if access has been granted, this right may always be removed. Acknowledgments The authors are grateful to the anonymous referees and to Jan Vitek for very valuable comments on the content and presentation of this paper. This work was support by the Swiss National Science Foundation, under grant FNRS 20-53399.98. --R Balloon Types: Controlling Sharing of State in Data Types. The Java Programming Language. Confined Types. The Real-Time Specification for Java The Javaseal Mobile Agent Ker- nel Protection in the Hydra Operating System. Application Isolation in the Java Virtual Ma- chine Inside Java 2 platform security: architecture Islands: Aliasing Protection in Object-Oriented Languages The Geneva Convention on the Treatment of Object Alias- ing A Language Extension for Controlling Access to Shared Data. Security in the Ajanta Mobile Agent System. A Note on the Confinement Problem. Security properties of typed ap- plets Dynamic Class Loading in the Java Virtual Machine. Access Control in Parallel Programs. Commercialization of Electronic Information. JFlow: practical mostly-static information flow control Providing Fine-Grained Access Control for Java Programs The Protection of Information in Computer Systems. Optimizing Java Bytecode using the Soot framework: Is it feasible? --TR Islands The Geneva convention on the treatment of object aliasing The use of name spaces in Plan 9 Security properties of typed applets Dynamic class loading in the Java virtual machine The Java programming language (2nd ed.) JFlow Inside Java 2 platform security architecture, API design, and implementation Confined types Commercialization of electronic information Application isolation in the Java Virtual Machine A note on the confinement problem J-Kernel The Real-Time Specification for Java Providing Fine-grained Access Control for Java Programs Optimizing Java Bytecode Using the Soot Framework Signing, Sealing, and Guarding Java Objects The JavaSeal Mobile Agent Kernel Protection in the Hydra Operating System --CTR Chris Hawblitzel , Thorsten von Eicken, Luna: a flexible Java protection system, Proceedings of the 5th symposium on Operating systems design and implementation Due to copyright restrictions we are not able to make the PDFs for this conference available for downloading, December 09-11, 2002, Boston, Massachusetts Chris Hawblitzel , Thorsten von Eicken, Luna: a flexible Java protection system, ACM SIGOPS Operating Systems Review, v.36 n.SI, Winter 2002 Laurent Dayns , Grzegorz Czajkowski, Lightweight flexible isolation for language-based extensible systems, Proceedings of the 28th international conference on Very Large Data Bases, p.718-729, August 20-23, 2002, Hong Kong, China Grzegorz Czajkowski , Laurent Dayns, Multitasking without comprimise: a virtual machine evolution, ACM SIGPLAN Notices, v.36 n.11, p.125-138, 11/01/2001 Krzysztof Palacz , Jan Vitek , Grzegorz Czajkowski , Laurent Daynas, Incommunicado: efficient communication for isolates, ACM SIGPLAN Notices, v.37 n.11, November 2002

access control;protection domains;sharing;aliasing

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Merging and Splitting Eigenspace Models.

AbstractWe present new deterministic methods that given two eigenspace modelseach representing a set of $n$-dimensional observationswill: 1) merge the models to yield a representation of the union of the sets and 2) split one model from another to represent the difference between the sets. As this is done, we accurately keep track of the mean. Here, we give a theoretical derivation of the methods, empirical results relating to the efficiency and accuracy of the techniques, and three general applications, including the construction of Gaussian mixture models that are dynamically updateable.

Introduction The contributions of this paper are: (1) a method for merging eigenspace models; (2) a method for splitting eigenspace models. These represent an advance in that previous methods for incremental computation of eigenspace models considered only the addition (or subtraction) of a single observation to (or from) an eigenspace model [1, 2, 3, 4, 5, 6]. Our method also allows the origin to be updated, unlike most other methods. Thus, our methods allow large eigenspace All authors are with the Department of Computer Science, University of Wales, Cardiff, PO Box 916, Cardiff CF2 3XF, Wales UK: peter@cs.cf.ac.uk models to be updated more quickly and more accurately than when using previous methods. A second advantage is that very large eigenspace models may now be reliably constructed using a divide-and- conquer approach. Limitations of existing techniques are mentioned in [7]. Eigenspace models have a wide variety of applica- tions, for example: classification for recognition systems [8], characterising normal modes of vibration for dynamic models, such as the heart [9], motion sequence analysis [10], and the temporal tracking of signals [4]. Clearly, in at least some of these appli- cations, merging and splitting of eigenspace models will be useful, as the data is likely to be presented or processed in batches. For example, a common paradigm for machine learning systems is to identify clusters [11], our methods allow clusters to be split and merged, and dynamically updated as new data arrive. As another example, an image database of university students may require as much as one quarter of its records to be replaced each year. our methods permit this without the need to recompute the eigenspace model ab initio. This paper is solely concerned with deriving a theoretical framework for merging and splitting eigenspaces, and an empirical evaluation of these new techniques, rather than their particular application in any area. An eigenspace model is a statistical description of a set of N observations in n-dimensional space; such a model may be regarded as a multi-dimensional Gaussian distribution. From a geometric point of view, an eigenspace model can be thought of as a hyperellipsoid that characterises a set of observations: its centre is the mean of the observations; its axes point in directions along which the spread of observations is maximised, subject to them being orthogonal; the surface of the hyperellipsoid is a contour that lies at one standard deviation from the mean. Often, the hyperellipsoid is almost flat along certain directions, and thus can be modelled as having lower lower dimension than the space in which it is embedded. Eigenspace models are computed using either eigenvalue decomposition (also called principal component analysis) or singular-value decomposition. We wish to distinguish between batch and incremental computation. In batch computation all observations are used simultaneously to compute the eigenspace model. In an incremental computation, an existing eigenspace model is updated using new observations. Previous research in incremental computation of eigenspace models has only considered adding exactly one new observation at a time to an eigenspace model [1, 2, 3, 4, 5, 6]. A common theme of these methods is that none require the original observations to be retained. Rather, a description of the hyperellipsoid is sufficient information for incremental computation of the new eigenspace model. Each of previous these approaches allows for a change in dimensionality of the hyperellipsoid, so that a single additional axis is added if necessary. Only our previous work allows for a shift of the centre of the hyperellipsoid [6]: other methods keep it fixed at the origin. This proves crucial if the eigenspace model is to be used for classification, as explained in [6]: a set of observations whose mean is far from the origin is clearly not well modelled by a hyperellipsoid centred at the origin. When using incremental methods previous observations need not be kept - thus reducing storage requirements and making large problems computationally feasible. Incremental methods must be used if not all observations are available simultaneously. For example, a computer may lack the memory resources required to store all observations. This is true even if low-dimensional methods are used to compute the eigenspace [5, 12]. (We will mention low-dimensional methods later, but they give an advantage when the number of observations of less than the dimensionality of the space, N ! n, which is often true when observations are images.) Even if all observations are available, it is usually faster to compute a new eigenspace model by incrementally updating an existing one rather than using batch computation [3]. This is because the incremental methods typically compute p eigenvectors, with p - min(n; N ). The disadvantage of incremental methods is their accuracy compared to batch methods. When only few incremental updates are made the inaccuracy is small, and is probably acceptable for the great majority of applications [6]. When many thousands of updates are made, as when eigenspace models are incremented with a single observation at a time, the inaccuracies build up, although methods exist to circumvent this problem [4]. In contrast, our methods allow a whole new set of observations to be added in a single step, thus reducing the total number of updates to an existing model. Section 2 defines eigenspace models in detail, standard methods for computing them, and how they are used for representing and classifying observations. Section 3 discusses merging of eigenspace models, while Section 4 treats splitting. Section 5 presents empirical results, and Section 6 gives our conclusions. Eigenspace models In this section, we describe what we mean by eigenspace models, briefly discuss standard methods for their batch computation, and how observations can be represented using them. Firstly, we establish our notation for the rest of the paper. Vectors are columns, and denoted by a single un- derline. Matrices are denoted by a double under- line. The size of a vector, or matrix, is often im- portant, and where we wish to emphasise this size, it is denoted by subscripts. Particular column vectors within a matrix are denoted by a superscript, and a superscript on a vector denotes a particular observation from a set of observations, so we treat observations as column vectors of a matrix. As an example, A i mn is the ith column vector in an (m \Theta n) matrix. We denote matrices formed by concatenation using square brackets. Thus [A mn b] is an (m \Theta (n+1) matrix, with vector b appended to A mn , as a last column 2.1 Theoretical background Consider N observations, each a column vector x We compute an eigenspace model as follows: The mean of the observations is and their covariance is Note that C nn is real and symmetric. The axes of the hyperellipsoid, and the spread of observations over each axis are the eigenvectors and eigenvalues of the eigenproblem or, equivalently, the eigenvalue decomposition (EVD) of C nn is nn nn where the columns of U nn are eigenvectors, and nn is a diagonal matrix of eigenvalues. The eigenvectors are orthonormal, so that U T nn U nn I nn , the (n \Theta n) identity matrix. The ith eigenvector U i and ith eigenvalue ii nn are associated; the eigenvalue is the length of the eigen- vector, which is the ith axis of the hyperellipsoid. Typically, only p - min(n; N) of the eigenvectors have significant eigenvalues, and hence only p of the eigenvectors need be retained. This is because the observations are correlated so that the covariance matrix is, to a good approximation rank-degenerate: small eigenvalues are presumed to be negligible. Thus an eigenspace model often spans a p-dimensional sub-space of the n-dimensional space in which it is embedded Different criteria for discarding eigenvectors and eigenvalues exist, and these suit different applications and different methods of computation. Three common methods are: (1) stipulate p as a fixed inte- ger, and so keep the p largest eigenvectors [5]; (2) those p eigenvectors whose size is larger than an absolute threshold [3]; (3) keep the p eigenvectors such that a specified fraction of energy in the eigenspectrum (computed as the sum of eigenvalues) is retained. Having chosen to discard certain eigenvectors and eigenvalues, we can recast Equation 4 using block form matrices and vectors. Without loss of general- ity, we can permute the eigenvectors and eigenvalues such that U np are those eigenvectors that are kept, and pp their eigenvalues. If nd and dd are those discarded. We may rewrite Equation 4 as: nn U nd dd [U np U nd nd (5) Hence with error U nd dd U T nd , which is small if dd - 0 nd . Thus, we define an eigenspace model, \Omega\Gamma as the mean, a (reduced) set of eigenvectors, their eigen- values, and the number of observations: 2.2 Low-dimensional computation of eigenspace models Low-dimensional batch methods are often used to compute eigenspace models, and are especially important when the dimensionality of the observations is very large compared to their number. Thus, they may be used to compute eigenspace models that would otherwise be infeasible. Incremental methods also use a low dimensional approach. In principle, computing an eigenspace model requires that we construct an (n \Theta n) matrix, where n is the dimension of each observation. In practice, the model can be computed by using an (N \Theta N) ma- trix, where N is the number of observations. This is an advantage in applications like image processing where, typically, N - n. We show how this can be done by first considering the relationship between eigenvalue decomposition (EVD) and singular value decomposition (SVD). This leads to a simple derivation for a low-dimensional batch method for computing the eigenspace model. The same results were obtained, at greater length, by [5], see also [12]. Let Y nN be the set of observations shifted to the mean, so that Y x. Then a SVD of Y nN is: where U nn are the left singular vectors, which are identical to the eigenvectors previously given; \Sigma nN is a matrix with singular values on its leading diagonal, with nn are right singular vectors. Both U nn and V NN are orthonormal matrices This can now be used to compute eigenspace models in a low-dimensional way, as follows: Y nN is an (N \Theta N) eigenproblem. S NN is the same as nn =N , except for the presence of extra trailing zeros on the main diagonal of nn . If we discard the small singular values, and their singular vectors, following the above, then remaining eigenvectors vectors are U np pp This result formed the basis of the incremental technique developed by [5] but they did not allow for a change in origin, nor does their approach readily generalise to merging and splitting. Others [3] observe that a solution based on the matrix product , as above, is likely to lead to inaccurate results because of conditioning problems, and they develop a method for incrementally updating SVD solutions with a single observation. Although their SVD method has proven more accurate (see [6]), it is, again, very difficult to generalise, especially if a change of origin is to be taken into account. SVD methods were actually proposed quite early in the development of incremental eigenproblem analysis [2]. This early work included a proposal to delete single observations, but did not extend to merging and splitting. SVD also formed the basis of a proposal to incrementally update an eigenspace with several observations at one step [10]. However, contrary to our method, a possible change in the dimension of the solution eigenspace was not considered. Further- more, none of these methods considered a change in origin. Our incremental method is based on the matrix product C nN , and specifically its approximation as in Equation 6. It is a generalisation of our earlier work [6], which now appears naturally as the special case of adding a single observation. 2.3 Representing and classifying observation High-dimensional observations may be approximated by a low-dimensional vector using an eigenspace model. Eigenspace models may also be used for clas- sification. We briefly discuss both ideas, prior to using them in our results section. An n-dimensional observation x n is represented using an eigenspace N) as a p-dimensional vector This shifts the observation to the mean, and then represents it by components along each eigenvec- tor. This is also called the Karhunen-Lo'eve transform [13]. The n-dimensional residue vector is defined by: and h n is orthogonal to every vector in U np . Thus, h n is the error in the representation of x n with respect The likelihood associated with the same observation is given by: Clearly, the above definition cannot be used directly in cases where N - n, as C nn is then rank degenerate. In such cases we use an alternative definition due to Moghaddam and Pentland [8] is appropriate (a full explanation of which is beyond the scope of this paper). Merging Eigenspace models We now turn our attention to one of the two main contributions of this paper, merging eigenspace models We derive a solution to the following problem. Let and Y nM be two sets of observations. Let their eigenspace models respectively. The problem is to compute the eigenspace model \Phi (-z; W nr for Z n(N+M) Y nM using only\Omega and \Psi. Clearly, the total number of new observations is The combined mean is: -z =(N +M) The combined covariance matrix is: where C nn and D nn are the covariance matrices for and Y nM , respectively. We wish to compute the s eigenvectors and eigen-values that satisfy: ns ss ns where some eigenvalues are subsequently discarded to give r non-negligible eigenvectors and eigenvalues. The problem to be solved is of size s, and this is necessarily bounded by We explain the perhaps surprising additional 1 in the upper limit later(Section 3.1.1), but briefly, it is needed to allow for the vector difference between the means, - y. 3.1 Method of solution This problem may be solved in three steps: 1. Construct an orthonormal basis set, \Upsilon ns , that spans both eigenspace models and - x\Gamma-y. This basis differs from the required eigenvectors, W ns , by a rotation, R ss , so that: ns \Upsilon ns R ss 2. Use \Upsilon ns to derive an intermediate eigenproblem. The solution of this problem provides the eigen- values, \Pi ss , needed for the merged eigenmodel. The eigenvectors, R ss , comprise the linear transform that rotates the basis set \Upsilon ns 3. Compute the eigenvectors W ns , as above, and discard any eigenvectors and eigenvalues using the chosen criteria (as discussed above) to yield and \Pi rr We now give details of each step. 3.1.1 Construct an orthonormal basis set To construct an orthonormal basis for the combined eigenmodels we must chose a set of orthonormal vectors that span three subspaces: (1) the sub-space spanned by eigenvectors U np ; (2): the sub-space spanned by eigenvectors V nq ;(3) the subspace spanned by (-x \Gamma - y). The last of these is a single vec- tor. It is necessary because the vector joining the centre of the two eigenspace models need not belong to either eigenspace. This accounts for the additional 1 in the upper limit of the bounds of s (Equation 17). For example, consider the case in which each of the eigenspaces is a 2D ellipse in a 3D space, and the ellipses are parallel but separated by a vector perpendicular to each of them. Clearly, a merged model should be a 3D ellipse because of the vector between their origins. A sufficient spanning set is: ns nt nt is an orthonormal basis set for that component of the eigenspace of \Psi which is orthogonal to the eigenspace of \Omega\Gamma and in addition accounts for that component of (-x \Gamma - y) orthogonal to both eigenspaces; To construct - nt we start by computing the residues of each of the eigenvectors in V nq with respect to the eigenspace of \Omega\Gamma The H nq are all orthogonal to U np in the sense that In general, however, some of the H nq are zero vectors, because such vectors represent the intersection of the two eigenspaces. We also compute the residue h of - x with respect to the eigenspace of \Omega\Gamma using Equation 12. - nt can now be computed by finding an orthonormal basis for [H nq ; h], which is sufficient to ensure that \Upsilon ns is orthonormal. Gramm-Schmidt orthonor- malisation may be used to do this: nt 3.1.2 Forming a intermediate eigenproblem We now form a new eigenproblem by substituting Equation 19 into Equation and the result together with Equation 15 into Equation 16 to obtain: nt ss \Pi ss R T ss nt Multiplying both sides on the left by [U nt on the right by [U np nt and using the fact that [U np nt T is a left inverse of [U np nt we obtain: nt nt ss \Pi ss R T ss which is a new eigenproblem whose solution eigenvectors constitute the R ss we seek, and whose eigenvalues provide eigenvalues for the combined eigenspace model. We do not know the covariance matrices C nn or D nn , but these can be eliminated as follows: The first term in Equation 24 is proportional to: nt nt nt C nn U np - T nt C nn - nt By Equation 6 U T U np . Also, U T nt pt by construction, and again, using Equation 6 we conclude: [U np nt [U np nt The second term in Equation 24 is proportional to: nt nt np D nn U np U T np D nn - nt nt nt D nn - nt We have D nn , which on substitution gives: U np nt nt nt nt From Equation 20 we have G pq . nt V nq . We obtain: nt nt Now consider the final term in Equation 24: nt nt nt nt nt nt Setting nt becomes: " So, the new eigenproblem to be solved may be approximated by G pq G pq ss ss R T ss Each matrix is of size Thus we have eliminated the need for the original covariance matrices. Note this also reduces the size of the central matrix on the left hand side. This is of crucial computational importance because it makes the eigenproblem tractable. 3.1.3 Computing the eigenvectors The matrix \Pi ss is the eigenvalue matrix we set out to compute. The eigenvectors R ss comprise a rotation for \Upsilon ns . Hence, we use Equation to compute the eigenvectors for \Pi ss . However, not all eigenvectors and eigenvalues need be kept, and some of them) may be discarded using a criterion as previously discussed in Section 2. This discarding of eigen-vectors and eigenvalues should usually be carried out each time a pair of eigenspace models is merged. 3.2 Discussion on the form of the so- lution We now briefly justify that the solution obtained is of the correct form by considering several special cases. First, suppose that both eigenspace models are null, that is each is specified by (0; 0; 0; 0). Then the system is clearly degenerate and null eigenvectors and zero eigenvalues are computed. If exactly one eigenspace model is null, then the non-null eigenspace model is computed and returned by this process. To see this, suppose that \Psi is null. Then, the second and third matrices on the left-hand side of Equation 31 both disappear. The first matrix reduces to pp exactly hence the eigen-values remain unchanged. In this case, the rotation R ss is the identity matrix, and the eigenvectors are also unchanged. If instead\Omega is a null model, then only the second matrix will remain (as - nt and V nq will be related by a rotation (or else identical). The solution to the eigenproblem then computes the inverse of any such rotation, and the eigenspace model remain unchanged. Suppose \Psi has exactly one observation, then it is specified by (y; 0; 0; 1). Hence the middle term on the left of Equation 31 disappears, and - nt is the unit vector in the direction x. Hence is a scalar, and the eigenproblem becomes which is exactly the form obtained when one observation is explicitly added, as we have proven elsewhere [6]. This special case has interesting properties too: if the new observation lies within the subspace spanned by U np , then any change in the eigenvectors and eigenvalues can be explained by rotation and scaling caused by g p g T . Furthermore, in the unlikely event that - y, then the right matrix disappears altogether, in which case the eigenvalues are scaled by N=(N +1), but the eigenvectors are un- changed. Finally, as indicating a stable model in the limit. If the\Omega has exactly one observation, then it is specified by (x; 0; 0; 1). Thus the first matrix on the left of Equation 31 disappears. The G pq is a zero matrix, and - nt = [V nq h], where h is the component of - which is orthogonal to the eigenspace of \Psi. Hence the eigenproblem is: Given that in this case \Gamma tq , then has the form of \Delta qq , but with a row and column of zeros appended. Also, fl t Substitution of these terms shows that in this case too, the solution reduces to the special case of adding a single new observation: Equation 33 is of the same form as Equation 32, as can readily shown. If the\Omega and \Psi models are identical, then - y. In this case the third term on the left of Equation 31 disappears. Furthermore, \Gamma tq is a zero matrix, and G pq U np is the identity matrix, with Hence, the first and second matrices on the left of Equation 31 are identical, with reduce to the matrices of eigenvalues. Hence, adding two identical eigenmodels yields an identical third. Finally, notice that for fixed M , as N !1 so the solution tends to the\Omega model; for fixed N the reverse is true; and if M and N tend to 1 simultaneously, then the final term loses its significance. 3.3 Algorithm Here, for completeness, we now express the mathematical results obtained above, for merging models, in the form of an algorithm for direct computer im- plementation; see Figure 1 3.4 Complexity Computing an eigenspace model of size N as a single batch incurs a computational cost O(N 3 ). Examination of our merging algorithm shows that it also requires an intermediate eigenvalue problem to be solved, as well as other steps; again overall giving cubic computational requirements. Nevertheless, let us suppose with N observations, can be represented by p eigenvectors, and that \Psi, with M Function Merge( - returns (-z; W; \Pi; P ) BEGIN y for each column vector of H discard this column, if it is of small magnitude. endfor of of \Delta number of basis vectors in - construct LHS of Equation 31 eigenvalues of A eigenvectors of A discard small eigensolutions, as appropriate END Figure 1: Algorithm for merging two eigenspace models observations, can be represented by q eigenvectors. Typically p and q are (much) less than N and M , respectively. To compute an overall model with the batch method requires O((N +M) 3 ) operations. Assuming that both models to be merged are already known, our merging method requires at most O((p the problem to be solved becomes smaller the greater the amount of overlap between the eigenspaces of\Omega and \Psi. (In fact, the number of operations required is O(s 3 see the end of Section 3.1.1.) If one, or both, of the models to be merged are unknown initially, then we incur an extra cost of O(N 3 ), reduces any advan- tage. Nevertheless, in one typical scenario, we might expect\Omega to be known (an existing large database of N observations), while \Psi is a relatively small batch of M observations to be added to it. In this case, the extra penalty, of O(M 3 ), is of little significance compared to O((N +M) 3 ). while an exact analysis is complicated and indeed data dependent, we expect efficiency gains in both time and memory resources in practice. Furthermore, if computer memory is limited, sub-division of the initial set may be unavoidable in order to manage eigenspace model computation. We have now provided a tractable solution to this problem. Splitting eigenspace models Here we show how to split two eigenspace models. Given an eigenspace model remove M) from it to give a third use \Pi rr , because ss is not available in general. Although splitting is essentially the opposite of merging, this is not completely true as it is impossible to regenerate information which has been discarded when the overall model is created (whether by batch methods or otherwise). Thus, if we split one eigenspace model from a larger one, the eigenvectors of the remnant must still form some subspace of the larger. We state the results for splitting without proof. Clearly, . The new mean is: As in the case of merging, new eigenvalues and eigen-vectors are computed via an intermediate eigenprob- lem. In this case it is: G rp rp r rr x). The eigenvalues we seek are the q non-zero elements on the diagonal of rr . Thus we can permute R rr and rr , and write without loss of generality: R rr rr R rt [R rp R rt Hence we need only identify the eigenvectors in R rr with non-zero eigenvalues, and compute the U np as: U np In terms of complexity, splitting must always involve the solution an eigenproblem of size r. An algorithm for splitting may readily be derived using a similar approach to that for merging. This section describes various experiments that we carried out to compare the computational efficiency of a batch method and our new methods for merging and splitting, and the eigenspace models produced. We compared models in terms of Euclidean distance between the means, mean angular deviation of corresponding eigenvectors, and mean rela- tiveabsolute difference between corresponding eigen- values. In doing so, we took care that both models had the same number of dimensions. As well as the simple measures above, other performance measures may be more relevant when eigenspace models are used for particular ap- plications, and thus other tests were also per- formed. Eigenspace models may be used for approximating high-dimensional observations with a low-dimensional vector; the error is the size of the residue vector. The sizes of such residue vectors can readily be compared for both batch and incremental meth- ods. Eigenspace models may also be used for classifying observations, giving the likelihood that an observation belongs to a cluster. Different eigenspace models may be compared by relative differences in likelihoods. We average these differences over all corresponding observations. We used a database of 400 face images (each of 10304 pixels) available on-line 1 in the tests reported here ; similar results were obtained in tests with randomly generated data. The gray levels in the images were scaled into the range [0; 1] by division only, but no other preprocessing was done. We implemented all functions using commercially available software (Matlab) on a computer with standard configuration (Sun Sparc Ultra 10, 300 Hz, 64 Mb RAM). The results we present used up to 300 images, as the physical resources of our computer meant that heavy paging started to occur beyond this limit for the batch method, although such paging did not affect the incremental method. For all tests, the experimental procedure used was to compute eigenspace models using a batch method [12], and compare these to models produced by merging or splitting other models also produced by the batch method. In each case, the largest of the three data sets contained 300 images. These were partitioned into two data sets, each containing a multiple of 50 images. We included the degenerate cases when one model contained zero images. Note that we tested both smaller models merged with larger ones, and vice-versa. The number of eigenvectors retained in any model, including a merged model, was set to be 100 as a maximum, for ease of comparing results. (Initial tests using other strategies indicate that the resulting eigenspace model is little effected.) 5.1 Timing When measuring CPU time we ran the same code several times and chose the smallest value, to minimise the effect of other concurrently running process. Initially we measured time taken to compute a model using the batch methods, for data sets of different sizes. Results are presented in Figure 2 and show cubic complexity, as predicted. 1 The Olivetti database of faces: http://www.cam-orl.co.uk/facedatabase.html Number of input images time in seconds observed data cubic fit Figure 2: Time to compute an eigenspace model with a batch method versus the number of images, N . The time is approximated by the cubic: 5:3 \Theta 10 \Gamma4 N 5.1.1 Merging We then measured the time taken to merge two previously constructed models. The results are shown in Figure 3. This shows that time complexity is approximately symmetric about the point half the number of input images. This result may be surprising because the algorithm given for merging is not symmetric with respect to its inputs, despite that fact that the mathematical solution is independent of order. The approximate symmetry in time-complexity can be explained by assuming independent eigenspaces with a fixed upper-bound on the number of eigenvectors: suppose the numbers of eigenvectors in the models are N and M . Then complexities of the main steps are approximately as fol- lows: computing a new spanning set, - is O(M 3 ); solving an eigenproblem is O(N 3 +M 3 ); rotating the new eigenvectors is O(N 3 +M 3 ). Thus the time com- plexity, under the stated conditions, is approximately O(N 3 +M 3 ), which is symmetric. Next, the times taken to compute an eigenspace model from 300 images in total, using the batch method and our merging method, are compared in Number of images in first model time in seconds incremental time joint Figure 3: Time to merge two eigenspace models of images versus the number of im- ages, N , in \Omega\Gamma The number of images in \Phi is 300 \Gamma N . Hence, the total number of different images used to compute \Phi is constant 300. Figure 4. The incremental time is the time needed to compute the eigenspace model to be merged, and merge it with a pre-computed existing one. The joint time is the time to compute both smaller eigenmodels and then merge them. As might be expected, incremental time falls as the additional number of images required falls. The joint time is approximately con- stant, and very similar to the total batch time. While the incremental method offers no time saving in the cases above, it does use much less memory. This could clearly be seen when a model was computed using 400 images: paging effects set in when a batch method was used and the time taken rose to over 800 seconds. The time to produce an equivalent model by merging two sub-models of size 200, however, took less than half that. 5.1.2 Splitting Time complexity for splitting eigenspaces should depend principally on the size of the large eigenspace which from which the smaller space is being removed, and the size of the smaller eigenspace should have Number of images in first model time in seconds incremental time joint batch time Figure 4: Time to make a complete eigenspace model for a database of 300 images. The incremental time is the addition of the time to construct only the eigenspace to be added. The joint time is the time to compute both eigenspace models and merge them. little effect. This is because the size of the intermediate eigenproblem to be solved depends on the size of the larger space, and therefore dominates the complexity. These expectations are borne out experi- mentally. We computed a large eigenmodel using 300 images, as before. We then removed smaller models of sizes between 50 and 250 images inclusive, in steps of 50 images. At most, 100 eigenvectors were kept in any model.The average time taken was approximately constant, and ranged between 9 and 12 seconds, with a mean time of about 11.4 seconds. These figures are much smaller than those observed for merging because the large eigenspace contains only 100 eigenvectors. Thus the matrices involved in the computation were of size (100 \Theta 100), whereas in merging the size was at least (150 \Theta 150), and other computations were involved (such as computing an orthonormal basis). 5.2 Similarity and performance The measures used for assessing similarity and performance of batch and incremental methods were described above. 5.2.1 Merging We first compared the means of the models produced by each method using Euclidean distance. This distance is greatest when the models to be merged have the same number of input images (150 in this case), as fall smoothly to zero when either of the models to be merged is empty. The value at maximum is typically very small, and we measured it to be 3:5 \Theta 10 \Gamma14 units of gray level. This compares favourably with the working precision of Matlab, which is 2:2 \Theta 10 \Gamma16 . We next compared the directions of the eigenvectors produced by each method. The error in eigenvector direction was measured by the mean angular deviation, as shown in Figure 5. Ignoring the degenerate cases, when one of the models is empty, we see that angular deviation has a single minimum when the eigenspace models were built with about the same number of images. This may be because when a small model is added to a large model its information tends to be swamped. These results show angular deviation to be very small on average. Number of input images eigenvector mean angular deviation in degrees Figure 5: Angular deviation between eigenvectors produced by batch and incremental methods versus the number of images in the first eigenspace model. The sizes of eigenvalues from both methods were compared next. In general we observed that the smaller eigenvalues had larger errors, as might be expected as they contain relatively little information and so are more susceptible to noise. In Figure 6 we given a mean absolute difference in eigenvalue. This rises to a single peak when the number of input images in both models is the same. Even so, the maximal value is small, 7 \Theta 10 \Gamma3 units of gray level. The largest eigenvalue is typically about 100. Number of input images eigenvalue mean absolute deviation Figure Difference between eigenvalues produced by batch and incremental methods versus the number of images in the first eigenspace model. We now turn to performance measures. The merged eigenspaces represent the image data with little loss in accuracy, as measured by the mean difference in residue error, Figure 7. This performance measure is typically small, about 10 \Gamma6 units of gray level per pixel, clearly below any noticeable effect. Finally we compared differences in likelihood values (Equation produced by the two methods. This difference is again small, typically of the order 8 shows; this should be compared with a mean likelihood over all observations of the Again the differences in classifications that would be made by these models would be very small. Number of input images difference in mean residue Figure 7: Difference in reconstruction errors per pixel produced by batch and incremental methods versus the number of images in the first eigenspace model. Number of input images mean class difference Figure 8: Difference in likelihoods produced by batch and incremental methods versus the number of images in the first eigenspace model. 5.2.2 Splitting Similar measures for splitting were computed using exactly those conditions described for testing the timing of splitting, and for exactly those characteristics described for merging. In each case a model to be subtracted was computed by a batch method, and removed from the overall model by our splitting pro- cedure. Also, a batch model was made for purposes of comparison from the residual data set. In all that follows the phrase "size of the removed eigenspace" means the number of images used to construct the eigenspace removed from the eigenspace built from 300 images. The Euclidean distance between the means of the models produced by each method grows monotonically as the size of the removed eigenspace falls, and never exceeds about 1:5 \Theta 10 \Gamma13 gray-level units. Splitting is slightly less accurate in this respect than merging. The mean angular deviation between corresponding eigenvector directions rises in similar fashion, from about 0.6 degrees when the size of the removed eigenspace is 250, to about 1.1 when the removed eigenmodel is of size 100. This represented a maximum in the deviation error, because an error of about degree was obtained when the removed model is of size 50. Again, these angular deviations are somewhat larger than those for merging. The mean difference in eigenvalues shows the same general trend. Its maximum is about 0.5 units of gray level, when the size of the removed eigenspace is 50. This is a much larger error than in the case of merging, but is still relatively small compared to a maximum eigenvalue of about 100. As in the case of merging, the deviation in eigenvalue grows larger as the size (importance) of the eigenvalue falls. Difference in reconstruction error rises as the size of the removed eigenspace falls. Its size is of the units of gray level per pixel, which again is negligible. The difference in likelihoods is significant, the relative difference some cases being factors of 10 or more. After conducting further experiments, we found that this relative difference is sensitive to the errors introduced when eigenvectors and eigenvalues are dis- carded. This is not a surprise, given that likelihood differences are magnified exponentially. We found that changing the criteria for discarding eigenvectors very much reduced: relative difference in likelihood of the order 10 \Gamma14 were achieved in some cases. We conclude that should an application require not only splitting, but also require classification, then eigen-vectors and eigenvalues must be discarded with care. We suggest that keeping eigenvectors which exceed some significance threshold be kept. Overall the trend is clear; accuracy and performance grew worse, against any measure we used, as the size of the eigenmodel being removed falls. 6 Conclusion We have shown that merging and splitting eigenspace models is possible, allowing a batch of new observations to be processed as a whole. This theoretical result is novel. Our experimental results show that the methods are wholly practical: computation times are feasible, the eigenspaces are very similar, and the performance characteristics differ little enough to not matter to applications. time advantage is obtained over batch methods whenever one of the eigenspace models exists already. A typical scenario is the addition of a set of observations (a new year's in take of student faces, say) to an existing, large, database. Our merging method is even more advantageous when both eigenspace models exist already. A typical scenario is dynamic clustering for classification, in which two eigenspace models can be merged, perhaps to create a hierarchy of eigenspace models. --R An eigenspace update algorithm for image analysis. Kumar. Natural basis functions and topographic memory for face recognition. Probabilistic visual learning for object representa- tion 3rd Edition. On the generalised karhunen-loeve expansion --TR --CTR Luis Carlos Altamirano , Leopoldo Altamirano , Matas Alvarado, Non-uniform sampling for improved appearance-based models, Pattern Recognition Letters, v.24 n.1-3, p.521-535, January Ko Nishino , Shree K. Nayar , Tony Jebara, Clustered Blockwise PCA for Representing Visual Data, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.27 n.10, p.1675-1679, October 2005 Jieping Ye , Qi Li , Hui Xiong , Haesun Park , Ravi Janardan , Vipin Kumar, IDR/QR: an incremental dimension reduction algorithm via QR decomposition, Proceedings of the tenth ACM SIGKDD international conference on Knowledge discovery and data mining, August 22-25, 2004, Seattle, WA, USA Hoi Chan , Thomas Kwok, An Autonomic Problem Determination and Remediation Agent for Ambiguous Situations Based on Singular Value Decomposition Technique, Proceedings of the IEEE/WIC/ACM international conference on Intelligent Agent Technology, p.270-275, December 18-22, 2006 Tae-Kyun Kim , Ognjen Arandjelovi , Roberto Cipolla, Boosted manifold principal angles for image set-based recognition, Pattern Recognition, v.40 n.9, p.2475-2484, September, 2007 Jieping Ye , Qi Li , Hui Xiong , Haesun Park , Ravi Janardan , Vipin Kumar, IDR/QR: An Incremental Dimension Reduction Algorithm via QR Decomposition, IEEE Transactions on Knowledge and Data Engineering, v.17 n.9, p.1208-1222, September 2005 Xiang Sean Zhou , Dorin Comaniciu , Alok Gupta, An Information Fusion Framework for Robust Shape Tracking, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.27 n.1, p.115-129, January 2005 John P. Collomosse , Peter M. Hall, Cubist Style Rendering from Photographs, IEEE Transactions on Visualization and Computer Graphics, v.9 n.4, p.443-453, October

principal component analysis;eigenspace models;model merging;model splitting;gaussian mixture models

353382

Enforceable security policies.

A precise characterization is given for the class of security policies enforceable with mechanisms that work by monitoring system execution, and automata are introduced for specifying exactly that class of security policies. Techniques to enforce security policies specified by such automata are also discussed.

Introduction A security policy defines execution that, for one reason or another, has been deemed unacceptable. For example, a security policy might concern access control, and restrict what operations principals can perform on objects, ffl information flow, and restrict what things principals can infer about objects from observing other aspects of system behavior, or ffl availability, and prohibit a principal from being denied use of a resource as a result of execution by other principals. Supported in part by ARPA/RADC grant F30602-96-1-0317 and AFOSR grant F49620-94-1-0198. The views and conclusions contained herein are those of the author and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of these organizations or the U.S. Government. To date, general-purpose security policies, like those above, have attracted most of the attention. But, application-dependent and special-purpose security policies are increasingly important. A system to support mobile code, like Java [5], might prevent information leakage by enforcing a security policy that bars messages from being sent after files have been read. To support electronic commerce, a security policy might prohibit executions in which a customer pays for a service but the seller does not provide that service. And finally, electronic storage and retrieval of intellectual property is governed by complicated rights-management schemes that restrict not only the use of stored materials but also the use of any derivatives [17]. The practicality of any security policy depends on whether that policy is enforceable and at what cost. In this paper, we address those questions for the class of enforcement mechanisms, which we call EM, that work by monitoring a target system and terminating any execution that is about to violate the security policy being enforced. Class EM includes security kernels, reference monitors, and all other operating system and hardware-based enforcement mechanisms that have appeared in the literature. Thus, understanding what can and cannot be accomplished using mechanisms in EM has value. Excluded from EM are mechanisms that use more information than is available from monitoring the execution of a target system. Therefore, compilers and theorem-provers, which analyze a static representation of the target system to deduce information about all of its possible executions, are excluded from EM. The availability of information about all possible target system executions gives power to an enforcement mechanism-just how much power is an open question. Also excluded from EM are mechanisms that modify a target system before executing it. Presumably the modified target system would be "equivalent" to the original except that it satisfies the security policy of interest; a definition for "equivalent" is needed in order to analyze this class of mechanisms. We proceed as follows. In x2, a precise characterization is given for security policies that can be enforced using mechanisms in EM. An automata-based formalism for specifying those security policies is the subject of x3. Mechanisms in EM for enforcing security policies specified by automata are described in x4. Next, x5 discusses some pragmatic issues related to specifying and enforcing security policies as well as the application of our enforcement mechanisms to safety-critical systems. 2 Characteristics of EM Enforcement Mechanisms Formally, we represent executions by finite and infinite sequences, where \Psi is the set of all possible such sequences. 1 Thus, a target system S defines a subset \Sigma S of \Psi, and a security policy is a predicate on sets of executions. Target system S satisfies security policy P if and only if P (\Sigma S ) equals true. Notice that, for sets \Sigma and \Pi of executions, we do not require that if \Sigma satisfies P and \Pi ae \Sigma holds, then \Pi satisfies P. Imposing such a requirement on security policies would disqualify too many useful candidates. For instance, the requirement would preclude information flow (as defined informally in x1) from being considered a security policy-set \Psi of all executions satisfies information flow, but a subset \Pi containing only those executions in which the value of a variable x in each execution is correlated with the value of y (say) does not. In particular, when an execution is known to be in \Pi then the value of variable x reveals information about the value of y. By definition, enforcement mechanisms in EM work by monitoring execution of the target system. Thus, any security policy P that can be enforced using a mechanism from EM must be equivalent to a predicate of the form P is a predicate on executions. b P formalizes the criteria used by the enforcement mechanism for deciding to terminate an execution that would otherwise violate the policy being enforced. In [1], a set of executions that can be defined by checking each execution individually is called a property. Using that terminology, we conclude from (1) that a security policy must characterize sets that are properties in order for the policy to have an enforcement mechanism in EM. Not every security policy characterizes sets that are properties. Some predicates on sets of executions cannot be put in form (1) because they cannot be defined in terms of criteria that individual executions must each satisfy in isolation. For example, the information flow policy discussed above characterizes sets that are not properties (as is proved in [11]). Whether information flows from x to y in a given execution depends, in part, on what values y takes in other possible executions (and whether those values are correlated with the value of x). A predicate to characterize such sets of executions cannot be constructed only using predicates defined on single executions. 1 The manner in which executions are represented is irrelevant here. Finite and infinite sequences of atomic actions, of higher-level system steps, of program states, or of state/action pairs are all plausible alternatives. Enforcement mechanisms in EM cannot base decisions on possible future execution, since that information is, by definition, not available to mechanisms in EM. This further restricts what security policies can be enforced by mechanisms from EM. In particular, consider security policy P of (1), and suppose - is the prefix of some execution - 0 where b enforcement mechanism for P must prohibit - even though extension - 0 satisfies b because otherwise execution of the target system might terminate before - is extended into - 0 , and the enforcement mechanism would then have failed to enforce P. We can formalize this requirement as follows. For oe a finite or infinite execution having i or more steps, and - a finite execution, let: oe[::i] denote the prefix of oe involving its first i steps - oe denote execution - followed by execution oe and define \Pi \Gamma to be the set of all finite prefixes of elements in set \Pi. Then, the above requirement concerning execution prefixes violating b policy P defined by (1), is: P(- oe))) (2) Finally, note that any execution rejected by an enforcement mechanism must be rejected after a finite period. This is formalized by: policies satisfying (2) and (3) are satisfied by sets that are safety properties [7], the class of properties that stipulate no "bad thing" happens during an execution. Formally, a property S is defined [8] to be a safety property if and only if for any finite or infinite execution oe, oe holds. This means that S is a safety property if and only if S is characterized by a set of finite executions that are excluded and, therefore, the prefix of no execution in S . Clearly, a security policy P satisfying (2) and (3) has such a set of finite prefixes-the set of prefixes - 2 \Psi \Gamma such that : b holds-so P is satisfied by sets that are safety properties according to (4). Our analysis of enforcement mechanisms in EM has established: Unenforceable Security Policy: If the sets of executions characterized by a security policy P are not safety properties, then an enforcement mechanism from EM does not exist for P. Obviously, the contrapositive holds as well: all EM enforcement mechanisms enforce safety properties. But, as discussed in x4, the converse-that all safety properties have EM enforcement mechanisms-does not hold. Revisiting the three application-independent security policies described in x1, we find: ffl Access control defines safety properties. The set of proscribed partial executions contains those partial executions ending with an unacceptable operation being invoked. ffl Information flow does not define sets that are properties (as argued above), so it does not define sets that are safety properties. Not being safety properties, there are no enforcement mechanisms in EM for exactly this policy. 2 ffl Availability defines sets that are properties but not safety properties. In particular, any partial execution can be extended in a way that allows a principal to access a resource, so availability lacks a defining set of proscribed partial executions that every safety property must have. Thus, there are no enforcement mechanisms in EM for availability-at least as that policy is defined in x1. 3 3 Security Automata Enforcement mechanisms in EM work by terminating any target-system execution after seeing a finite prefix oe such that : b P(oe) holds, for some predicate b defined by the policy being enforced. We established in x2 that the set of executions satisfying b also must be a safety property. Those being the only constraints on b P , we conclude that recognizers for sets of 2 Mechanisms from EM purporting to prevent information flow do so by enforcing a security policy that implies, but is not equivalent to, the absence of information flow. Given security policies P and Q for which P ) Q holds, a mechanism that enforces does suffice for enforcing Q. And, there do exist security policies that both imply restrictions on information flow and define sets that are safety properties. However, a policy P that implies Q might rule out executions that do not violate Q, so using the stronger policy is not without adverse consequences. 3 There are alternative formulations of availability that do characterize sets that are safety properties. An example is "one principal cannot be denied use of a resource for more than D steps as a result of execution by other principals". Here, the defining set of partial executions contains intervals that exceed D steps and during which a principal is denied use of a resource. FileRead not FileRead not Send Figure 1: No Send after F ileRead executions that are safety properties can serve as the basis for enforcement mechanisms in EM. A class of automata for recognizing safety properties is defined (but not named) in [2]. We shall refer to these recognizers as security automata; they are similar to ordinary non-deterministic finite-state automata [6]. Formally, a security automaton is defined by: - a (finite) set Q of automaton states, - a set Q 0 ' Q of initial automaton states, - a (countable) set I of input symbols, and - a transition function, I is dictated by the security policy being enforced; the symbols in I might correspond to system states, atomic actions, higher-level actions of the sys- tem, or state/action pairs. In addition, the symbols of I might also have to encode information about the past-for some safety properties, the transition function will require that information. To process a sequence s 1 of input symbols, the automaton starts with its current state set equal to Q 0 and reads the sequence, one symbol at a time. As each symbol s i is read, the automaton changes its current state set Q 0 to the set Q 00 of automaton states, where If ever Q 00 is empty, the input is rejected; otherwise the input is accepted. Notice that this acceptance criterion means that a security automaton can accept sequences that have infinite length as well as those having finite length. A(prin, obj, oper) Figure 2: Access Control Figure 1 depicts a security automaton for a security policy that prohibits execution of Send operations after a F ileRead has been executed. The automaton's states are represented by the two nodes labeled q nfr (for "no file read" ) and q fr (for "file read"). Initial states of the automaton are represented in the figure by unlabeled incoming edges, so automaton state is the only initial automaton state. In the figure, transition function is specified in terms of edges labeled by transition predicates, which are Boolean-valued effectively computable total functions with domain I. Let ij denote the predicate that labels the edge from node q i to node q j . Then, the security automaton, upon reading an input symbol s when Q 0 is the current state set, changes its current state set to In Figure transition predicate not FileRead is assumed to be satisfied by system execution steps that are not file read operations, and transition predicate not Send is assumed to be satisfied by system execution steps that are not message-send operations. Since no transition is defined from q fr for input symbols corresponding to message-send execution steps, the security automaton if Figure 1 rejects inputs in which a message is sent after a file is read. Another example of a security automaton is given in Figure 2. Different instantiations for transition predicate A(prin; obj; oper) allow this automaton to specify either discretionary access control [9] or mandatory access control [3]. Discretionary Access Control. This policy prohibits operations according to an access control matrix. Specifically, given access control matrix M , a principal P rin is permitted by the policy to execute an operation Oper involving object Obj only if Oper 2 M [P rin; Obj] holds. To specify this policy using the automaton of Figure 2, transition predicate A(prin; obj; oper) would be instantiated by: Mandatory Access Control. This policy prohibits execution of operations according to a partially ordered set of security labels that are associated with system objects. Information in objects assigned higher labels is not permitted to be read and then stored into objects assigned lower labels. For example, a system's objects might be assigned labels from the set ftopsecret; secret; sensitive; unclassifiedg ordered according to: topsecret - secret - sensitive - unclassified Suppose two system operations are supported-read and write. A mandatory access control policy might restrict execution of these operations according to: (i) a principal p with label -(p) is permitted to execute read(F ), which reads a file F with label -(F ), only if -(p) -(F ) holds. (ii) a principal p with label -(p) is permitted to execute write(F ), which writes a file F with label -(F ), only if -(F ) -(p) holds. To specify this policy using the automaton of Figure 2, transition predicate A(prin; obj; oper) is instantiated by: As a final illustration of security automata, we turn to electronic com- merce. We might, for example, desire that a service-provider be prevented from engaging in actions other than delivering the service for which a customer has paid. This requirement is a security policy; it can be formalized in terms of the following predicates, if executions are represented as sequences of operations: not pay(C) Figure 3: Security automaton for fair transaction customer C requests and pays for service customer C is rendered service The security policy of interest proscribes executions in which the service-provider executes an operation that does not satisfy serve(C) after having engaged in operation that satisfies pay(C). A security automaton for this policy is given in Figure 3. Notice, the security automaton of Figure 3 does not stipulate that payment guarantees service. The security policy it specifies only limits what the service-provider can do once a customer has made payment. In partic- ular, the security policy that is specified allows a service-provider to stop executing (i.e. stop producing input symbols) rather than rendering a paid- for service. We do not impose the stronger security policy that service be guaranteed after payment because that is not a safety property-there is no defining set of proscribed partial executions-and therefore, according to x2, it is not enforceable using a mechanism from EM. 4 Using Security Automata for Enforcement Any security automaton can serve as the basis for an enforcement mechanism in class EM, as follows. Each step the target system will next take is represented by an input symbol and sent to an implementation of the security automaton. (i) If the automaton can make a transition on that input symbol, then the target system is allowed to perform that step and the automaton state is changed according to its transition predicates. (ii) If the automaton cannot make a transition on that input symbol, then the target system is terminated. In fact, any security policy enforceable using a mechanism from EM can be enforced using such a security-automaton implementation. This is because all safety properties have specifications as security automata, and Unenforceable Security Policy of x2 implies that EM enforcement mechanisms enforce safety properties. Consequently, by understanding the limitations of security-automata enforcement mechanisms, we can gain insight into the limitations of all enforcement mechanisms in class EM. Implicit in (ii) is the assumption that the target system can be terminated by the enforcement mechanism. Specifically, we assume that the enforcement mechanism has sufficient control over the target system to stop further automaton input symbols from being produced. This control requirement is subtle and makes certain security policies-even though they characterize sets that are safety properties-unenforceable using mechanisms from EM. For example, consider the following variation on availability of x1: Real-Time Availability: One principal cannot be denied use of a resource for more than D seconds. Sets satisfying Real-Time Availability are safety properties-the "bad thing" is an interval of execution spanning more than D seconds during which some principal is denied the resource. The input symbols of a security automaton for Real-Time Availability must therefore encode time. However, the passage of time cannot be stopped, so a target system with real-time clocks cannot be prevented from continuing to produce input symbols. Real-Time Availability simply cannot be enforced using one of our automata-based enforcement mechanisms, because target systems lack the necessary controls. And, since the other mechanisms in EM are no more powerful, we conclude that Real-Time Availability cannot be enforced using any mechanism in EM. Two mechanisms are involved in our security-automaton implementation of an enforcement mechanism. Automaton Input Read: A mechanism to determine that an input symbol has been produced by the target system and then to forward that symbol to the security automaton. Automaton Transition: A mechanism to determine whether the security automaton can make a transition on a given input and then to perform that transition. Their aggregate cost can be quite high. For example, when the automaton's input symbols are the set of program states and its transition predicates are arbitrary state predicates, a new input symbol is produced for each machine-language instruction that the target system executes. The enforcement mechanism must be invoked before every target-system instruction. However, for security policies where the target system's production of input symbols coincides with occurrences of hardware traps, our automata-based enforcement mechanism can be supported quite cheaply by incorporating it into the trap-handler. One example is implementing an enforcement mechanism for access control policies on system-supported objects, like files. Here, the target system's production of input symbols coincides with invocations of system operations, hence the production of input symbols coincides with occurrences of system-call traps. A second example of exploiting hardware traps arises in implementing memory protection. Memory protection implements discretionary access control with operations read, write, and execute and an access control matrix that tells how processes can access each region of memory. The security automaton of Figure 2 specifies this security policy. Notice that this security automaton expects an input symbol for each memory reference. But most, if not all, of the these input symbols cause no change to the security automaton's state. Input symbols that do not cause automaton state transitions need not be forwarded to the automaton, and that justifies the following optimization of Automaton Input Read: Automaton Input Read Optimization: Input symbols are not forwarded to the security automaton if the state of the automaton just after the transition would be the same as it was before the transition. Given this optimization, the production of automaton input symbols for memory protection can be made to coincide with occurrences of traps. The target system's memory-protection hardware-base/bounds registers or page and segment tables-is initialized so that a trap occurs when an input symbol should be forwarded to the memory protection automaton. Memory references that do not cause traps never cause a state transition or undefined transition by the automaton. Inexpensive implementation of our automata-based enforcement mechanisms is also possible when programs are executed by a software-implemented virtual machine (sometimes known as a reference monitor). The virtual machine instruction-processing cycle is augmented so that it produces input symbols and makes automaton transitions, according to either an internal or an externally specified security automaton. For example, the Java virtual machine[10] could easily be augmented to implement the Automaton Input Read and Automaton Transition mechanisms for input symbols that correspond to method invocations. Beyond Class EM Enforcement Mechanisms The overhead of enforcement can sometimes be reduced by merging the enforcement mechanism into the target system. One such scheme, which has recently attracted attention, is software-based fault isolation (SFI), also known as "sandboxing" [19, 16]. SFI implements memory protection, as specified by a one-state automaton like that of Figure 2, but does so without hardware assistance. Instead, a program is edited before it is executed, and only such edited programs are run by the target system. (Usually, it is the object code that is edited.) The edits insert instructions to check and/or modify the values of operands, so that illegal memory references are never attempted, SFI is not in class EM because SFI involves modifying the target sys- tem, and modifications are not permitted for enforcement mechanisms in EM. But viewed in our framework, the inserted instructions for SFI can be seen to implement Automaton Input Read by copying code for Automaton Transition in-line before each target system instruction that produces an input symbol. Notice that nothing prevents the SFI approach from being used with multi-state automata, thereby enforcing any security policy that can be specified as a security automaton. Moreover, a program optimizer should be able to simplify the inserted code and eliminate useless portions 4 although this introduces a second type of program analysis to SFI and requires putting further trust in automated program analysis tools. Finally, there is no need for any run-time enforcement mechanism if the target system can be analyzed and proved not to violate the security policy of interest. This approach has been employed for a security policy like what SFI was originally intended to address-a policy specified by one-state security automata-in proof carrying code (PCC) [13]. With PCC, a proof is supplied along with a program, and this proof comes in a form that can be checked mechanically before running that program. The security policy will not be violated if, before the program is executed, the accompanying proof is checked and found to be correct. The original formulation of PCC required that proofs be constructed by hand. This restriction can be relaxed. For Ulfar Erlingsson of Cornell has implemented a system that does exactly this for the Java virtual machine and for Intel x86 machine language. certain security policies specified by one-state security automata, a compiler can automatically produce PCC from programs written in high-level, type-safe programming languages[12, 14]. To extend PCC for security policies that are specified by arbitrary security automata, a method is needed to extract proof obligations for establishing that a program satisfies the property given by such an automaton. Such a method does exist-it is described in [2]. The utility of a formalism partly depends on the ease with which objects of the formalism can be read and written. Users of the formalism must be able to translate informal requirements into objects of the formalism. With security automata, establishing the correspondence between transition predicates and informal requirements on system behavior is crucial and can require a detailed understanding of the target system. The automaton of Figure 1, for example, only captures the informal requirement that messages are not sent after a file is read if it is impossible to send a message unless transition predicate Send is true and it is impossible to read a file unless transition predicate F ileRead is true. There might be many ways to send messages-some obvious and others buried deep within the bowels of the target system. All must be identified and included in the definition of Send; a similar obligation accompanies transition predicate F ileRead. The general problem of establishing the correspondence between informal requirements and some purported formalization of those requirements is not new to software engineers. The usual solution is to analyze the formalization, being alert to inconsistencies between the results of the analysis and the informal requirements. We might use a formal logic to derive consequences from the formalization; we might use partial evaluation to analyze what the formalization implies about one or another scenario, a form of testing; or, we might (manually or automatically) transform the formalization into a prototype and observe its behavior in various scenarios. Success with proving, testing, or prototyping as a way to gain confidence in a formalization depends upon two things. The first is to decide what aspects of a formalization to check, and this is largely independent of the formalism. But the second, having the means to do those checks, not only depends on the formalism but largely determines the usability of that formalism. To do proving, we require a logic whose language includes the formalism; to do testing, we require a means of evaluating a formalization in one or another scenario; and to do prototyping, we must have some way to transform a formalization into a computational form. As it happens, a rich set of analytical tools does exist for security au- tomata, because security automata are a class of Buchi automata [4], and Buchi automata are widely used in computer-aided program verification tools. Existing formal methods based either on model checking or on theorem proving can be employed to analyze a security policy that has been specified as a security automaton. And, testing or prototyping a security policy that is specified by a security automaton is just a matter of running the automaton. Guidelines for Structuring Security Automata Real system security policies are best given as collections of simpler policies, a single large monolithic policy being difficult to comprehend. The system's security policy is then the result of composing the simpler policies in the collection by taking their conjunction. To employ such a separation of concerns when security policies are specified by security automata, we must be able to compose security automata in an analogous fashion. Given a collection of security automata, we must be able to construct a single conjunction security automaton for the conjunction of the security policies specified by the automata in the collection. That construction is not difficult: An execution is rejected by the conjunction security automaton if and only if it is rejected by any automaton in the collection. Beyond comprehensibility, there are other advantages to specifying system security policies as collections of security automata. First, having a collection allows different enforcement mechanisms to be used for the different automata (hence the different security policies) in the collection. Second, security policies specified by distinct automata can be enforced by distinct system components, something that is attractive when all of some security automaton's input symbols correspond to events at a single system compo- nent. Benefits that accrue from having the source of all of an automaton's input symbols be a single component include: ffl Enforcement of a component's security policy involves trusting only that component. ffl The overhead of an enforcement mechanism is lower because communication between components can be reduced. For example, the security policy for a distributed system might be specified by giving a separate security automaton for each system host. Then, each host would itself implement Automaton Input Read and Automaton Transitions mechanisms for only the security automata concerning that host. The designer of a security automaton often must choose between encoding security-relevant information in the target system's state and in an automaton state. Larger automata are usually more complicated, hence more difficult to understand, and often lead to more expensive enforcement mechanisms. For example, our generalization of SFI involves modifying the target system by inserting code and then employing a program optimizer to simplify the result. The inserted code simulates the security automaton, and the code for a smaller security automaton will be smaller, cheaper to execute, and easier to optimize. Similarly, for our generalization of PCC, proof obligations derived according to [2] are fewer and simpler if the security automata is smaller. However, we conjecture that the cost of executing a reference monitor that implements Automaton Transition is probably insensitive to whether security-relevant state information is stored in the target system or in automaton states. This is because the predicates that must be evaluated in the two implementations will differ only in whether a state component has been associated with the security automaton or the target system, and the cost of reading that state information and of evaluating the predicates should be similar. Application to Safety-Critical Systems The idea that security kernels might have application in safety-critical systems is eloquently justified in [15] and continues to interest researchers [18]. Safety-critical systems are concerned with enforcing properties that are safety properties (in the sense of [8]), so it is natural to expect an enforcement mechanism for safety properties to have application in this class of systems. And, we see no impediments to using security automata or our security-automata based enforcement mechanisms for enforcing safety properties in safety-critical systems. The justification given in [15] for using security kernels in safety-critical systems involves a characterization of what types of properties can be enforced by a security kernel. As do we in this paper, [15] concludes that safety properties but not liveness properties are enforceable. However, the arguments given in [15] are informal and are coupled to the semantics of kernel-supported operations. The essential attributes of enforceability, which we isolate and formalize by equations (1), (2), and (3), are neither identified nor shown to imply that only safety properties can be enforced. In addition, because [15] concerns kernelized systems, the notion of property there is restricted to being sequences of kernel-provided functions. By allowing security automata to have arbitrary sets of input symbols, our results can be seen as generalizing those of [15]. And the generalization is a useful one, because it applies to enforcement mechanisms that are not part of a kernel. Thus, we can now extend the central thesis of [15], that kernelized systems have application beyond implementing security policies, to justify the use of enforcement mechanisms from EM when building safety-critical systems. Acknowledgments I am grateful to Robbert van Renesse, Greg Morrisett, ' Ulfar Erlingsson, Yaron Minsky, and Lidong Zhou for helpful feedback on the use and implementation of security automata and for comments on previous drafts of this paper. Helpful comments on an earlier draft of this paper were also provided by Earl Boebert, Li Gong, Robert Grimm, Keith Marzullo, and John Rushby. John McLean served as a valuable sounding board for these ideas as I developed them. Feedback from Martin Abadi helped to sharpen the formalism. And, the University of Tromso was a hospitable setting and a compelling excuse for performing some of the work reported herein. --R Defining liveness. Recognizing safety and liveness. Secure computer systems: Mathematical foundations. 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security automata;security policies;safety properties;proof carrying code;SASI;EM security policies;inlined reference monitors

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Integrating object-oriented programming and protected objects in Ada 95.

Integrating concurrent and object-oriented programming has been an active research topic since the late 1980's. There is a now a plethora of methods for achieving this integration. The majority of approaches have taken a sequential object-oriented language and made it concurrent. A few approaches have taken a concurrent language and made it object-oriented. The most important of this latter class is the Ada 95 language, which is an extension to the object-based concurrent programming language Ada 83. Arguably, Ada 95 does not fully integrate its models of concurrency and object-oriented programming. For example, neither tasks nor protected objects are extensible. This article discusses ways in which protected objects can be made more extensible.

Introduction Arguably, Ada 95 does not fully integrate its models of concurrent and object-oriented programming (Atkinson and Weller, 1993; Wellings et al., 1996; Burns and Wellings, 1998). For example, neither tasks nor protected objects are exten- sible. When Ada 95 was designed, the extensions to Ada 83 for object-oriented programming were, for the most part, considered separate to extensions to the concurrency model. Although some consideration was given to abandoning protected types and instead using Java-like synchronised methods in their place, there was no public debate of this issue. Similarly, there was no public debate on the issues associated with allowing protected types or tasks to be extended. The purpose of this paper is to discuss ways in which the Ada 95 concurrency model can be better integrated with object-oriented programming. The paper is structured as follows. Section 2 introduces the main problems associated with the integration of object-oriented and concurrent programming. Section 3 then describes the main features of the Ada 95 language that are relevant to this work. Section 4 argues that Ada 95 does not have a well-integrated object-oriented concurrency model. To achieve better integration, Section 5 proposes that Ada's protected type mechanism be made extensible and discusses the main syntactic and semantic issues. Sections 6 then considers how extensible protected types integrate with Ada's general model of abstraction and inheritance. Sections 7 and 8 discuss how the proposals address the inheritance anomaly and how they can be used in conjunction with the current object-oriented mechanisms. Section 9 presents some extended examples and Section 10 draws conclusions from this work. Concurrent Object-Oriented Programming Integrating concurrent and object-oriented programming has been an active research topic since the late 1980s. There is now a plethora of methods for achieving this integration (see (Wyatt et al., 1992) for a review). The majority of approaches have taken a sequential object-oriented language and made it concurrent (for example, the various versions of concurrent Eiffel (Meyer, 1993; Caromel, 1993; Karaorman and Bruno, 1993)). A few approaches have taken a concurrent language and made it object-oriented. The most important of this latter class is the Ada 95 language which is an extension to the object-based concurrent programming language Ada 83. A full discussion of this language will be given in the next section. In general, there are two main issues for concurrent object-oriented programming ffl the relationship between concurrent activities and objects - here the distinction is often between the concept of an active object (which by definition will execute concurrently with other active objects, for example (Maio et al., 1989; Mitchell and Wellings, 1996; Newman, 1998)) and where concurrent execution is created by the use of asynchronous method calls (or early returns from method calls) ((Yonezawa et al., 1986; Yokote and Tororo, 1987; Corradi and Leonardi, 1990)) ffl the way in which concurrent activities communicate and synchronise (and yet avoid the so-called inheritance anomaly (Matsuoka and Yonezawa, see (Mitchell and Wellings, 1996) for a summary of the various proposals. Perhaps the most interesting recent development in concurrent object-oriented programming is Java (Lea, 1997; Oaks and Wong, 1997). Here we have, notion- ally, a new language which is able to design a concurrency model within an object-oriented framework without worrying about backward compatibility is- sues. The Java model integrates concurrency into the object-oriented framework by the combination of the active object concept and asynchronous method calls. All descendants of the pre-defined class Thread have the pre-defined methods run and start. When start is called, a new thread is created, which executes run. Subclassing Thread and overriding the run method allows an application to express active objects. (It is also possible to obtain start and run by implementing the interface Runnable.) Other methods available on the Thread class allow for a wide range of thread control. Communication and synchronisation is achieved by allowing any method of any object to be specified as 'synchronised'. Synchronised methods execute with a mutual exclusion lock associated with the object. All classes in Java are derived from the Object class that has methods which implement a simple form of condition synchronisation. A thread can, therefore, wait for notification of a single event. When used in conjunction with synchronised methods, the language provides the functionality similar to that of a simple monitor (Hoare, 1974). Arguably, Java provides an elegant, although simplistic, model of object-oriented concurrency. 3 The Ada 95 Programming Language The Ada 83 language allowed programs to be constructed from several basic building blocks: packages, subprograms (procedures and functions), and tasks. Of these, only tasks were considered to be types and integrated with the typing model of the language. Just as with any other type in Ada, many instances of a task type can be declared, tasks can be placed in arrays and records, and pointers to tasks can be declared and created. Ada 83 fully integrated its concurrency model into the sequential components of the language. Tasks can encapsulate data objects as well as other tasks. They are built using a consistent underlying type model. 3.1 Data-Oriented Synchronization: Protected Types Ada 95 extends the facilities of Ada 83 in areas of the language where weaknesses were perceived. One of the innovations was the introduction of data-oriented communication and synchronization through protected types. Instances of a protected type are called protected objects; they are basically monitors (Hoare, but avoid the disadvantages associated with the use of low-level condition variables. Instead, protected types may have guarded entries similar to those provided by conditional critical regions (Brinch-Hansen, 1972). protected type in Ada 95 encapsulates some data items, which can only be accessed through the protected type's operations. It is declared as shown in the following example: protected type Shared_Int is - Public operations procedure function Get return entry Wait_Until_Zero; private Encapsulated data Current - Private operations might follow here The operations of this protected type are implemented in a corresponding body: protected body Shared_Int is procedure in Integer) is begin Current := Value; function Get return Integer is begin return Current; entry Wait_Until_Zero when Current = 0 is - Entry barrier (guard) begin Instances of this protected type, i.e. protected objects, can be declared just like any other variable: protected object named 'X' Operations on this shared object can be invoked in the following way: Some_Variable := X.Get; Calls to the operations of a protected type are so-called protected actions and guarantee mutually exclusive access to a protected object with the usual semantics of multiple readers (function calls, which are read-only) or one writer (procedure and entry calls). When an entry is called and its barrier is false, the call is queued and the calling task is blocked until the call has been finally executed. Otherwise, the call is accepted and executed in a protected action. At the end of each procedure or entry call, the barriers of all entries are examined. If a barrier has become true, a possibly queued call is then executed as part of the same protected action, i.e. without relinquishing the mutual exclusion in between. This servicing of entry queues is repeated until either there are no more queued calls or all their barriers are false. The protected action then terminates. The following example illustrates the use of entries with a simple bounded buffer, where items can only be taken from the buffer when it is not empty, and items can be put into it only when it is not full. protected type Integer_Bounded_Buffer is entry entry Get private array (1 . 10) of Integer; First, Last : Natural := Nof_Items : Natural := 0; protected body Integer_Bounded_Buffer is entry when Nof_Items < Buffer'Length is begin Buffer (Last) := I; Last := Last mod Buffer'Length Nof_Items entry Get when Nof_Items > 0 is begin I := Buffer(First); First := First mod Buffer'Length Nof_Items := Nof_Items - If Get is called when Nof Items is zero, the call is queued. When another task calls Put, Nof Items will be incremented. When the entry queues are serviced after the call to Put has finished, the barrier of Get is now true and the queued call is allowed to proceed, thus unblocking the task that made that call. A requeue statement of the form requeue Target_Entry; allows an entry to put a call, which it has already begun processing, back on the same or some other entry queue again. A requeue immediately leaves the current entry, requeues the call and then initiates entry queue servicing. Once the requeued call has been executed, control is returned to the task that made the original call. An example of the requeue statement can be found in section 9.2. Within the operations of a protected type, the attribute E'Count represents the number of calls in the queue of entry E. Potentially blocking calls, in particular entry calls, are forbidden within a protected action. This language rule helps avoid deadlocks due to the nested monitors problem and also avoids a possible unbounded priority inversion that might otherwise occur. This means that a procedure of a protected type may call other procedures or functions of the same or some other protected object, but not entries. Functions of a protected type may only call other protected functions of the same protected object to avoid that they circumvent the read-only restriction. However, they may call both protected functions and procedures of other protected objects. Entries may call procedures or functions, but not other entries; they may only requeue to another entry. 3.2 Object-Orientation: Tagged Types One of the other main extensions to Ada 83 was the introduction of object-oriented programming facilities. Here the designers of Ada 95 were faced with a dilemma. Ada 83's facility for encapsulation was the package. Unfortunately, packages (unlike tasks) were not fully integrated into the typing model: there were no package types. Rather than introduce a class-like construct into the language (as had been done by almost all other object-oriented languages), Ada 95 followed the Oberon (Wirth, 1988) approach and achieved object-orientation by type extension. The designers argued that Ada 83 already had the ability to derive types from other types and override their operations. Consequently, object-orientation was achieved via the introduction of "tagged types". Tagged types in Ada 95 are record types that can be extended. Thus a class in Ada is represented by the following: package Objects is type Class is tagged record data attributes of the class - the following are the primitive operations of the type procedure Method1 (O: in Class; Params: Some_Type); procedure Method2 (O: in out Class; Params : Some_Type); Objects of the class can be created and used by: with Objects; use Objects; Params: Some_Type; begin Contrast this to a call to an object's method in the more typical object-oriented paradigm where the call is of the form: Object.Method1(Params). The difference is purely syntactical; both forms have the same expressive power and denote the same language construct, namely, a call to a primitive operation of a tagged type or a call of a method of a class, respectively. Inheritance in Ada 95 is achieved by extending the parent type and overriding the primitive operations. with Objects; use Objects; package Extended_Objects is type Extended_Class is new Class with new data attributes overridden primitive operations procedure Method1 (O: in Extended_Class; Params: Some_Type); procedure Method2 (O: in out Extended_Class; Params : Some_Type); new primitive operation procedure Method3 (O: in out Extended_Class; Params : Some_Type); Polymorphism in Ada 95 is achieved by the use of class-wide types or pointers to class-wide types. It is possible, for example, to declare a pointer to a hierarchy of tagged types rooted at a place in the tree of type extensions. This pointer can then reference any object in the type hierarchy. When a primitive method is called passing the de-referenced pointer, run-time dispatching occurs to the correct operation: type Pointer is access Object.Class'Class; 'Class indicates a class-wide type Ap: Pointer := new .; - some object derived from Object.Class; dispatches to appropriate method In Ada 95, dispatching only occurs when the actual parameter of a call to a primitive operation is of a class-wide type. This contrasts with some other object-oriented programming languages where dispatching is the default (e.g. Java). In order to force dispatching in Ada, the parameter must be explicitly converted to a class-wide type when invoking the primitive operation. This situation often occurs when one primitive operation of an object wants to dispatch to some other primitive operation of the same object. This is called re-dispatching, and can be achieved by converting the operand to a class-wide type, as shown in the following example: type T is tagged record .; procedure P (X: T) is .; procedure Q (X: T) is begin re-dispatch type new T with record .; procedure P (X: T1); Here, procedure Q does a re-dispatch, by explicitly converting the parameter X to a class-wide type before invoking P. If this conversion had been omitted and just called P(X), then the call would be statically bound to the procedure P of T, regardless of what actual parameter was passed to Q. It should be noted that Ada allows calls to overridden operations to be statically bound from outside the defining tagged type. For example, although the Extended Objects package (defined earlier) has extended the Class tagged type and overridden Method1, it is possible for a client to write the following: Eo: Extended_Class; and call the overridden method explicitly. Arguably this has now broken the Extended Class abstraction, and perhaps should be disallowed. Such explicit conversions can only be safely done from within the overridden method itself when it wishes to call its parent method. 3.3 Object-Oriented Programming and Concurrency Although task types and protected types are fully integrated into the typing model of Ada 95, it is not possible to create a tagged protected type or a tagged task type. The designers shied away from this possibility partly because they felt that fully integrating object-oriented programming and concurrency was not a well-understood topic and, therefore, not suitable for an ISO standard professional programming language. Also, there were inevitable concerns that the scope of potential language changes being proposed was too large for the Ada community to accept. In spite of this, there is some level of integration between tagged types and tasks and protected objects. Tagged types after all are just part of the typing mechanism and therefore can be used by protected types and tasks types in the same way as other types. Indeed paradigms for their use have been developed (see (Burns and Wellings, 1998) chapter 13). However, these approaches cannot get around the basic limitation that protected types and task types cannot be extended. Making Concurrent Programming in Ada 95 more Object-Oriented Now that the dust is beginning to settle around the Ada 95 standard, it is important to begin to look to the future. The object-oriented paradigm has largely been welcomed by the Ada community. Even the real-time community, which was originally sceptical of the facilities and worried about the impact they would have on predictability, is beginning to see some of the advantages. Furthermore, as people become more proficient in the use of the language they begin to realise that better integration between the concurrency and object-oriented features would be beneficial. The goal of this paper is to continue the debate on how best to achieve full integration in any future version of the language. There are the following classes of basic types in Ada: ffl scalar types - such as integer types, enumeration types, real types, etc. ffl structured types - such as record types and array types ffl protected types ffl task types ffl access types Access types are special as they provide the mechanism by which pointers to the other types can be created. Note that, although access types to subprograms (procedures and functions) can be created, subprograms are not a basic type of the language. In providing tagged types, Ada 95 has provided a mechanism whereby a structured type can be extended. It should be stressed, though, that only record types can be extended, not array types. This is understandable as the record is the primary mechanism for grouping together items which will represent the heterogeneous attributes of the objects. Furthermore, variable length array manipulation is already catered for in the language. Similarly, scalar types can already be extended using subtypes and derived types. Allowing records to be extended thus is consistent with allowing variable length arrays, subtypes and derived types. protected type is similar to a record in that it groups items together. (In the case of a protected type, these items must be accessed under mutual exclusion.) It would be consistent, then, to allow a protected type to be extended with additional items. The following sections will discuss some of the issues in allowing extensible protected types. The issues associated with extensible task types are the subject of on-going research. 5 Extensible Protected Types The requirements for extensible protected types are easy to articulate. In par- ticular, extensible (tagged) protected types should allow: ffl new data fields to be added, ffl new functions, procedures and entries to be added, ffl functions, procedures and entries to be overridden, class-wide programming to be performed. These simple requirements raise many complex semantic issues. Further- more, any proposed extensions should be fully integrated with the Ada model of object-oriented programming. 5.1 Declaration and Primitive Operations For consistency with the usage elsewhere in Ada, the word 'tagged' indicates that a protected type is extensible. As described in section 3.1, a protected type encapsulates the operations that can be performed on its protected data. Consequently, the primitive operations of a tagged protected type are, in effect, already defined. They are, of course, similar to primitive operations of other tagged types in spirit but not in syntax, since other primitive operations are defined by being declared in the same package specification as a tagged type. Consider the following example: protected type T is tagged procedure W (.); function X (.) return .; entry Y (.); private data attributes of T W, X, and Y can be viewed as primitive operations on T. Interestingly, the call O.X takes a syntactic form similar to that in most object-oriented languages. Indeed, Ada's protected object syntax is in conflict with the language's usual representation of an 'object' (see Section 3.2). 5.2 Inheritance Tagged protected types can be extended in the same manner as tagged types. Hence, protected type T1 is new T with procedure W (.); - override T.W procedure Z (.); - a new method private new attributes of The issue of overriding protected entries will be considered in section 5.4. One consideration is whether or not private fields in the parent type (T) can be seen in the child type (T1). In protected types, all data has to be declared as private so that it can not be changed without first obtaining mutual exclusion. There are four possible approaches to this visibility issue: 1. Prevent a child protected object from accessing the parent's data. This would limit the child's power to modify the behaviour of its parent object, it only being allowed to invoke operations in its parent. 2. Allow a child protected object full access to private data declared in its parent. This would be more flexible but has the potential to compromise the parent abstraction. 3. Provide an additional keyword to distinguish between data that is fully private and data that is private but visible to child types. This keyword would be used in a similar way to private (much like C++ uses its key-word 'protected' to permit descendent classes direct access to inherited data items). 4. Allow child protected types to access private components of their parent protected type if they are declared in a child of the package in which their parent protected type is declared. This would be slightly inconsistent with the way protected types currently work in Ada because protected types do not rely on using packages to provide encapsulation. The remainder of this paper will assume the second method, as it provides the most flexibility and requires no new keywords. It is also consistent with normal tagged types. If a procedure in a child protected type calls a procedure or function in its parent, it should not have to wait to obtain the lock on the protected object before entering the parent, otherwise deadlock would occur. There is one lock for each instance of a protected type and the same lock should be used when the protected object is converted to a parent type. This is consistent with the current Ada approach when one procedure/function calls another in the same protected object. 5.3 Dispatching and re-dispatching Given a hierarchy of tagged protected types, it is possible to create class-wide types and accesses to class-wide types; for example: type Pt is access protected type T'Class; P: Pt := new .; - some type in the hierarchy dispatches to the appropriate projected object. Of course from within P.W, it should be possible to convert back to the class-wide type and re-dispatch to another primitive operation. Unfortunately, an operation inside a tagged protected type does not have the option of converting the object (on which it was originally dispatched) to a class-wide type because this object is passed implicitly to the operation. There are two possible strategies which can be taken: 1. make all calls to other operations from within a tagged protected type dispatching, or 2. use some form of syntactic change to make it possible to specify whether to re-dispatch or not. The first strategy is not ideal because it is often useful to be able to call an operation in the same type or a parent type without re-dispatching. In addition, the first strategy is inconsistent with ordinary tagged types where re-dispatching is not automatic. A solution according to the second strategy uses calls of the form type.operation, where type is the type to which the implicit protected object should be con- verted. The following is an example of this syntax for a re-dispatch: protected body T is procedure P (.) is begin T'Class indicates the type to which the protected object (which is in the hierarchy of type T'Class but which is being viewed as type T) that was passed implicitly to P should be view converted. This allows it to define which Q procedure to call. This syntax is also necessary to allow an operation to call an overridden operation in its parent, for example: protected body T1 is - an extension of T procedure W (.) is - overrides the W procedure of T begin calls the parent operation This new syntax does not conflict with any other part of the language because it is strictly only a type that precedes the period. If it could be an instance of a protected type then the call could be mis-interpreted as an external call: the An alternative syntactic representation might be type'operation. Ada Reference Manual (Intermetrics, 1995) distinguishes between external and internal calls by the use, or not, of the full protected object name (Burns and Wellings, 1998). The call would then be a bounded error. Requeuing can also lead to situations where re-dispatching is desirable. Just as with procedures, re-dispatching would only occur when explicitly requested, so for example, in a protected type T, requeue E would not dispatch whereas requeue T'Class.E would dispatch. Requeuing to a parent entry would require barrier re-evaluation. Requeues from other protected objects or from accept statements in tasks could also involve dispatching to the correct operation in a similar way. 5.4 Entry Calls Allowing entries to be primitive operations of extensible protected types raises many inter-related complex issues. These include: 1. Can a child entry call its parent's entry? - From an object-oriented per- spective, it is essential to allow the child entry to call its parent. This is how reuse is achieved. Unfortunately, from the protected object perspec- tive, calling an entry is a potentially suspending operation and these are not allowed within the body of a protected operation (see section 3.1). It is clear that a compromise is required and that a child entry must be able to extend the facilities provided by its parent. 2. What is the relationship, if any, between the parent's barrier and the child's barrier? - There are three possibilities: no relationship, the child can weaken the parent's barrier, or the child can strengthen the parent's barrier. Fr-lund (Fr-lund, 1992) suggests that as the child method extends the parent's method, the child must have more restrictive synchronisation constraints, in order to ensure that the parent's state remains consistent y . However, he also indicates that if the behaviour of the child method totally redefines that of the parent, it should be possible to redefine the synchronisation constraints. Alternatively, it can also be argued that the synchronisation constraints of the child should weaken those of the par- ent, not strengthen them, in order to avoid violating the substitutability property of subtypes (Liskov and Wing, 1994). 3. How many queues does an implementation need to maintain for an over-ridden - If there is no relationship between the parent and the child barrier, it is necessary to maintain a separate entry queue for each over-ridden entry. If there is more than one queue, the 'Count attribute should reflect this. Hence 'Count might give different values when called from the parent or when called from the child. A problem with using separate entry queues with different barriers for overridden and overriding entries is that it is harder to theorise about the order of entries being serviced. Normally entries are serviced in first-in, first-out (FIFO) order but with separate queues, each with a separate barrier, this might not be possible. For example, a later call to an overridden entry will be accepted before an y It perhaps should be noted that where the child has access to its parent's state, barrier strengthening is not a sufficient condition to ensure the consistency of that state, as the child can make the barrier false before calling the entry. See also the discussion in section 5.4.1. earlier call to an overriding entry if the barrier for the overridden entry becomes true with the overriding entry's barrier remaining false. 4. What happens if a parent entry requeues to another entry? - When an entry call requeues to another entry, control is not returned to the calling entry but to the task which originally made the entry call (see section 3.1). This means that when a child entry calls its parent and the parent entry requeues, control is not returned to the child. Given that the code of the parent is invisible to the child, this would effectively prohibit the child entry from undertaking any post-processing. In order to reduce the number of options for discussion, it is assumed that child entries must strengthen their parent's barrier for the remainder of the paper. The syntax and when is used to indicate this z . To avoid having the body of a child protected object depend on the body of its parent, it is necessary to move the declaration of the barrier from the body to the specification of the protected type (private part). Consider protected type T is tagged private I: Integer := 0; barrier given in the private part protected type T1 is new T with private entry E and when I > 0; If a call was made to A.E, this would be statically defined as a call to T1.E and would be subject to its barrier (E'Count ? 1 and I ? 0). The barrier would be repeated in the entry body. Even with barrier strengthening, the issue of barrier evaluation must be addressed. Consider the case where a tagged protected object is converted to its parent type (using a view conversion external to the protected type) and then an entry is called on that type. It is not clear which barrier needs to be passed. There are three possible strategies that can be taken: 1. Use the barrier associated with the exact entry which is being called, ignoring any barrier associated with an entry which overrides this exact entry. As the parent type does not know about new data added in the child, it could be argued that allowing an entry in the parent to execute when the child has strengthened the barrier for that entry should be safe. Unfortunately, this is not the case. Consider a bounded buffer which has been extended so that the Put and Get operations can be locked. Here, if the lockable buffer is viewed converted to a normal buffer and Get/Put called with only the buffer barriers evaluated, a locked buffer will be accessible even if it is locked. Furthermore, this approach would z Short circuit control forms such as and then when could also be made available. also mean that there would be separate entry queues for overridden entries. The problems associated with maintaining more than one entry queue per overridden entry have already been mentioned. 2. Use the barrier associated with the entry to which dispatching would occur if the object was converted to a class wide type (i.e., the barrier of the entry of the object's actual type). This is the strongest barrier and would allow safe re-dispatching in the entry body. This method results in one entry queue per entry instead of one for each entry and every overridden entry. However, it is perhaps misleading as it is the parent's code which is executed but the child's barrier expression that is evaluated. 3. Allow view conversions from inside the protected object but require that all external calls are dispatching calls. Hence, there is only one entry queue, and all external calls would always invoke the primitive operations of the object's actual type. The problem with this approach is that currently Ada does not dispatch by default. Consequently, this approach would introduce an inconsistency between the way tagged types and extensible protected types are treated. For the remainder of this paper, it is assumed that external calls to protected objects always dispatch x . 5.4.1 Calling the Parent Entry and Parent Requeues So far this section has discussed the various issues associated with overridden entry calls. However, details of how the child entry actually calls its parent have been left unspecified. The main problem is that Ada forbids an entry from explicitly calling another entry (see section 3.1). There are several approaches to this problem. 1. Use requeue. - Although Ada forbids nested entry calls, it does allow an entry call to be requeued. Hence, the child can only requeue to the parent. Requeue gives the impression of calling the parent but it is not possible for the child to do any post-processing once the parent entry has executed (as the call returns to the caller of the child entry). As a requeue, the parent's barrier would have to be re-evaluated. Given that the child barrier has strengthened the parent's barrier, the parent's barrier would normally be open. If this is not the case, an exception is raised (to queue the call would require more than one entry queue). - Furthermore, if atomicity is to be maintained and the parent requeue is to be part of the same protected action, the parent entry must be serviced before any other entries whose barriers also happen to be open. Hence, this requeue has slightly different semantics from a requeue between unrelated entries. 2. Allow the child entry to call the parent entry and treat that call as a procedure call. - It is clear that calling the parent entry is different from a x To harmonize with regular tagged types a new pragma could be introduced called "Ex- ternal Calls Always Dispatch" which can be applied to regular tagged types. - With the requeue approach and multiple entry queues, there need not be any relationship between the parent and the child barriers. Such an approach has already been ruled out in the previous subsection. normal entry call; special syntax has already been introduced to facilitate it (see section 5.3). In this approach, the parent call is viewed as a procedure call and therefore not a potentially suspending operation. However, the parent's barrier is still a potential cause for concern. One option is to view the barrier as an assertion and raise an exception if it is not true. k The other option is not to test the barrier at all, based on the premise that the barrier was true when the child was called and, therefore, need not be re-evaluated until the whole protected action is completed. With either of these approaches, there is still the problem that control is not returned to the child if the parent entry requeues requests to other entries for servicing. This, of course, could be made illegal and an exception raised. However, requeue is an essential part of the Ada 95 model and to effectively forbid its use with extensible protected types would be a severe restriction. The remainder of this paper will assume a model where parent calls are treated as procedure calls (the issue of the assertion is left open) and requeue in the parent is allowed. A consequence of this is that no post-processing is allowed after a parent call. 6 Integration into the Full Ada 95 Model The above section has considered the basic extensible protected type model. Of course, any proposal for the introduction of such a facility must also consider the full implications of its introduction. This section considers the following topics: ffl private types, abstract types, and ffl generics and mix-in inheritance 6.1 Private Types The encapsulation mechanism of Ada 95, the package, gives the programmer great control over the visibility of the entities declared in a package. In par- ticular, Ada 95 supports the notion of private and limited private types, i.e. types whose internal structure is hidden for clients of the packages (where the types are declared) and that can be modified only through the primitive operations declared in these packages (for these types). A protected type is a limited type, hence it is necessary to show how extensible protected types integrate into limited private types. The following illustrates how this is easily achieved. In order to make a type private, its full definition is moved to the private part of the package. This can also be done for extensible protected types: package Example1 is protected type Pt0 is tagged private; private k Special consideration would need to be given to barriers which use the 'Count attribute in the parent, since these will clearly change when the child begins execution. protected type Pt0 is tagged primitive operations. private data items etc. Note that in this example, the primitive operations of type Pt0 are all declared in the private part of the package and are thus visible only in child packages of package Example1. Other packages cannot do anything with type Pt0, because they do not have access to the type's primitive operations. Nevertheless, this construct can be useful for class-wide programming using access types, e.g. through type Pt_Ref is access Pt0'Class; Private types can also give a finer control over visibility. One might declare a type and make some of its primitive operations publicly visible while other primitive operations would be private (and thus visible only to child packages). For example: package Example2 is protected type Pt1 is tagged primitive operations, visible anywhere with private data items etc. private protected type Pt1 is tagged private primitive operations, visible only in child packages private data items etc. Note that the public declaration of type Pt1 uses "with private" instead of only "private" to start its private section. This is supposed to give a syntactical indication that the public view of Pt1 is an incomplete type that must be completed later on in the private part of the package. Alternatively a protected type can be declared to have a private extension. Given a protected type Pt2: package Base is protected type Pt2 is tagged private A private extension can then be written as: with Base; package Example3 is protected type Pt3 is new Base.Pt2 with private; private protected type Pt3 is new Base.Pt2 with Additional primitive operations private Additional data items Here, only the features inherited from Pt2 are publicly visible, the additional features introduced in the private part of the package are private and hence visible only in child packages of package Example3. Private types can be used in Ada 95 to implement hidden and semi-hidden inheritance, two forms of implementation inheritance (as opposed to interface in- heritance, i.e. subtyping). For instance, one may declare a tagged type publicly as a root type (i.e., not derived from any other type) while privately deriving it from another tagged type to reuse the latter's implementation. This hidden inheritance is also possible with extended protected types. Given the above package Base, hidden inheritance from Pt2 can be implemented as follows: with Base; package Example4 is - the public view of Pt4 is a root type protected type Pt4 is tagged primitive operations, visible anywhere with private data items etc. private - the private view of Pt4 is derived from Pt2 protected type Pt4 is new Base.Pt2 with additional primitive operations, visible only in child packages with private additional data items etc. The derivation of Pt4 from Pt2 is not publicly visible: operations and data items inherited from Pt2 cannot be accessed by other packages. If some of the primitive operations inherited from Pt2 should in fact be visible in the public view of Pt4, too, Pt4 must re-declare them and implement them as call-throughs to the privately inherited primitive operations of Pt2. In child packages of package Example4, the derivation relationship is exposed and hence these inherited features are accessible in child packages. Semi-hidden inheritance is similar in spirit, but exposes part of the inheritance relation. Given an existing hierarchy of extensible protected types: package Example5_Base is protected type Pt5 is tagged private protected type Pt6 is new Pt5 with private One can now declare a new type Pt7 that uses interface inheritance from Pt5, but implementation inheritance from some type derived from Pt5, e.g. from with Example5_Base; use Example5_Base; package Example5 is protected type Pt7 is new Pt5 with with private private protected type Pt7 is new Pt6 with private As these examples show, extensible protected types offer the same expressive power concerning private types as ordinary tagged types. In fact, because protected types are an encapsulation unit in their own right (in addition to the encapsulation provided by packages), extensible protected types offer an even greater visibility control than ordinary tagged types. Primitive operations of an extensible protected type declared in the type's private section are visible only within that type itself or within a child extension of that type. Combining this kind of visibility (which is similar to Java's `protected' declarator) with the visibility rules for packages gives some visibility specifications that do not exist for ordinary tagged types. There is one difficulty with this scheme, though. It is currently possible in Ada 95 to define a limited private type that is implemented as a protected type. This raises the question whether the following should be legal: package Example6 is type T is tagged limited private; private protected type T is tagged private Here, although child packages could treat T as an extensible protected type, other client packages could do very little with the type. Furthermore, the mixture of protected and non-protected views of one and the same type may give rise to incalculable implementation problems because in some cases accesses to an object would have to be done under mutual exclusion even if the view of the object's type was not protected, simply because its full view was a protected type. Consequently, the kind of private completion shown in Example6 is probably best disallowed. 6.2 Abstract Extensible Protected Types Ada 95 allows tagged types and their primitive operations to be abstract. This means that instances of the type cannot be created. An abstract type can be an extension of another abstract type. A concrete tagged type can be an extension from an abstract type. An abstract primitive operation can only be declared for an abstract type. However, an abstract type can have non-abstract primitive operations. The Ada 95 model can easily be applied to extensible protected types. The following examples illustrate the integration: protected type Ept is abstract tagged - Concrete operations: function F (.) return .; procedure P (.); Abstract operations: function F1 (.) return . is abstract; procedure P1 (.) is abstract; entry E1 (.) is abstract; private The one issue that is perhaps not obvious concerns whether an abstract entry can have a barrier. On the one hand, an abstract entry cannot be called so any barrier is superfluous. On the other hand, the programmer may want to define an abstraction where it is appropriate to guard an abstract entry. For example: protected type Lockable_Operation is abstract tagged procedure Lock; procedure Unlock; entry Operation (.) is abstract; private entry Operation (.) when not Locked; The bodies of Lock and Unlock set the Locked variable to the corresponding values. Now because of the barrier strengthening rule, the when not Locked barrier will automatically be enforced on any concrete implementation of the operation. The remainder of the paper will assume that abstract entries do not have bar- riers. The above example can be rewritten with a concrete entry for Operation that has a null body. It should be noted, however, that with a concrete null- operation, one cannot force concrete children to supply an implementation for the entry. With an abstract entry, one can. 6.3 Generics and Mix-in Inheritance Ada 95 does not support multiple inheritance. However, it does support various approaches which can be used to achieve the desired affect. One such approach is mix-in inheritance where a generic package which can take a parameter of a tagged type is declared. A version of Ada with extensible protected types must also allow them to be parameters to generics and hence take part in mix-in inheritance. As with normal tagged types, two kinds of generic formal parameters can be defined: type Base_Type is [abstract] protected tagged private; type Derived_From is [abstract] new protected Derived [with private]; In the former, the generic body has no knowledge of the extensible protected type actual parameter. In the latter, the actual type must be a type in the tree of extensible protected types rooted at Derived. Unfortunately, these facilities are not enough to cope with situations involving entries. One of the causes of the inheritance anomaly (Matsuoka and Yonezawa, 1993)(see also section 7) is that adding code in a child object affects the synchronisation code in the parent. Consider the case of a predefined lock which can be mixed in with any other protected object to define a lockable version. Without extra functionality, there is no way to express this. For these reasons, the generic modifier entry !? is used to mean all the entries of the actual parameter. The lockable mix-in type can now be achieved: type Base_Type is [abstract] protected tagged private; package Lockable_G is protected type Lockable_Type is new Base_Type with procedure Lock; procedure Unlock; private entry <> and when not Locked; Lockable_G; The code entry !? and when not Locked indicates that all entries in the parent protected type should have their barriers strengthened by the boolean expression not Locked. The entry !? feature makes it possible to modify the barriers of entries that are unknown at the time the generic unit is written. At the time the generic unit is instantiated, the entries of the actual generic parameter supplied for Base Type are known, and entry !? then denotes a well-defined set of primitive operations. This generic barrier modifier is similar to Fr-lund's ``all-except'' specifier (Fr-lund, 1992), except that the latter also applies to primitive operations that are added later on in further derivations, whereas entry !? does not. If new primitive operations are added in further derivations, it is the programmer's responsibility to make sure that these new entries get the right barriers (i.e., include when not Locked). Clearly, the effect is limited to entries while procedures are unaffected. This gives rise to the following anomaly: If all the barriers need to be strengthened by adding the condition not Locked, it may well be that the inherited procedures need to be similarly guarded. This cannot be done without introducing a mechanism for overriding procedures with entries. This is an Ada-specific inheritance anomaly, which is discussed in the next section. 7 Inheritance Anomaly The combination of the object-oriented paradigm with mechanisms for concurrent programming may give rise to the so-called "inheritance anomaly" (Matsuoka and Yonezawa, 1993). An inheritance anomaly exists if the synchronization between operations of a class is not local but may depend on the whole set of operations present for the class. When a subclass adds new operations, it may therefore become necessary to change the synchronization defined in the parent class to account for these new operations. This section examines how extensible protected types can deal with this inheritance anomaly. Synchronization for extensible protected types is done via entry barriers. An entry barrier can be interpreted in two slightly different ways: ffl as a precondition (which must become a guard when concurrency is introduced in an object-oriented programming language, as (Meyer, 1997) ar- gues). In this sense, entries are the equivalent of partial operations (Herlihy and Wing, 1994). ffl as a synchronization constraint. The use of entry barriers (i.e., guards) for synchronization makes extended protected types immune against one of the kinds of inheritance anomalies identified by (Matsuoka and Yonezawa, 1993): guards are not subject to inheritance anomalies caused by a partitioning of states. To avoid a major break of encapsulation, it is mandatory for a concurrent object-oriented programming language to have a way to re-use existing synchronization code defined for a parent class and to incrementally modify this inherited synchronization in a child class. In our proposal, this is given by the and when clause, which incrementally modifies an inherited entry barrier and hence the inherited synchronization code. Inheritance anomalies in Ada 95 with extended protected types can still occur, though. As (Mitchell and Wellings, 1996) argue, the root cause of inheritance anomalies lies in a lack of expressive power of concurrent object-oriented programming languages: if not all five criteria identified by (Bloom, 1979) are fulfilled, inheritance anomalies may occur. Ada 95 satisfies only three of these criteria; synchronization based on history information cannot be expressed directly using entry barriers (local state must instead be used to record execution history), and synchronization based on request parameter values also is not possible directly in Ada 95. The example for the resource controller shown in section 9.2 exhibits both of these inheritance anomalies. Because the barrier of entry Allocate N cannot depend on the parameter N itself, an internal re- queue to Wait For N must be used instead. The synchronization constraint for Wait For N itself is history-sensitive: the operation should be allowed only after a call to Deallocate has freed some resources. As a result, Deallocate must be overridden to record this history information in local state, although both the synchronization constraints for Deallocate itself as well as its functionality remain unchanged. In addition to that, extensible protected types may suffer from an Ada- specific inheritance anomaly. As synchronization is done via barriers, only entries can be synchronised, but not procedures. If the synchronization constraints of a subtype should restrict an inherited primitive operation that was implemented as a procedure in the parent type, the subtype would have to over-ride this procedure by an entry. However, when using class-wide programming, a task may assume that a protected operation is implemented as a procedure (as that is what the base type indicates) and is therefore non-blocking. At run-time the call might dispatch to an entry and block on the barrier, which would make the call illegal if it occurred within a protected action. For these reasons, overriding procedures with entries should not be allowed for extensible protected types. As discussed in section 6.3, further Ada-specific inheritance anomalies that might arise when mix-in inheritance is used can be avoided by providing additional functionality for generics. The new generic barrier modifier entry !? alone is not sufficient to avoid the introduction of new Ada-specific inheritance anomalies. Because the generic mix-in class must define the synchronization for the complete class resulting from the combination of the mix-in class with some a priori unknown base class, the entry !? barrier modifier was introduced. It allows the mix-in class to impose its own synchronization constraints on an unknown set of inherited operations. However, it is also necessary to have a way for the mix-in class to adapt the synchronization of its additional primitive operations to the synchronization constraints imposed by an actual base type. or and then when When the generic mix-in is instantiated with some base type to create a new result type, it must be possible to parametrise the mix-in's synchronization based upon the base type in order to obtain the correct synchronization for the new result type. How such a parametrisation could be obtained is still a topic of on-going research. 8 Interaction with Tagged Types So far, the discussion has focused on how protected types can be extended. This section now considers the interaction between tagged types and protected tagged types. Consider the following which defines a simple buffer: package Simple_Buffer is type Data_T is tagged private; procedure Write (M : in out procedure Read (M : in private type Data_T is tagged record Such a buffer can only be used safely in a sequential environment. To make a pre-written buffer safe for concurrent access requires it to be encapsulated in a protected type. The following illustrates how this can easily be achieved. protected type Buffer is tagged procedure Write procedure Read (X : out Integer); private The buffer can now only be accessed through its protected interface. Of course if the Buffer protected type is extended, the following will dispatch on the buffer. type B is access Buffer'Class; new .; Alternatively, Simple Buffer.Data T can be made protected but not encapsulated by the following: protected type Buffer is tagged procedure Write (M : in out procedure Read (M : in out private This would allow the buffer to be accessed directly (without the protection overheads) where the situation dictates that it is safe to do so. Combining extensible protected types with class-wide tagged types allow for even more powerful paradigms. Consider protected type Buffer is tagged procedure Write (M : in out Simple_Buffer.Data_T'Class; procedure Read (M : in out Single_Data_T'Class; private Here, both the protected type and the tagged type can be easily extended. The program can arrange for dispatching on the Buffer and from within the Write/Read routines. Further, by using access discriminants the data can be encapsulated and protected from any concurrent use. type Ad is access Simple_Buffer.Data_T'Class; protected type tagged - a normal discriminant procedure Write procedure Read (X : out Integer); private type B is access Buffer'Class; new Buffer(new Simple_Buffer.Data_T).; Here, B1 will dispatch to the correct buffer and Write/Read will dispatch to the correct data which will be encapsulated. 9 Examples This section presents two examples illustrating the principles discussed in this paper. They assume all external calls dispatch, there is no post-processing after parent calls, no checking of parents' barriers, and that the child has access to the parent's state. 9.1 Signals In concurrent programming, signals are often used to inform tasks that events have occurred. Signals often have different forms: there are transient and persistent signals, those that wake up only a single task and those that wake up all tasks. This sections illustrates how these abstractions can be built using extensible protected types. Consider first, an abstract definition of a signal. package Signals is protected type Signal is abstract procedure Send; entry Wait is abstract; private Signal_Arrived type All_Signals is access Signal'Class; package body Signals is protected body Signal is abstract procedure Send is begin Signal_Arrived := True; Now to create a persistent signal: with Signals; use Signals; package Persistent_Signals is protected type Persistent_Signal is new Signal with entry Wait; private entry Wait when Signal_Arrived; package body Persistent_Signals is protected body type Persistent_Signal is entry Wait when Signal_Arrived is begin Signal_Arrived := False; To create a transient signal with Signals; use Signals; package Transient_Signals is protected type Transient_Signal is new Signal with procedure Send; entry Wait; private entry Wait when Signal_Arrived; package body Transient_Signals is protected body type Transient_Signal is procedure Send is begin return; entry Wait when Signal_Arrived is begin Signal_Arrived := False; To create a signal which will release all tasks. type Base_Signal is new protected Signal; package Release_All_Signals is protected type Release_All_Signal is new Base_Signal with entry Wait; private entry Wait and when True; package body Release_All_Signals is protected body Release_All_Signal entry Wait and when True is begin if Wait'Count /= 0 then return; Base_Signal.Wait; Now, of course, My_Signal : All_Signals := .; will dispatch to the appropriate signal handler. 9.2 Advanced Resource Control Resource allocation is a fundamental problem in all aspects of concurrent pro- gramming. Its consideration exercises all Bloom's criteria (see section 7) and forms an appropriate basis for assessing the synchronisation mechanisms of concurrent languages, such as Ada. Consider the problem of constructing a resource controller that allocates some resource to a group of client agents. There are a number of instances of the resource but the number is bounded; contention is possible and must be catered for in the design of the program. (Mitchell and Wellings, 1996) propose the following resource controller problem as a benchmark for concurrent object-oriented programming languages. Implement a resource controller with 4 operations: ffl Allocate: to allocate one resource, ffl Deallocate: to deallocate a resource (which thus becomes available again for allocation) ffl Hold: to inhibit allocation until a call to ffl Resume: which allows allocation again. There are the following constraints on these operations: 1. Allocate is accepted when resources are available and the controller is not held (synchronization on local state and history) 2. Deallocate is accepted when resources have been allocated (synchronization on local state) 3. calls to Hold must be serviced before calls to Allocate (syn- chronization on type of request) 4. calls to Resume are accepted only when the controller is held (synchronization on history information). As Ada 95 has no deontic logic operators, not all history information can be expressed directly in barriers. However, it is possible to use local state variables to record execution history. The following solution simplifies the presentation by modelling the resources by a counter indicating the number of free resources. Requirement 2 is interpreted as meaning that an exception can be raised if an attempt is made to deallocate resources which have not yet been allocated. package Rsc_Controller is Max_Resources_Available : constant Natural := 100; - For example No_Resources_Allocated raised by deallocate protected type Simple_Resource_Controller is tagged entry Allocate; procedure Deallocate; entry Hold; entry Resume; private entry Allocate when Free > 0 and not Locked and - req. 1 entry Hold when not Locked; entry Resume when Locked; - req. 4 The body of this package simply keeps track of the resources taken and freed, and sets and resets the Locked variable. package body Rsc_Controller is protected body Simple_Resource_Controller is entry Allocate when Free > 0 and not Locked and begin Free := Free -1; - allocate resource Taken procedure Deallocate is begin raise No_Resources_Allocated; Free return resource Taken := Taken - entry Hold when not Locked is begin Locked := True; entry Resume when Locked is begin Locked := False; Rsc_Controller (Mitchell and Wellings, 1996) then extend the problem to consider the impact of inheritance: Extend this resource controller to add a method: Allocate N which takes an integer parameter N and then allocates N resources. The extension is subject to the following additional requirements: 5. Calls to Allocate N are accepted only when there are at least available resources. 6. Calls to Deallocate must be serviced before calls to Allocate or Allocate N. The additional constraint that calls must be serviced in a FIFO Within Priorities fashion is ignored here. (Mitchell and Wellings, 1996) also do not implement this, and in Ada 95, it would be done through pragmas. Note that this specification is flawed, and the implementation shown in (Mitchell and Wellings, 1996) also exhibits this flaw: if Deallocate is called when no resources are allocated, the resource controller will deadlock and not service any calls to Deallocate, Allocate, or Allocate N. In this implemen- tation, this has been corrected implicitly, because calling Deallocate when no resources are allocated is viewed as an error and an exception is raised. Requirement 5 is implemented by requeueing to Wait For N if not enough resources are available. Requirement 6 is implicitly fulfilled because calls to Deallocate are never queued since Deallocate is implemented as a procedure. with use Rsc_Controller; package Advanced_Controller is protected type Advanced_Resource_Controller is new Simple_Resource_Controller with entry Allocate_N (N : in Natural); procedure Deallocate; - Ada-specific anomaly: because barriers cannot access parameters, - we must also override this method so that we can set 'Changed' - (see below). private entry Allocate_N when Free > 0 and not Locked and - req. 1 - Note: Ada does not allow access to parameters in a barrier (purely - for efficiency reasons). Such cases must in Ada always be imple- - mented by using internal suspension of the method through a statement. Everything below is just necessary overhead - in Ada 95 to implement the equivalent of having access to - parameters in barriers. - Indicates which of the two 'Wait_For_N' entry queues is the one - that currently shall be used. (Two queues are used: one queue - is used when trying to satisfy requests, requests that cannot - be satisfied are requeued to the other. Then, the roles of the - two queues are swapped. This avoids problems when the calling tasks have different priorities.) Changed something is deallocated. Needed for correct - implementation of 'Allocate_N' and 'Wait_For_N'. Reset each outstanding calls to these routines have been serviced. actually encodes the history information "Wait_For_N" - is only accepted after a call to 'Deallocate'. entry Wait_For_N (for Queue in Boolean) (N : in Natural); entry Wait_For_N (for Queue in Boolean) when not Locked and Hold'Count = 0 and This private entry is used by 'Allocate_N' to requeue to if - less than N resources are currently available. package body Advanced_Controller protected body Advanced_Resource_Controller is procedure Deallocate is - Overridden to account for new history information encoding - needed for access to parameter in the barrier of Allocate_N. begin Changed := True; entry Allocate_N (N : in Natural) when Free > 0 and not Locked and begin if Free >= N then Free := Free - N; Taken else requeue Wait_For_N(Current_Queue); entry Wait_For_N (for Queue in Boolean)(N : in Natural) when not Locked and Hold'Count = 0 and Changed is begin Current_Queue := not Current_Queue; Changed := False; if Free >= N then Free := Free - N; Taken else requeue Wait_For_N(not Queue); Conclusions This paper has argued that Ada 95's model of concurrency is not well integrated with its object-oriented model. It has focussed on the issue of how to make protected types extensible and yet avoid the pitfalls of the inheritance anomaly. The approach adopted has been to introduce the notion of a tagged protected type which has the same underlying philosophy as normal tagged types. Although the requirements for extensible protected types are easily articu- lated, there are many potential solutions. The paper has explored the major issues and, where appropriate, has made concrete proposals. Ada is an extremely expressive language with many orthogonal features. The paper has shown that the introduction of extensible protected types does not undermine that orthogonality, and that the proposal fits in well with limited private types, generics and normal tagged types. The work presented here, however, has not been without its difficulties. The major one is associated with overridden entries. It is a fundamental principle of object-oriented programming that a child object can build upon the functionality provided by its parent. The child can call its parent to access that functionality, and therefore extend it. In Ada, calling an entry is a potentially suspending operation and this is not allowed from within a protected object. Hence, overriding entries gives a conflict between the object-oriented and the protected type models. Furthermore, Ada allows an entry to requeue a call to another entry. When the requeued entry is serviced, control is not returned to the entry which issued the requeue request. Consequently, if a parent entry issues a requeue, control is never returned to the child. This again causes a conflict with the object-oriented programming model, where a child is allowed to undertake post-processing after a parent call. The paper has discussed these conflicts in detail and has proposed a range of potential compromise solutions. Ada 95 is an important language - the only international standard for object-oriented real-time distributed programming. It is important that it continues to evolve. This paper has tried to contribute to the growing debate of how best to fully integrate the protected type model of Ada into the object-oriented model. It is clear that introducing extensible protected types is a large change to Ada and one that is only acceptable at the next major revision of the language. Many of the complications come from the ability to override entries. One possible major simplification of the proposal made here would be not to allow these facilities. Entries would be considered 'final' (using Java terminology). Such a simplification might lead to an early transition path between current Ada and a more fully integrated version. Acknowledgements The authors gratefully acknowledge the contributions of Oliver Kiddle and Kristina Lundqvist to the ideas discussed in this paper. We also would like to acknowledge the participants at the 9th International Workshop on Real-Time Ada Issues who gave us some feedback on some of our initial ideas. --R Integrating Inheritance and Synchronisation in Ada9X Evaluating synchronisation mechanisms Structured multiprogramming Concurrency in Ada Toward a method of object-oriented concurrent program- ming Parellism in object-oriented programming languages Inheritance of synchronization constraints in cocur- rent object-oriented programming languages Linearizability: A correctness criterion for concurrent objects Extended protected types Concurrent Programming in Java A behavioral notion of subtyping DRAGOON: An Ada-based object oriented language for concurrent Analysis of inheritance anomaly in object-oriented concurrent programming languages Systematic concurrent object-oriented programming Extendable dispatchable task communication mechanisms The classiC programming language and design of synchronous concurrent object oriented languages Java Thread The programming language Oberon Parallelism in object-oriented languages: a survey Concurrent programming in concurrents- malltalk --TR Object-oriented concurrent programming ABCL/1 Concurrent programming in concurrent Smalltalk The programming language Oberon Linearizability: a correctness condition for concurrent objects Systematic concurrent object-oriented programming Toward a method of object-oriented concurrent programming Introducing concurrency to a sequential language Analysis of inheritance anomaly in object-oriented concurrent programming languages Integrating inheritance and synchronization in Ada9X A behavioral notion of subtyping Concurrency in Ada Java Threads Object-oriented software construction (2nd ed.) Extensible protected types Concurrency and distribution in object-oriented programming The ClassiC programming language and design of synchronous concurrent object oriented languages Extendable, dispatchable task communication mechanisms Monitors Structured multiprogramming Parallelism in Object-Oriented Languages Inheritance of Synchronization Constraints in Concurrent Object-Oriented Programming Languages Evaluating synchronization mechanisms --CTR Rodrigo Garca Garca , Alfred Strohmeier, Experiences report on the implementation of EPTs for GNAT, ACM SIGAda Ada Letters, v.XXII n.4, December 2002 Albert M. K. Cheng , James Ras, The implementation of the Priority Ceiling Protocol in Ada-2005, ACM SIGAda Ada Letters, v.XXVII n.1, p.24-39, April 2007 Knut H. Pedersen , Constantinos Constantinides, AspectAda: aspect oriented programming for ada95, ACM SIGAda Ada Letters, v.XXV n.4, p.79-92, December 2005 Gustaf Naeser , Kristina Lundqvist , Lars Asplund, Temporal skeletons for verifying time, ACM SIGAda Ada Letters, v.XXV n.4, p.49-56, December 2005 Aaron W. Keen , Tingjian Ge , Justin T. Maris , Ronald A. Olsson, JR: Flexible distributed programming in an extended Java, ACM Transactions on Programming Languages and Systems (TOPLAS), v.26 n.3, p.578-608, May 2004

concurrency;concurrent object-oriented programming;ada 95;inheritance anomaly

353944

Data Dependence Analysis of Assembly Code.

Determination of data dependences is a task typically performed with high-level language source code in today's optimizing and parallelizing compilers. Very little work has been done in the field of data dependence analysis on assembly language code, but this area will be of growing importance, e.g., for increasing instruction-level parallelism. A central element of a data dependence analysis in this case is a method for memory reference disambiguation which decides whether two memory operations may access (or definitely access) the same memory location. In this paper we describe a new approach for the determination of data dependences in assembly code. Our method is based on a sophisticated algorithm for symbolic value propagation, and it can derive value-based dependences between memory operations instead of just address-based dependences. We have integrated our method into the Salto system for assembly language optimization. Experimental results show that our approach greatly improves the precision of the dependence analysis in many cases.

Introduction The determination of data dependences is nowadays most often done by parallelizing and optimizing compiler systems on the level of source code, e.g. C or FORTRAN 90, or some intermediate code, e.g. RTL [21]. Data dependence analysis on the level of assembly code aims at increasing instruction level parallelism. Using various scheduling techniques like list scheduling [6], trace scheduling [9], or percolation scheduling [17], a new sequence of instructions is constructed with regard to data and control dependences, and properties of the target processor. Most of today's instruction schedulers only determine data dependences between register accesses and consider memory to be one cell, so that every two memory accesses must be assumed as data dependent. Thus, analyzing memory accesses becomes more important while doing global instruction scheduling [3]. In this paper, we describe an intraprocedural value-based data dependence analysis, (see Maslov [14] for details about address-based and value-based data dependences), implemented in the context of the SALTO tool [19]. SALTO is a framework to develop optimization and transformation techniques for various processors. The user describes the target processor using a mixture of RTL and C language. A program written in assembly code can then be analyzed and modified using an interface in C++. SALTO has already implemented some kind of conflict analysis [12], but their approach only determines address-based dependences between register accesses and assumes memory to be one cell. When analyzing data dependences in assembly code we must distinguish between accesses to registers and those to memory. In both cases we derive data dependence from reaching definitions and reaching uses information that we obtain by a monotone data flow analysis. Register analysis makes no complications: the set of used and defined registers in one instruction can be established easily, because registers do not have aliases. Therefore, determination of data dependences between register accesses is not in the scope of this paper. For memory references we have to solve the aliasing problem [22]: whether two memory references access the same location. See Landi and Ryder [11] for more details on aliasing. We have to prove that two references always point to the same location (must-alias) or must show that they never refer to the same location. If we cannot prove this, we would like to have a conservative approximation of all alias pairs (may-alias), i.e., memory references that might refer to the same location. To derive all possible addresses that might be accessed by one memory instruction, we use a symbolic value propagation algorithm. To compare memory addresses we use a modification of the GCD test [23]. Experimental results indicate that in many cases our method can be more accurate in the determination of data dependences than other previous methods. 2. Programming Model and Assumptions In the following we assume a RISC instruction set. Memory is only accessed through load (ld) and store (st) instructions. Memory references can only have the following offset. Use of a scaling factor is not provided in this model, but an addition would not be difficult. Memory accesses normally read or write a word of four bytes. For global memory access, the address (which is a label) first has to be moved to a register. Then it can be read or written using a memory instruction. Initialization of registers or copying the contents of one register to another can be done using the mv instruction. All logic and arithmetic operators have the following dest. The operation op is executed on operand src 1 and operand the result is written to register dest. An operand can be a register or an integer constant. Control flow is modeled using unconditional (b) or conditional (bcc) branch instructions. Runtime-memory can be divided into three classes [1]: static or global memory, stack, and heap mem- ory. When an address unequivocally references one of these classes, some simple memory reference disambiguation is feasible (see section 3). Unfortunately it is not easy to prove that an address always references the stack, when no inter-procedural analysis is done from which one can obtain information about the frame pointer. In our approach we do not make such assumptions. 3. Alias Analysis of Assembly Code In this section we briefly review techniques for alias analysis of memory references. Doing no alias analysis leads to the assumption that a store instruction is always dependent on a load or store instruction. A common technique in compile-time instruction schedulers is alias analysis by instruction inspection, where the scheduler looks at two instructions to see if it is obvious that different memory are referenced. With this technique independence of the memory references in Fig. 1 (a) and (b) can be proved, because the same base register but different offsets are used (a), or different memory classes are referenced (b). Fig. 1 (c) shows an example where this technique fails. By looking only at register %o1 it must be assumed that register can point to any memory location, and therefore we have to determine that S3 is data dependent on S2. This local analysis disables notice of the definition of register %o1 in the first statement. This example makes it clear that a two-fold improvement is needed. First, we need to save information about address arithmetic, and secondly we need some kind of copy-propagation. Provided that we have such an algorithm, it would be easy to show that in statement S2 register %o1 has the value %fp \Gamma 20 and therefore there is no overlap between the 4 byte memory blocks starting at 4. Symbolic Value Set Propagation In this section we present an extension of the well-known constant propagation algorithm [23]. Our target is the determination of possible symbolic value sets (contents) for each register and each program statement. In a subsequent step of the analysis this information will be used for the determination of data dependences between storage memory accesses-meaning store and load instructions. The calculation of symbolic value sets is performed by a data flow analysis [10]. Therefore, we have to model our problem as a data flow framework (L; -; F ), where L is called the data flow information set, - is the union operator, and F is the set of semantic functions. If the semantic functions are monotone and (L; -) forms a bounded semi-lattice with a one element and a zero element, we can use a general iterative algorithm [10] that always terminates and yields the least fix-point of the data flow system. 4.1. Data Flow Information Set Our method describes the content of a register in the form of symbolic values. Therefore, we have to define the initialization points of program P. A statement j is called an initialization point R i;j of P if j is a load instruction that defines the content of r i , a call node, or an entry node of a procedure. The finite set of all initialization points of P is given by init(P ). The finite set SV of all symbolic values consists of the symbol ? and all proper symbolic values that are polynomials: a i;j A variable R i;j of a symbolic value represents the value that is stored in register r i at initialization point R i;j . We use the value ? when we cannot make any assumptions on the content of a register. As we are performing a static analysis, we are not able to infer the direction of branches taken during program execu- tion. Therefore, it could happen that for a register r i , more than one symbolic value is valid at a specific program point. As a consequence, we must then describe possible register contents by the so-called k-bounded symbolic value sets. The limitation of the sets is to ensure the termination of the analysis. Let k 2 N be arbitrary, but fixed. Then a k-bounded symbolic value set is a set 1: ld [%fp-4],%o1 2: st %o2,[%fp-8] 1: ld [%fp-4],%o1 2: sethi %hi(.LLC0),%o2 3: st %o3,[%o2+%lo(.LLC0)] 1: add %fp,-20,%o1 2: st %o2,[%o1-4] 3: ld [%fp-20],%o3 (a) (b) (c) Figure 1. Sample code for different techniques of alias detection: (a) and (b) can be solved by instruction inspection, whereas (c) needs a sophisticated analysis. In the following let REGS stand for the set of all registers. We call a total map ff a state. By this means the data flow information set we use for the calculation of symbolic value sets is given by the set of possible states SVS. 4.2. Union Operator If a node in a control flow graph has more than one pre- decessor, we must integrate all information stemming from these predecessors. In data flow frameworks, joining paths in the flow graph is implemented by the union operator. Let then the union operator - of our data flow problem is defined as shown in Fig. 2. The union operator - is a simple componentwise union of sets. Additionally, to ensure the well-definition of the operator, we map arising sets with cardinality greater than k to the special value ?. We have proven, that for a fixed k 2 N, the set of states SVS in conjunction with this union operator constitutes a bounded semi-lattice with a one element and a zero element. 4.3. Semantic Functions In the control flow graph chosen for the analysis, each node stands for a uniquely labeled program statement. Therefore we can unambiguously assign a semantic function to each of the nodes; this semantic function will be used to update the symbolic value sets assigned to each reg- ister. In Fig. 3 we specify some semantic functions used by our method. In this specification, ff stands for a state before the execution of the semantic function, and ff 0 for the corresponding state after the execution of the semantic function. After the execution of an initialization point R i;j , we have no knowledge about the defined value of register r i . The main idea of our method is to describe the register content of r i after such a definition as a symbolic value. As mentioned before, entry nodes of a procedure as well as load instructions are initialization points. The semantic function of an entry node n initializes the symbolic value set for each register r i with its corresponding initialization point R i;n . By doing so, after the execution of n, the symbolic value R i;n stands for the value which is stored in r i before the execution of any procedure code. The semantic function assigned to a load instruction initializes the symbolic value set of the register, whose value will be defined by the operation, similar to the description above of the corresponding initialization point. As opposed to entry nodes, such an initialization is only valid if the initialization point is safe. We call an initialization point safe if the corresponding statement is not part of a loop. In con- trast, an initialization point inside a loop is called unsafe. The problem with unsafe initialization points is that the value of the affected register may change at each loop itera- tion. Therefore, we cannot make a safe assumption about its initialization value. To obtain a safe approximation in such a case the symbolic value set of the register is set to the special value ?. In Fig. 3 we use the operator \Phi which is an extension of the add operator for polynomials. The result of an application A \Phi is a pairwise addition of the terms of A and B. To ensure the well-definition of the operator, resulting sets with cardinality greater than k will be mapped to ?. Further, if one of the operands has the value ?, the operator returns ?. As we have proven that the semantic functions are monotone, the general iterative algorithm [10] can be used to solve our data flow problem. 5. Improvement of Value Set Propagation Without limiting the cardinality of symbolic value sets our propagation algorithm may lead to infinite sets. Registers whose contents could change at each loop iteration are responsible for this phenomenon. The calculated symbolic value set for these registers comprises only the special value ?. Such an inaccuracy in the analysis cannot be accepted in practice. Therefore, we propose an improvement of the symbolic value set propagation algorithm by using registers. 5.1. NSV Registers In this section we introduce the concept of non-symbolic value registers, hereafter called NSV registers. A NSV register of a loop G 0 is a register r i used in G 0 , which content Figure 2. The union operator for symbolic value sets. n: entry n: mv a,%rj (copy a 2 Z into register r j ) 2. ff 0 (r j ) := fag n: add %ri,%rj,%rm (add value of r i and r j and store the result in r m ) 2. ff n: ld [mem],%rj (load value from address mem into register r j ) 2. ff 0 (r j ) := ae fR j;n g if R j;n is a safe initialization point Figure 3. Semantic functions for some instructions. can change. The modified propagation algorithm works as follows: 1. First, we have to determine the NSV registers for all loops of the program. The sets of NSV registers contain among other things induction registers and registers which will be defined by a load instruction in G 0 . 2. Thereafter, for each NSV register r i we insert additional nodes into the control flow graph. At the beginning of the loop body we attach a statement where n 0 is a unique and unused statement number. At the end of the loop body, and before each node of the control flow graph that can be reached after execution of the loop we insert a statement 3. After this, we perform the symbolic value set propagation on the modified control flow graph. Further, for the inserted nodes we have defined semantic functions which set the symbolic value set of r i to the initialization point R i;n 0 (init) resp. to ? (setbot). Now we consider every initialization point as safe. The improved version of our algorithm has two advantages: The number of iterations of the general iterative algorithm, which we use for data flow analysis, will be reduced. Ad- ditionally, we can compare memory addresses even though they depend on NSV registers. 5.2. Determination of NSV Registers In the following let G 0 be a loop and S a statement inside of G 0 . A statement S is called loop invariant if the destination register r i is defined with the same value in each loop iteration. The determination of loop invariant statements of G 0 can be performed in two steps [1]: 1. Mark all statements as loop invariant, which only use constants as operands or operands defined outside of G 0 . 2. Iteratively, mark all untagged statements of G 0 as loop invariant which only use operands that are defined only by a loop invariant statement. The algorithm terminates if no further statement can be marked. By using the concept of loop invariants we can determine the NSV registers of a loop G 0 in a simple way. For this, a register r i is a NSV register in G 0 iff r i is defined by a statement in G 0 that is not a loop invariant statement in G 0 . Fig. 4 shows the results of an improved symbolic value set propagation for a simple program. The NSV registers of the loop are %r1, %r2, %r3, and %r4. For each NSV register an init instruction resp. setbot instructions is inserted into the program. As a consequence, the data flow algorithm terminates after the third iteration. The concept of registers allows a more accurate analysis of memory references inside the loop. Without NSV registers the value of register %r1 would have been set to ? eventually. In contrast, an improved symbolic value propagation always leads to proper values. 6. Data Dependence Analysis The determination of data dependences can be achieved by different means. The most commonly used is the calculation of reaching definitions resp. reaching uses for all statements. This can be described as the problem of de- termining, for a specific statement and memory location, all statements where the value of this memory location has been written last resp. has been used last. Once the reaching definitions and uses have been determined, we are able to infer def-use, def-def, and use-def associations; a def-use pair of statements indicates a true dependence between them, a def-def pair an output dependence, and an use-def pair an anti-dependence. For scalar variables the determination of reaching definitions can be performed by a well-known standard algorithm described in [1]. To use this algorithm for data dependence analysis of assembly code we have to derive the may-alias information, i.e., we have to check whether two storage accesses could refer to the same storage object. To improve the accuracy of the data dependence analysis the must-alias information is needed, i.e., we have to check whether two storage accesses refer always to the same storage object. To achieve all this information we need a mechanism which checks whether the index expressions of two storage could represent the same value. We solve this problem by applying a modified GCD test [23]. Therefore, we replace the appearances of registers in X and Y with elements of their corresponding symbolic value sets, and check for all possible combinations whether the equation has a solution. For an example, we refer to Fig. 4. Obviously, instruction 5 is a reaching use of memory in instruction 8. The derived memory addresses are R respectively. With the assumption that both instructions are executed in the same loop iteration, we can prove that different memory addresses will be accessed. This means, there is no loop-independent data dependence between these two instructions. When the instructions are executed in different loop it- erations, R 1;12 may have different values. The modified GCD test shows that both instructions may reference the same memory location. Therefore, we have to assume a loop-carried data dependence between instructions 5 and 8. 7. Implementation and Results The method for determining of data dependences in assembly code presented in the last sections was implemented as a user function in SALTO on a Sun SPARC 10 workstation running Solaris 2.5. Presently, only the assembly code for the SPARC V7 processor can be analyzed, but an extension to other processors will require minimal technical effort. Results of our analysis can be used by other tools in SALTO. For evaluation of our method we have taken a closer look at two aspects: 1. Comparison of the number of data dependences using our method against the method implemented in SALTO; this shows the difference between address-based and value-based dependence analysis concerning register accesses. 2. Comparison between the number of data dependences using address-based and value-based dependence analysis for memory accesses. As a sample we chose 160 procedures out of the sixth public release of the Independent JPEG Group's free JPEG software, a package for compression and decompression of JPEG images. We distinguish between the following four levels of accuracy: In level 1 we determine address-based dependences between register accesses, memory is modeled as one cell, so that every pair of memory accesses is assumed to be data dependent. Level 2 models the memory the same way as in level 1, and does value-based dependence analysis for register accesses. From level 3 on, register accesses are determined the same way as in level 2, and we analyze memory accesses with our symbolic value set propagation, but in level 3 the derivation of dependence is address-based. In level 4 we perform value-based dependence analysis. Level 1 analysis is performed by SALTO [19], but SALTO does not consider control flow. Two instructions are assumed to be data dependent, even if they cannot be executed one after another. Level 2 is a common technique used by today's instruction schedulers, e.g. the 1. iteration 2. iteration 14 init %r3 3 ld [%r1-40],%r3 5 ld [%r1-80],%r4 8 st %r3,[%r1-40] 9 add %r1,4,%r1 19 setbot %r4 ble .LL11 22 setbot %r4 Figure 4. Symbolic value set propagation. Registers r i that are not mentioned have the value fR i;0 g. one in gcc [21] or the one used by Larus et. al. [20]. Systems that do some kind of value propagation, but only determine address-based dependences, are classified in level 3. In section 8 we will have a closer look at other techniques for value propagation. Our method is classified in level 4. As yet, we know of no other method which also determines value-based dependences. The table only contains those 39 procedures in which an improvement, i.e., less de- pendences, was noticeable from level 3 to level 4. Fig. 5 shows the number of dependences (sum of true, anti-, and output dependences), where we distinguish different levels of accuracy, as well as register and memory accesses. Fig. 5 also shows in the two rightmost columns the effect of a value-based analysis against an address-based analysis. For every procedure it is clear to see the proportion of data dependences that our method disproves. 8. Related Work So far, only some work has been done in the field of memory reference disambiguation. Ellis [8] presented a method to derive symbolic expressions for memory addresses by chasing back all reaching definitions of a symbolic register, the expression is simplified using rules of al- gebra, and two expressions are compared using the GCD test. The method is implemented in the Bulldog compiler, but it works on an intermediate level close to high-level lan- guage. Other authors were inspired by Ellis, e.g. Lowney et. al. [13], B-ockle [4], and Ebcio-glu et. al. [15]. The approach presented by Ebcio-glu is implemented in the Chameleon compiler [16] and works on assembly code. First, a procedure is transformed into SSA form [5], and loops are nor- malized. For gathering possible register values the same Procedure Name LOC Level 1 Level 2 Level 3 Level 4 Improvement Reg. Mem. Reg. Mem. Reg. Mem. Reg. Mem. Reg. Mem. test3function is shifting signed 33 178 38 87 38 87 31 87 22 51% 29% jpeg CreateCompress 126 4273 1945 423 1945 423 1664 423 1619 90% 3% jpeg suppress tables 74 1127 396 143 396 143 229 143 184 87% 20% jpeg finish compress 144 10432 2333 1121 2333 1121 2210 1121 2197 89% 1% emit byte 42 433 214 119 214 119 189 119 184 73% 3% emit dqt 125 4794 1097 575 1097 575 771 575 726 88% 6% emit dht 134 5219 1461 589 1461 589 980 589 870 89% 11% emit sof 100 6389 1282 661 1282 661 1087 661 1077 90% 1% emit sos 100 5252 1285 574 1285 574 873 574 840 89% 4% any marker 41 561 175 184 175 184 110 184 106 67% 4% write frame header 142 4309 1368 679 1368 679 870 679 744 84% 14% scan header 86 3656 626 934 626 934 486 934 459 74% 6% tables only 83 2495 390 716 390 716 324 716 267 71% 18% jpeg abort 38 268 84 93 84 93 67 93 63 65% 6% jpeg CreateDecompress 124 4878 1972 507 1972 507 1716 507 1659 90% 3 % jpeg start decompress 135 4097 902 674 902 674 860 674 856 84% 1% post process 2pass 111 2583 1385 278 1385 278 907 278 878 89% 3% jpeg read coefficients 113 3783 897 538 897 538 853 538 851 86% 1% select file name 104 5631 1146 473 1146 473 714 473 644 92% 10% jround up 20 jcopy sample rows read 1 byte 48 653 84 186 84 186 70 186 67 72% 4% read 2 bytes 93 3115 360 555 360 555 297 555 285 82% 4% next marker 42 567 137 305 137 305 112 305 98 46% 12% first marker 84 1989 259 360 259 360 197 360 187 82% 5% process COM 107 6901 979 1147 979 1147 697 1147 592 83% 15% process SOFn 75 4545 729 670 729 670 601 670 598 85% 1% scan JPEG header 34 804 82 306 82 306 78 306 77 62% 1% read byte 43 415 105 129 105 129 102 129 97 69% 5% read colormap 67 2221 668 305 668 305 583 305 568 86% 3% read non rle pixel 40 368 93 125 93 125 84 125 83 66% 1% read rle pixel 80 976 289 268 289 268 280 268 279 73% 1% jcopy sample rows flush packet 44 468 187 131 187 131 187 131 182 72% 3% start output tga 215 12870 3272 974 3272 974 2937 974 2876 92% 2% Figure 5. Number of dependences (sum of true, anti-, and output dependences) found in four levels of accuracy. The results are divided into register-based and memory-based dependences. The two rightmost columns show the improvement of a value-based dependence analysis on an address-based dependence analysis. technique as in the Bulldog compiler is used. If a register has multiple definitions, the algorithm described in [15] can chase all reaching definitions, whereas the concrete implementation in the Chameleon compiler seems to not support this. Comparing memory addresses makes use of the GCD test and the Banerjee inequalities [2, 23]. The results of their method are alias information. Debray et. al. [7] present an approach close to ours. They use address descriptors to represent abstract addresses, i.e., addresses containing symbolic registers. An address descriptor is a where I is an instruction and M is a set of mod \Gamma k residues. M denotes a set of offsets relative to the register defined in instruction I . Note that an address descriptor only depends on one symbolic register. A data flow system is used to propagate values through the control flow graph. mod \Gamma k sets are used as a bounded semi-lattice is needed (in the tests it is 64). However this leads to an approximation of address representation that makes it impossible to derive must-alias information. The second drawback is that definitions of the same register in different control flow paths are not joined in a set, but mapped to ?. Comparing address descriptors can be reduced to a comparison of mod\Gammak sets, using some dominator information to handle loops cor- rectly. They do not derive data dependence information. 9. Conclusions In this paper we presented a new method to detect data dependences in assembly code. It works in two steps: First we perform a symbolic value set propagation using a monotone data flow system. Then we compute reaching definitions and reaching uses for register and memory access, and derive value-based data dependences. For comparing memory references we use a modification of the GCD test. All known approaches for memory reference disambiguation do not propagate values through memory cells. Remember that loading from memory causes the destination register to have a symbolic value. When we compare two memory references we must have in mind that registers defined in different instructions may have different values, even if they were loaded from the same memory address. To handle this situation we plan to extend our method to propagate values through memory cells. Software pipelining will be one major application of the present work in the near future; this family of techniques overlaps the execution of different iterations from an original loop, and therefore requires a very precise dependence analysis with additional information about the distance of the dependence. Development of this work entails in particular discovering induction variables, which is possible as a post-pass, as soon as loop invariants are known. Then coupling with known dependence tests, such as Banerjee test or Omega test [18] can be considered. Finally, extending our method to interprocedural analysis would lead to a more accurate dependence analysis. Presently we have to assume that the contents of almost all registers and all memory cells may have changed after the evaluation of a procedure call. As a first step, we could make assumptions about the use of global memory loca- tions, and we could derive exact dependences. Acknowledgments We thank the referees for their comments which helped in improving this paper. --R Dependence analysis for supercomputing. Global instruction scheduling for superscalar machines. Exploitation of Fine-Grain Parallelism An efficient method of computing static single assignment form. Some experiments in local microcode compaction for horizontal machines. Alias analysis in executable code. A Compiler for VLIW Architectures. Trace scheduling: A technique for global microcode compaction. Monotone data flow analysis frameworks. Detecting conflicts between structure accesses. Lazy array data-flow dependence analysis A study on the number of memory ports in multiple instruction issue machines. Compiler/architecture interaction in a tree-based VLIW processor Percolation scheduling: A parallel compilation technique. The Omega test: a fast and practical integer programming algorithm for dependence analysis. SALTO: System for assembly-language transformation and optimization Instruction scheduling and executable editing. The GNU instruction scheduler. Limits of instruction-level parallelism Supercompilers for parallel and vector computers. --TR Compilers: principles, techniques, and tools Detecting conflicts between structure accesses Array expansion An efficient method of computing static single assignment form Dependence flow graphs: an algebraic approach to program dependencies Pointer-induced aliasing: a problem taxonomy Limits of instruction-level parallelism A practical algorithm for exact array dependence analysis Abstract interpretation and application to logic programs Binary translation Instruction-level parallel processing The multiflow trace scheduling compiler A hierarchical approach to instruction-level parallelization Abstract interpretation Instruction scheduling and executable editing A study on the number of memory ports in multiple instruction issue machines Alias analysis of executable code Path-sensitive value-flow analysis Advanced compiler design and implementation Dependence Analysis for Supercomputing The Design and Analysis of Computer Algorithms Walk-Time Techniques An Exact Method for Analysis of Value-based Array Data Dependences Data Dependence Analysis of Assembly Code Percolation Scheduling: A Parallel Compilation Technique Bulldog --CTR Thomas Reps , Gogul Balakrishnan , Junghee Lim, Intermediate-representation recovery from low-level code, Proceedings of the 2006 ACM SIGPLAN symposium on Partial evaluation and semantics-based program manipulation, January 09-10, 2006, Charleston, South Carolina Saurabh Chheda , Osman Unsal , Israel Koren , C. Mani Krishna , Csaba Andras Moritz, Combining compiler and runtime IPC predictions to reduce energy in next generation architectures, Proceedings of the 1st conference on Computing frontiers, April 14-16, 2004, Ischia, Italy Patricio Buli , Veselko Gutin, An extended ANSI C for processors with a multimedia extension, International Journal of Parallel Programming, v.31 n.2, p.107-136, April

assembly code;memory reference disambiguation;value-based dependences;monotone data flow frameworks;data dependence analysis

353947

Loop Shifting for Loop Compaction.

The idea of decomposed software pipelining is to decouple the software pipelining problem into a cyclic scheduling problem without resource constraints and an acyclic scheduling problem with resource constraints. In terms of loop transformation and code motion, the technique can be formulated as a combination of loop shifting and loop compaction. Loop shifting amounts to moving statements between iterations thereby changing some loop independent dependences into loop carried dependences and vice versa. Then, loop compaction schedules the body of the loop considering only loop independent dependences, but taking into account the details of the target architecture. In this paper, we show how loop shifting can be optimized so as to minimize both the length of the critical path and the number of dependences for loop compaction. The first problem is well-known and can be solved by an algorithm due to Leiserson and Saxe. We show that the second optimization (and the combination with the first one) is also polynomially solvable with a fast graph algorithm, variant of minimum-cost flow algorithms. Finally, we analyze the improvements obtained on loop compaction by experiments on random graphs.

Introduction Modern computers now exploit parallelism at the instruction level in the microprocessor itself. A sequential microprocessor is not anymore a simple unit that processes instructions following a unique stream. The processor may have multiple independent functional units, some pipelined and possibly some other non-pipelined. To take advantage of these parallel functionalities in the processor, it is not sucient to exploit instruction-level parallelism only inside basic blocks. To feed the functional units, it may be necessary to consider, for the schedule, instructions from more than one basic block. Finding ways to extract more instruction-level parallelism (ILP) has led to a large amount of research from both a hardware and a software perspective (see for example [12, Chap. 4] for an overview). A hardware solution to this problem is to provide support for speculative execution on control as it is done on superscalar architectures and/or support for predicated execution as for example in the IA-64 architecture [8]. A software solution is to schedule statements across conditional branches whose behavior is fairly predictable. Loop unrolling and trace scheduling have this eect. This is also what the software pipelining technique does for loop branches: loops are scheduled so that each iteration in the software-pipelined code is made from instructions that belong to dierent iterations of the original loop. Software pipelining is a NP-hard problem when resource are limited. For this reason, a huge number of heuristic algorithms has been proposed, following various strategies. A comprehensive survey is available in the paper by Allan et al. [2]. They classify these algorithms roughly in three dierent categories: modulo scheduling [16, 24] and its variations ([23, 13, 18] to quote but a few), kernel recognition algorithms such as [1, 21], and move-then-schedule algorithms [14, 19, 5]. Briey speaking, the ideas of these dierent types of algorithms are the following. Modulo scheduling algorithms look for a solution with a cyclic allocation of resources: every clock cycles, the resource usage will repeat. The algorithm thus looks for a schedule compatible with an allocation modulo , for a given (called the initiation interval): the value is incremented until a solution is found. Kernel recognition algorithms simulate loop unrolling and scheduling until a pattern appears, that means a point where the schedule would be cyclic. This pattern will form the kernel of the software-pipelined code. Move-then-schedule algorithms use an iterative scheme that alternatively schedules the body of the loop (loop compaction) and moves instructions across the back-edge of the loop as long as this improves the schedule. The goal of this paper is to explore more deeply the concept of move-then-schedule algorithms, in particular to see how moving instructions can help for loop compaction. As explained by B. Rau in [22], although such code motion can yield improvements in the schedule, it is not always clear which operations should be moved around the back edge, in which direction and how many times to get the best results.] How close it gets, in practice, to the optimal has not been studied, and, in fact, for this approach, even the notion of optimal has not been dened. Following the ideas developed in decomposed software pipelining [9, 27, 4], we show how we can nd directly in one pre-processing step a good 1 loop shifting (i.e. how to move statements across iterations) so that the loop compaction is more likely to be improved. The general idea of this two-step heuristic is the same as for decomposed software pipelining: we decouple the software pipelining problem into a cyclic scheduling problem without resource constraints (nding the loop shifting) and an acyclic scheduling problem with resource constraints (the loop compaction). The rest of the paper is organized as follows. In Section 2, we recall the software pipelining problem and well-known results such as problem complexity, lower bounds for the initiation interval, etc. In Section 3, we explain why combining loop shifting with loop compaction can give better performance than loop compaction alone. Loop shifting changes some loop independent dependences into loop carried dependences and vice versa. Then, loop compaction schedules the body of the loop considering only loop independent dependences, but taking into account the details of the target architecture. A rst optimization is to shift statements so as to minimize the critical path for loop compaction as it was done in [4] using an algorithm due to Leiserson and Saxe. A second optimization is to minimize the number of constraints for loop compaction, i.e. shift statements so that there remain as few loop independent edges as possible. This optimization is the main contribution of the paper and is presented with full details in Section 4. Section 5 discusses some limitations of our technique and how we think we could overcome them in the future. 2 The software pipelining problem As mentioned in Section 1, our goal is to determine, in the context of move-then-schedule software pipelining algorithms, how to move statements so as to make loop compaction more ecient. We thus need to be able to discuss about optimality and about performances relative to an optimum. For that, we need a model for the loops we are considering and a model for the architecture we are targeting that are both simple enough so that optimality can be discussed. These simplied models are presented in Section 2.1. To summarize, from a theoretical point of view, we assume a simple loop (i.e. with no conditional branches 2 ), with constant dependence distances, and a nite number We put good in quotation marks because the technique remains of course a heuristic. Loop compaction itself is NP-complete in the case of resource constraints Optimality for arbitrary loops is in general not easy to discuss as shown by Schwiegelshohn et al. [25]. of non pipelined homogeneous functional units. Despite this simplied model, our algorithm can still be used in practice for more sophisticated resource models (even if the theoretical guarantee that we give is no longer true). Indeed, the loop shifting technique that we develop is the rst phase of the process and does not depend on the architecture model but only on dependence constraints. It just shifts statements so that the critical path and the number of constraints for loop compaction are minimized. Resource constraints are taken into account only in the second phase of the algorithm, when compacting the loop, and a specic and aggressive instruction scheduler can be used for this task. Handling conditional branches however has not been considered yet, although move-then-schedule algorithms usually have this capability [19, 27]. We will thus explore in the future how this feature can be integrated in our shifting technique. We could also rely on predicated execution as modulo scheduling algorithms do. 2.1 Problem formulation We consider the problem of scheduling a loop with a possibly very large number of iterations. The loop is represented by a nite, vertex-weighted, edge-weighted directed multigraph w). The vertices V model the statements of the loop body: each v 2 V represents a set of operations one for each iteration of the loop. Each statement v has a delay (or latency) . The directed edges E model dependence constraints: each edge weight w(e) 2 N, the dependence distance, that expresses the fact that the operation (u; instance of statement u at iteration k) must be completed before the execution of the operation (the instance of statement v at iteration k In terms of loops, an edge e corresponds to a loop independent dependence if to a loop carried dependence otherwise. A loop independent dependence is always directed from a statement u to a statement v that is textually after in the loop body. Thus, if G corresponds to a loop, it has no circuit C of zero weight (w(C) 6= 0). The goal is to determine a schedule for all operations (v; k), i.e. a function respects the dependence constraints and the resource constraints: if p non pipelined homogeneous resources are available, no more than p operations should be being processed at any clock cycle. The performance of a schedule is measured by its average cycle time dened by: Among all schedules, schedules that exhibit a cyclic pattern are particularly interesting. A cyclic schedule is a schedule such that (v; N. The schedule has period : the same pattern of computations occurs every units of time. Within each period, one and only one instance of each statement is initiated: is, for this reason, called the initiation interval in the literature. It is also equal to the average cycle time of the schedule. 2.2 Lower bounds for the average cycle time and complexity results The average cycle time of any schedule (cyclic or not) is limited both by the resource and the dependence constraints. We denote by 1 (resp. p ) the minimal average cycle time achievable by a schedule (cyclic or not) with innitely many resources (resp. p resources). Of course, p 1 . For a circuit C, we denote by (C) the duration to distance ratio w(C) and we let We have the following well-known lower bounds: Dependence constraints: 1 max Resource constraints: p resource constraints, the scheduling problem is polynomially solvable. Indeed, max and there is an optimal cyclic schedule (possibly with a fractional initiation interval if 1 is not integral). Such a schedule can be found with standard minimum ratio algorithms [10, pp. 636- 641]. With d w(e), the complexity is O(jV jjEj log(jV j)), we look for max , and we look for d max e. When d Karp's minimum mean-weight cycle algorithm [15] can be used with complexity O(jV jjEj). With resource constraints however, the decision problem associated to the problem of determining a schedule with minimal average cycle time is NP-hard. It is open whether it belongs to NP or not. When restricting to cyclic schedules, the problem is NP-complete. See [11] for an overview of the cyclic scheduling problem. 3 Loop shifting and loop compaction In this section, we explain how loop shifting can be used to improve the performances of loop compaction. We rst formalize loop compaction and study the performances of cyclic schedules obtained by loop compaction alone. 3.1 Performances of loop compaction alone Loop compaction consists in scheduling the body of the loop without trying to mix up iterations. The general principle is the following. We consider the directed graph A(G) that captures the dependences lying within the loop body, in other words the loop independent dependences. These correspond to edges e such that The graph A(G) is acyclic since G has no circuit C such that can thus be scheduled using techniques for directed acyclic graphs, for example list scheduling. Then, the new pattern built for the loop body is repeated to dene a cyclic schedule for the whole loop. Resource constraints and dependence constraints are respected inside the body by the list scheduling, while resource constraints and dependence constraints between dierent iterations are respected by the fact that the patterns do not overlap. The algorithm is the following. Algorithm 1 (Loop compaction) w) be a dependence graph. 1. Dene 0g. 2. Perform a list scheduling a on A(G). 3. Compute the makespan of a : 4. Dene the cyclic schedule by: 8v 2 V; 8k 2 N; (v; Because of dependence constraints, loop compaction is limited by critical paths, i.e. paths P of maximal delay d(P ). We denote by (G) the maximal delay of a path in A(G). Whatever the schedule chosen for loop compaction, the cyclic schedule satises (G). Furthermore, if a list scheduling is used, Coman's technique [6] shows that there is a path P in A(G) such that How is this related to optimal initiation intervals 1 and p ? We know that For the acyclic scheduling problem, (G) is a lower bound for the makespan of any schedule. Thus, list scheduling is a heuristic with a worst-case performance ratio 2 1=p. Here unfortunately, (G) has - a priori - nothing to do with the minimal average cycle time p . This is the reason why loop compaction alone can be arbitrarily bad. Our goal is now to mix up iterations (through loop shifting, see the following section) so that the resulting acyclic graph A(G) the subgraph of loop independent dependences is more likely to be optimized by loop compaction. 3.2 Loop shifting We rst dene loop shifting formally. Loop shifting consists in the following transformation. We dene for each statement v a shift r(v) that means that we delay operation (v; by r(v) iterations. In other words, instead of considering that the vertex in the graph G represents all the operations of the form (v; k), we consider that it represents all the operations of the form (v; k r(v)). The new dependence distance w r (e) for an edge since the dependence is from (u; k r(u)) to (v; k This denes a transformed graph G w r ). Note that the shift does not change the weight of circuits: for all circuits C, the two operations in dependence are computed in dierent iterations in the transformed code: if w r (e) > 0, the two operations are computed in the original order and the dependence is now a loop carried dependence. If w r both operations are computed in the same iteration, and we place the statement corresponding to u textually before the statement corresponding to v so as to preserve the dependence as a loop independent dependence. This reordering is always possible since the transformed graph G circuit (G and G r have the same circuit weights). If w r (e) < 0, the loop shifting is not legal. Note that G r and G are two representations of the same problem. Indeed, there is a one-to-one correspondence between the schedules for G and the schedules for G r : r is a schedule for G r if and only if the function , dened by (v; r(v)), is a schedule for G. In other words, reasoning on G r is just a change of representation. It cannot prevent us to nd a schedule. The only dierence between the original code and the shifted code is due to loop bounds: the shifted code is typically a standard loop plus a prologue and an epilogue (see the example in Section 4.3). Such a function r is called a legal retiming in the context of synchronous VLSI circuits [17]. Each vertex v represents an operator, with a delay d(v). The weight w(e) of an edge e is interpreted as a number of registers. Retiming amounts to suppress r(u) registers to the weight of each edge leaving u and to add r(v) registers to each edge entering v. The constraint w(e) means that a negative number of registers is not allowed for a legal retiming. The graph A(G) that we used in loop compaction (Algorithm 1) is the graph of edges without register. What we called (G) is now the largest delay of a path without register, called the clock period of the circuit. This link between loop shifting and circuit retiming is not new. It has been used in several algorithms on loop transformations (see for example [5, 3, 4, 7]), including software pipelining. 3.3 Selecting loop shifting for loop compaction How can we select a good shifting for loop compaction? Let us rst consider the strategies followed by the dierent move-then-schedule algorithms. Enhanced software pipelining and its extensions [19], circular software pipelining [14], and rotation software pipelining [5] use similar approaches: they do loop compaction, then they shift backwards (or forwards) the vertices that appear at the beginning (resp. end) of the loop body schedule. In other words, candidates for backwards shifting (i.e. are the sources of A(G) and candidates for forwards shifting (i.e. are the sinks of A(G). This rotation is performed as long as there are some benets for the schedule, but no guarantee is given for such a technique. In decomposed software pipelining, the principle is slightly dierent. The algorithm is not an iterative process that uses loop shifting and loop compaction alternately. Loop shifting is chosen once, following a mathematically well-dened objective, and then the loop is scheduled. Two intuitive objectives may be to shift statements so as to minimize: the maximal delay (G) of a path in A(G) since it is tightly linked to the guaranteed bound for the list scheduling. As shown in Section 3.1, it is a lower bound for the performances of loop compaction and reducing (G) reduces the performance upper bound when list scheduling is used. the number of edges in the acyclic graph A(G), so as to reduce the number of dependence constraints for loop compaction. Intuitively, the fewer constraints, the more freedom for exploiting resources. Until now, all eort has been put into the rst objective. In [9] and [27], the loop is rst software- pipelined assuming unlimited resources. The cyclic schedule obtained corresponds to a particular retiming r which is then used for compacting the loop. In can be shown that, with this technique, the maximal critical path in A(G r ) is less than d In [4], the shift is chosen so that the critical path for loop compaction is minimal, using the retiming algorithm due to Leiserson and Saxe [17] for clock-period minimization (see Algorithm 2 below). The derived retiming r is such that (G r opt the minimum achievable clock period for G. It can also be shown that Both techniques lead to similar guaranteed performances for non pipelined resources. Indeed, the performances of loop compaction (see Equation 2) now become: Unlike for loop compaction alone, the critical path in A(G r ) is now related to p . The performances are not arbitrarily bad as for loop compaction alone. Both techniques however are limited by the fact that they optimize the retiming only in the critical parts of the graph (for an example of this situation, see Section 4.1). For this reason, they cannot address the second objective, trying to retime the graph so that as few dependences as possible are loop independent. In [4], an integer linear programming formulation is proposed to solve this problem. We show in the next section that a pure graph-theoretic approach is possible, as Calland et al. suspected. Before, let us recall the algorithm of Leiserson and Saxe. The technique is to use a binary search for determining the minimal achievable clock period opt . To test whether each potential clock period is feasible, the following O(jV jjEj) algorithm is used. It produces a legal retiming r of G such that G r is a synchronous circuit with clock period (G r ) , if such a retiming exists. The overall complexity is O(jV jjEj log jV j). Algorithm 2 (Feasible clock period) 1. For each vertex set r(v) to 0. 2. Repeat the following times: (a) Compute the graph G r with the existing values for r. (b) for any vertex v 2 V compute (v) the maximum sum d(P ) of vertex delays along any zero-weight directed path P in G r leading to v. (c) For each vertex v such that (v) > , set r(v) to r(v) + 1. 3. Run the same algorithm used for Step (2b) to compute (G r ). If (G r ) > then no feasible retiming exists. Otherwise, r is the desired retiming. 4 Minimizing the number of dependence constraints for loop com- paction We now deal with the problem of nding a loop shifting such that the number of dependence constraints for loop compaction is minimized. We rst consider the particular case where all dependence constraints can be removed (Section 4.1): we give an algorithm that either nds such a retiming or proves that no such retiming exists. Section 4.2 is the heart of the paper: we give an algorithm that minimizes by retiming the number of zero-weight edges of a graph. Then, in Section 4.3, we run a complete example to illustrate our technique. Finally, in Section 4.4, we extend this algorithm to minimize the number of zero-weight edges without increasing the clock period over a given constant (that can be for instance the minimal clock period). This allows us to combine both objectives proposed in Section 3.3: applying loop shifting so as to minimize both the number of constraints and the critical path for loop compaction. 4.1 A particular case: the fully parallel loop body We rst give an example that illustrates why minimizing the number of constraints for loop compaction may be useful. This example is also a case where all constraints can be removed. Example 1 do i=1,n The dependence graph of this example is represented in Figure 1(a): we assume an execution time of two cycles for the loads, one cycle for the addition, and three cycles for the multiplication. Because of the multiplication, the minimal clock period cannot be less than 3. Figure 1(b) depicts the retimed graph with a clock period of 3 and the retiming values found if we run Leiserson and Saxe algorithm (Algorithm 2). Figure 1(c) represents the graph obtained when minimizing the number of zero-weight edges still with a clock period of 3. load load d=3(a) load load (b) load load *1+1(c) Figure 1: (a) The dependence graph of Example 1, (b) After clock period minimization, (c) After loop-carried dependence minimization. (a) (b) (c) L/S ALU L/S ALU L/S ALU cycles load load load load load load Figure 2: (a) The loop compaction for Example 1, (b) After clock period minimization, (c) After loop-carried dependence minimization. Following the principle of decomposed software pipelining, once the retiming (shift) is found, we schedule the subgraph generated by the zero-weight edges (compaction) in order to nd the pattern of the loop body. We assume some limits on the resources for our schedule: we are given one load/store unit and two ALUs. As we can see on Figure 2(a), the simple compaction without shift gives a very bad result (with initiation interval 8) since the constraints impose a sequential execution of the operations. After clock period minimization (Figure 2(b)), the multiplication has no longer to be executed after the addition and a signicant improvement is found, but we are still limited by the single load/store resource associated with the two loop independent dependences which constrain the addition to wait for the serial execution of the two loads (the initiation interval is 5). Finally with the minimization of the loop-carried dependences (Figure 2(c)), there are no more constraints for loop compaction (except resource constraints) and we get an optimal result with initiation interval equal to 4. This can not be improved because of the two loads and the single load/store resource. In this example, resources can be kept busy all the time. Note that if we assume pipelined resources which can fetch one instruction each cycle (with the same delays for latency), we get similar results (see Figure 3). The corresponding initiation interval are respectively 7, 5 and 3, if we do not try to initiate the next iteration before the complete end of the previous one, and 3, 4, 3 if we overlap patterns (see Section 5 for more about what we mean by overlapping). (a) (b) (c) L/S ALU L/S ALU L/S ALU load load load cycles load load load Figure 3: (a) The loop compaction for Example 1 with pipelined resources, (b) After clock period minimization, (c) After loop-carried dependence minimization. Example 1 was an easy case to solve because it was possible to remove all constraints for loop compaction. After retiming, the loop body was fully parallel. In this case, the compaction phase is reduced to the problem of scheduling tasks without precedence constraints, which is, though NP- complete, easier: guaranteed heuristics with a better performance ratio than list scheduling exist. More formally, we say that the body of a loop is fully parallel when: When is it possible to shift the loop such that all dependences become loop carried? Let l(C) be the length (number of edges) of a circuit C in the graph G. The following proposition gives a simple and ecient way to nd a retiming that makes the body fully parallel. Proposition 1 Shifting a loop with dependence graph G into a loop with fully parallel body is possible if and only if w(C) l(C) for all circuits C of G. Proof: Assume rst that G can be retimed so that the loop has a fully parallel body, then there exists a retiming r such that 8e 2 E, w r (e) 1. Summing up these inequalities on each circuit C of G, we get w r (C) l(C), and since the weight of a circuit is unchanged by retiming, we get Conversely, assume that for all circuits C of G, w(C) l(C). Then the graph G dened by 8e 2 E, w no circuit of negative weight. We can thus dene for each vertex u, (u) the minimal weight of a path leading to u. By construction, is the weight of a path leading to v. In other words, is the desired retiming. From Proposition 1, we can deduce an algorithm that nds a retiming of G such that the loop body is fully parallel, or answers that no such retiming can be found. Algorithm 3 (Fully parallel body) by adding a new source s to G, setting apply the Bellman-Ford algorithm on G 0 to nd the shortest path from s to any vertex in V . Two cases can occur: the Bellman-Ford algorithm nds a circuit of negative weight, in this case return FALSE. the Bellman-Ford algorithm nds some values (u) for each vertex u of G 0 , in this case set return TRUE. The complexity of this algorithm is dominated by the complexity of the Bellman-Ford algorithm, which is O(jV jjEj). We can also notice that if the graph is an acyclic directed graph it can always be retimed so that the loop has a fully parallel body since it does not contain any circuit (and consequently, there is no circuit of negative weight in G 0 ). In this case, the algorithm can be simplied into a simple graph traversal instead of the complete Bellman-Ford algorithm, leading to a 4.2 Zero-weight edges minimization (general case) Since we cannot always make the loop body fully parallel, we must nd another solution to minimize the constraints for loop compaction. We give here a pure graph algorithm to nd a retiming for which as few edges as possible are edges with no register (i.e. as many dependences as possible are loop-carried after loop shifting). The algorithm is an adaptation of a minimal cost ow algorithm, known as the out-of-kilter method ([10, pp. 178-185]), proposed by Fulkerson in 1961. 4.2.1 Problem analysis Given a dependence graph and a retiming r of G, we dene, as in the previous sections, the retimed graph G for each edge want to count the number of edges e such that w r For that, we dene the cost v r (e) of an edge e as follows: We dene the cost of the retiming r as the number of zero-weight edges in the retimed graph. We say that r is optimal when r (e) is minimal, i.e. when r minimizes the number of zero-weight edges of G. We will rst give a lower bound for the cost of any retiming, then we will show how we can nd a retiming that achieves this bound. We will use ows in G dened as functions f A nonnegative ow is a ow such that 8e 2 E, f(e) 0. A ow f corresponds to a union of cycle (a multi-cycle) C f that traverses f(e) times each edge e: when f(e) > 0 the edge is used forwards, when f(e) < 0 the edge is backwards. When the ow is nonnegative, the ow corresponds to a union of circuits (a multi-circuit) [10, p. 163]. For a given legal retiming r and a given nonnegative ow f , we dene for each edge e 2 E its kilter index ki(e) by: It is easy to check that the kilter index is always nonnegative since v r and since the ow is nonnegative. We will show in Proposition 4 that, when all kilter indices are zero, we have found an optimal retiming. Before that, we need a independence property related to ows and retimings. We dene as the cost of a ow f for the graph G r . Of course, the cost of a ow depends on the edges of C f , i.e. the edges e such that f(e) 6= 0. The following proposition shows that the cost of a ow does not depend of the retiming, i.e. the cost of f for G and for G r are equal. Proposition 2 Proof: f(e)A since f is a ow. We are now ready to give a lower bound for the cost of any retiming. Proposition 3 For any legal retiming r: nonnegative ow g: Proof: (1 f(e)w(e)) since 8e 2 E; v r (e) 0 (v r because of Prop. 2 0 since 8e 2 E; ki(e) 0: Proposition 4 Let r be a legal retiming. If there is a ow f such that 8e 2 E, is optimal. Proof: if for each edge of G, then (v r and by Proposition 3, Proposition 4 alone does not show that the lower bound can be achieved. It remains to show that we can nd a retiming and a ow such that all kilter indices are zero. We now study when this happens, by characterizing the edges in terms of their kilter index. 4.2.2 Characterization of edges us represent the edges e by the pair (f(e); w r (e)): we get the diagram of Figure 4, called the kilter diagram. Edges e for which correspond to the black angled line. Below and above this line, ki(e) > 0. We call conformable the edges for which non conformable the edges for which ki(e) > 0. We assign a type to each edge e depending on the values w r (e) and f(e), as follows: Type 1: w r (e) > 1 and Type 3: w r Type 4: w r Type Type 7: w r conformable edges Type 2: w r (e) > 0 and f(e) > 1 or w r (e) > 1 and f(e) > 0 Type 5: w r non conformable edges If every edge is conformable, that is if for each edge e, then the optimal is reached. Furthermore, we can notice that for each conformable edge e, it is possible to modify by one unit, either w r (e) or f(e), while keeping it conformable, and for each non conformable edge, it is possible to decrease strictly its kilter index by changing either w r (e) or f(e) by one unit. More precisely, we have the following cases: if f(e) increases: edges of types 3, 6 and 7 become respectively of type 4, 7, and 7, and remain conformable. An edge of type 5 becomes conformable (of type 6). The kilter index of any other edge increases (strictly). if f(e) decreases: edges of type 4 and 7 become respectively of type 3 and 6 or 7, and remain conformable. An edge of type 2 becomes conformable (type 4 or 1) or its kilter index decreases (strictly). The kilter index of edges of type 6 increases (strictly). increases: the edges of type 1, 3 and 6 become respectively of type 1, 1 and 4, and remain conformable. An edge of type 5 becomes conformable (type 3). The kilter index of any other edge increases (strictly). decreases: the edges of type 1 and 4 become respectively of type 1 or 3 and 6, and remain conformable. An edge of type 2 becomes conformable (type 4 or 7) or its kilter index decreases (strictly). The kilter index of edges of type 3 increases (strictly). We are going to exploit these possibilities in order to converge towards an optimal solution, by successive modications of the retiming or of the ow. type 7 type 6 f(e)1r Figure 4: Kilter diagram and edge types. Black Black Black Red Green Uncoloured Green Figure 5: Coloration of the dierent types of edges. 4.2.3 Algorithm The algorithm starts from a feasible initial solution and makes it evolve towards an optimal solution. The null retiming and the null ow are respectively a legal retiming and a feasible ow: for them we have for each edge e 2 E, which means a kilter index equal to 1 for a zero-weight edge and 0 for any other edge. Notice that for this solution any edge is of type 1, 3 or 5. Only the edges of type 5 are non conformable. The problem is then to make the kilter index of type 5 edges decrease without increasing the kilter index of any other edge. To realize that, we assign to each type of edge a color (see Figure 5) that expresses the degree of freedom it allows: black for the edges of type 3, 5 and 6: f(e) and w r (e) can only increase. green for the edges of type 2 and 4: f(e) and w r (e) can only decrease. red for the edges of type 7: f(e) can increase or decrease but w r (e) should be kept constant. uncoloured for the edges of type 1: f(e) cannot be changed while w r (e) can increase or decrease. Note: actually, during the algorithm, we will not have to take into account the green edges of longer. Indeed, if we start from a solution that does not involve such edges, and if we make the kilter index of black edges of type 5 decrease without increasing the kilter index of any other edge, we will never create any other non conformable edge. In particular no edge of type 2 will be created. We can now use the painting lemma (due to Minty, 1966, see [10, pp. 163-165]) as in a standard out-of-kilter algorithm. (Painting lemma) Let E) be a graph whose edges are arbitrarily colored in black, green, and red. (Some edges may be uncoloured.) Assume that there exists at least one black edge e 0 . Then one and only one of the two following propositions is true: a) there is a cycle containing e 0 , without any uncoloured edge, with all black edges oriented in the same direction as e 0 , and all green edges oriented in the opposite direction. b) there is a cocycle containing e 0 , without any red edge, with all black edges oriented in the same direction as e 0 , and all green edges oriented in the opposite direction. R R G G Uncoloured G Figure The two cases of Minty's lemma Proof: The proof is a constructive proof based on a labeling process, see [10]. We end up with the following algorithm: Algorithm 4 (Minimize the number of zero-weight edges) 1. start with color the edges as explained above. 2. if 8e 2 E, (a) then return: r is an optimal retiming. (b) else choose a non conformable edge e 0 (it is black) and apply the painting lemma: i. if a cycle is found, then add one (respectively subtract one) to the ow of any edge oriented in the cycle in the direction of e 0 (resp. in the opposite direction). ii. if a cocycle is found, this cocycle determines a partition of the vertices into two sets. Add one to the retiming of any vertex that belongs to the same set as the terminal vertex of e 0 . (c) update the color of edges and go back to Step 2. Proof: By denition of the edge colors that express what ow or retiming changes are possible, it is easy to check that at each step of the algorithm an edge, whose retiming or ow value changes, becomes conformable if it was not, and remains conformable otherwise. Furthermore the retiming remains legal (w r (e) 0) and the ow nonnegative (f(e) 0). So, after each application of the painting lemma, at least one non conformable edge becomes conformable. By repeating this operation (coloration, then variation of the ow or of the retiming) until all edges become conformable, we end up with an optimal retiming. 4.2.4 Complexity Looking for the cycle or the cocycle in the painting lemma can be done by a marking procedure with complexity O(jEj) (see [10, pp. 164-165]). As, at each step, at least one non conformable edge becomes conformable, the total number of steps is less than or equal to the number of zero-weight edges in the initial graph, which is itself less than the total number of edges in the graph. Thus, the complexity of Algorithm 4 is O(jEj 2 ). Note: for the implementation, the algorithm can be optimized by rst considering each strongly connected component independently. Indeed, once each strongly connected component has been retimed, we can dene (as mentioned in Section 4.1) a retiming value for each strongly connected component such that all edges between dierent strongly connected components have now a positive weight. This can be done by a simple traversal of the directed acyclic graph dened by the strongly connected components. 4.3 Applying the algorithm We now run a complete example for which minimizing the number of zero-weight edges gives some benet. The example is a toy example that computes the oating point number a n given by the recursion: a The straight calculation of the powers is expensive and can be improved. A possibility is to make the calculation as follows (after initialization of rst Example 2 do Assume, for the sake of illustration, that the time for a multiply is twice the time for an add (one cycle). The dependence graph is depicted on Figure 7: the clock period is already minimal (equal to due to the circuit of length 3). The corresponding loop compaction is given, assuming two multipurpose units. The restricted resources impose the execution of the three multiplications in at least 4 cycles, and because of the loop independent dependences, the second addition has to wait for the last multiplication before starting, so we get a pattern which is 5 cycles long. The dierent steps of the algorithm are the following (see also Figure 8): we choose, for example, (d; a) as a non conformable edge, we nd the circuit (a; b; c; d; a) and we change the ow accordingly. we choose (t; a) as a non conformable edge, we nd the cocycle dened by the sets fag and and we change the retiming accordingly. we choose (c; t) as a non conformable edge, we nd the circuit (t; a; d; c; t) and we change the ow accordingly. Figure 7: Dependence graph and the associated schedule for two resources. we choose (b; t) as a non conformable edge, we nd the cocycle dened by the two sets ftg and V n ftg, and we change the retiming accordingly. all edges are now conformable, the retiming is optimal: equal Note that, at each step, we have to choose a non conformable edge: choosing a dierent one than ours can result in a dierent number of steps and possibly a dierent solution, but the result will still be optimal. Note also that, on this example, the clock period remains unchanged by the minimization, there is no need to use the technique that will be presented in Section 4.4. G G G G G R U G Green Red Uncoloured Flow unit Distance (register) unit G G c d G G G c d c c c d Non conformable (black) a a a a a Figure 8: The dierent steps of the zero-weight edge minimization. After the minimization (Figure 9), there remain two (instead of four) loop independent dependences that form a single path. We just have to ll the second resource with the remaining tasks. This results in a 4 cycles pattern which is (here) optimal because of the resource constraints. Figure 9: Dependence graph and associated schedule for two resources after transformation. The resulting shifted code is given below. Since the dierent retiming values are 1 and 0, an extra prelude and postlude have been added compared to the original code: do i=4,n (computed at clock cycle (computed at clock cycle 1) (computed at clock cycle 2) (computed at clock cycle (computed at clock cycle 2) 4.4 Taking the clock period into account We now show that the algorithm given in Section 4.2.3 can be extended so as to minimize the number of zero-weight edges, subject to a constraint on the clock period after retiming. In other words, given a dependence graph d), we want to retime G into a graph G r whose clock period (G r ) is less than or equal to a given constant (which must be a feasible clock period), and which has as few zero-weight edges as possible. We recall that the clock period (G) is dened as the largest delay of a zero-weight path of Gg. 4.4.1 Clock edges Following Leiserson and Saxe technique [17], we add a new constraint between each pair of vertices that are linked by a path whose delay is greater than the desired clock period, since such a path has to contain at least one register: In the above equation, W (u; v) denotes the minimal number of registers on any path between u and v, and D(u; v) denotes the maximal delay of a path from u to v with W (u; v) registers (see [17] for a detailed discussion about the clock period minimization problem and how some redundant constraints can be removed). This corresponds to adding new edges of weight W (u; v) 1 between each pair of vertices (u; v) such that D(u; v) > . We denote by E clock the set of new edges and we let We can notice that although the new edges (that we call clock edges) must have no eect on the retiming cost (since they are not part of the original graph, they should not be taken into account when counting the number of zero-weight edges), they may have some inuence on the ow cost since they constrain the register moves. 4.4.2 Changes in the algorithm We now show how to incorporate these new edges into the algorithm, and how to dene for them a coloration compatible with the previously dened colors. The problem is the following: given a dependence graph w), the goal is to nd a retiming r Z such that r is legal for and of course that minimizes the number of zero-weight edges in E. Summarizing all previous observations, we change the denition of the kilter index in the following way (the kilter index is still nonnegative): This leads us to change also the proofs of Propositions 3 and 4 for taking into account the new edges. We denote by C f (resp. C fclock ) the set of edges in E (resp. in E clock ) such that f(e) > 0, and by C 0 the multi-circuit of G 0 dened by f . Proposition 5 f f(e)w(e). Proof: f f (v r f f(e)w r (e) by Proposition 2 (v r ki(e) by denition of the kilter index Proposition 5 gives a new optimality condition that takes the clock edges into account. Proposition 6 If 8e then r is optimal and furthermore (G r ) . Proof: The proof is similar to the proof of Proposition 4. If (v r clock f(e)w r (e) f(e)w r (e) f f Thus, f f(e)w(e) and by Proposition 5, Furthermore, since the retiming is legal, the inequalities (4) are satised by construction of E clock . In other words, (G r ) . It remains to draw the kilter diagram of E Clock and to assign types and colors. The types are: Type 1: w r (e) > 0 and Type 3: w r Type 4: w r conformable edges Type 2: w r (e) > 0 and f(e) > 0 non conformable edges The colors are chosen as follows: black for type 3 edges, red for type 4 edges, green for type 2 edges, and uncoloured for type 1 edges (see Figure 10). Black f(e) Red r w (e) r Uncoloured Green Figure 10: Kilter diagram and coloration of clock edges. Note: we need to start from a legal retiming, i.e. a retiming for which (G r ) is less than , so that all edge weights are nonnegative (including the weights of clock edges). Such a retiming is computed using Algorithm 2. Then, initially, all the edges of E clock are conformable. As the algorithm never creates non conformable edges, green edges in E clock never appear. The nal algorithm is similar to Algorithm 4. It reaches an optimal retiming while keeping a clock period less than a given constant . The number of steps is still bounded by jEj since all clock edges are conformable and remain conformable, but the number of edges of G 0 is O(jV since G 0 can be the complete graph in the worst case because of the new clock edges. The marking procedure is thus now O(jV and the overall complexity of the algorithm O(jEjjV j 2 ). 5 Conclusions, limitations and future work As any heuristic, the shift-then-compact algorithm that we proposed can fall into traps, even if we rely on an optimal (exponential) algorithm for loop compaction. In other words, for a given problem instance, the shift that we select may not be the best one for loop compaction. Nevertheless, the separation between two phases, a cyclic problem without resource constraints and an acyclic problem with resource constraints, allows us to convert worst case performance results for the acyclic scheduling problem into worst case performance results for the software pipelining problem (see Equation 3). This important idea, due to Gasperoni and Schwiegelshohn [9], is one of the interests of decomposed software pipelining. No other method (including modulo-scheduling) has this property: how close they get to the optimal has not been proved. This separation into two phases has several other advantages. The rst advantage that we already mentioned is that we do not need to take the resource model into account until the loop compaction phase. We can thus design or use very aggressive (even exponential) loop compaction algorithms for specic architectures. A second advantage is that it is fairly easy to control how statements move simply by incorporating additional edges. This is how we enforced the critical path for loop compaction to be smaller than a given constant. This constant could be opt , but choosing a larger value can give more freedom for the second objective, the minimization of loop independent edges. As long as this value is less than holds. Another example is if we want to limit by a constant l(e) the maximal value w r (e) of an edge which is linked to the number of registers needed to store the value created by u: we simply add an edge from v to u with weight l(e) w(e). Taking these additional edges into account can be done as we did for clock edges. A third advantage (that we still need to explore) is that the rst phase (the loop shifting) can maybe be used alone as a pre-compilation step, even for architectures that are dynamically scheduled: minimizing the number of dependences inside the loop body for example will reduce the number of mispredictions for data speculation [20]. Despite the advantages mentioned above, our technique has still one important weakness, which comes from the fact that we never try to overlap the patterns obtained by loop compaction. In practice of course, instead of waiting for the completion of a pattern before initiating the following one (by choosing the initiation interval equal to the makespan of loop compaction, see Algorithm 1), we could choose a smaller initiation interval as long as resource constraints and dependence constraints are satised. For each resource p, we can dene end(p) the last clock cycle for which p is used in the acyclic schedule a . Then, given a , we can choose as initial interval the smallest that satises all dependences larger than p. This is for example how we found the initiation intervals equal to 3, 4 and 3 for Example 2 with pipelined resources (end of Section 4.3). While this overlapping approach may work well in practice, it does not improve in general the worst case performance bound of Equation 3. We still have to nd improvements in this direction, especially when dealing with pipelined resources. We point out that Gasperoni and Schwiegelshohn approach [9] does not have this weakness (at least for the worst case performance bound): by considering the shifting and the schedule for innite resources as a whole, they can use a loop compaction algorithm with release dates (tasks are not scheduled as soon as possible) that ensures a good overlapping. In the resulting worst case performance bound, d max can be replaced by 1 for the case of pipelined resources. The problem however is that it seems dicult to incorporate other objectives (such as the zero-weight edge minimization described in this paper) in Gasperoni and Schwiegelshohn approach since there is no real separation between the shifting and the scheduling problems. Nevertheless, this loop compaction with release dates looks promising and we plan to explore it. A possibility when dealing with pipelined resources is to change the graph so that d cutting an operation into several nodes: a rst real node corresponding to the eective resource utilization, and several virtual nodes, one for each unit of latency (virtual nodes can be scheduled on innite virtual resources). With this approach, we will nd a shorter clock period (because the delay of a node is now cut into several parts) equal to d 1 e. The situation is actually equivalent to the case of unit delays and unpipelined resources (and thus the bound of Equation 3 is the same with d max replaced by 1). The problem however is that the algorithm is now pseudo-polynomial because the number of vertices of the graph depends linearly on d max . This can be unacceptable if pipeline latencies are large. Another strategy is to rely, as several other heuristics do, on loop unrolling. We can hide the preponderance of d max (in the worst case bound and in the overlapping) by unrolling the loop a sucient number of times so that p is large compared to d max . Since resources are limited, unrolling the loop increases the lower bound due to resources and thus the minimal initiation interval. But, this is also not completely satisfying. We are thus working on better optimization choices for the initial shift or on an improved compaction phase to overcome this problem. Currently, our algorithm has been implemented at the source level using the source-to-source transformation library, Nestor [26], mainly to check the correctness of our strategies. But we found dicult to completely control what compiler back-ends do. In particular, lot of work remains to be done to better understand the link between loop shifting and low-level optimizations such as register allocation, register renaming, strength reduction, and even the way the code is translated! Just consider Example 2 where we write instead simple modication reduces 1 and opt to 3! Minimizing the number of zero-weight edges still lead to the best solution for loop compaction (with a clock period equal to 4), except if we simultaneously want to keep the clock period less than 3. Both programs are equivalent, but the software pipelining problem changes. How can we control this? This leads to the more general question: at which level should software pipelining be performed? --R Perfect pipelining Software pipelining. Circuit retiming applied to decomposed software pipelining. Rotation scheduling: a loop pipelining algorithm. Combining retiming and scheduling techniques for loop parallelization and loop tiling. Generating close to optimum loop schedules on parallel processors. Graphs and Algorithms. Cyclic scheduling on parallel processors: an overview. Computer architecture: a quantitative approach (2nd edition). Circular scheduling. A characterization of the minimum cycle mean in a digraph. Software pipelining Retiming synchronous circuitry. Swing modulo scheduling: a lifetime-sensitive approach Dynamic speculation and synchronization of data dependences. Iterative modulo scheduling: an algorithm for software pipelining. Iterative modulo scheduling. Some scheduling techniques and an easily schedulable horizontal architecture for high performance scienti The Nestor library: A tool for implementing Fortran source to source transformations. Decomposed software pipelining. --TR Graphs and algorithms Software pipelining: an effective scheduling technique for VLIW machines On optimal parallelization of arbitrary loops Circular scheduling An efficient resource-constrained global scheduling technique for superscalar and VLIW processors Lifetime-sensitive modulo scheduling Rotation scheduling Minimum register requirements for a modulo schedule Specification of software pipelining using Petri nets Decomposed software pipelining Software pipelining Circuit Retiming Applied to Decomposed Software Pipelining Computer architecture (2nd ed.) The IA-64 Architecture at Work Perfect Pipelining Some scheduling techniques and an easily schedulable horizontal architecture for high performance scientific computing Swing Modulo Scheduling --CTR Pemysl cha , Zdenk Hanzlek , Antonn Hemnek , Jan Schier, Scheduling of Iterative Algorithms with Matrix Operations for Efficient FPGA Design--Implementation of Finite Interval Constant Modulus Algorithm, Journal of VLSI Signal Processing Systems, v.46 n.1, p.35-53, January 2007

list scheduling;software pipelining;cyclic scheduling;circuit retiming

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Efficient Rule-Based Attribute-Oriented Induction for Data Mining.

Data mining has become an important technique which has tremendous potential in many commercial and industrial applications. Attribute-oriented induction is a powerful mining technique and has been successfully implemented in the data mining system DBMiner (Han et al. Proc. 1996 Int'l Conf. on Data Mining and Knowledge Discovery (KDD'96), Portland, Oregon, 1996). However, its induction capability is limited by the unconditional concept generalization. In this paper, we extend the concept generalization to rule-based concept hierarchy, which enhances greatly its induction power. When previously proposed induction algorithm is applied to the more general rule-based case, a problem of induction anomaly occurs which impacts its efficiency. We have developed an efficient algorithm to facilitate induction on the rule-based case which can avoid the anomaly. Performance studies have shown that the algorithm is superior than a previously proposed algorithm based on backtracking.

Introduction Data mining (also known as Knowledge Discovery in Databases) is the nontrivial extraction of implicit, previously unknown, and potentially useful information from data [12]. Over the past twenty years, huge amounts of data have been collected and managed in relational databases by industrial, commercial or public organizations. The growth in size and number of existing databases has far exceeded the human abilities to analyze such data with available technologies. This has created a need and a challenge for extracting knowledge from these databases. Many large corporations are investing into the data mining tools, and in many cases, they are coupled with the data warehousing technology to become an integrated system to support management decision and business planning [27]. Within the research community, this problem of data mining has been touted as one of the many great challenges [10, 12, 25]. Researches have been performed with different approaches to tackle this problem [2, 3, 4, 8, 9, 15, 16, 18, 19, 21, 28]. The research of the authors were supported in part by RGC (the Hong Kong Research Grants Council) grant HKU 286/95E. Research of the fourth author was supported in part by grants from the Natural Sciences and Engineering Research Council of Canada the Centre for Systems Science of Simon Fraser University. In the previous studies [16], a Basic Attribute-Oriented Induction (Basic AO Induction) method has been developed for knowledge discovery in relational databases. AO induction can discover many different types of rules. A representative type is the characteristic rule. For example, from a database of computer science students, it can discover a characteristic rule such as "if x is a computer science student, then there is a 45% chance that he is a foreign student and his GPA is excellent". Note that the concepts of "foreign student" and "excellent GPA" do not exist in the database. Instead, only lower level information such as "birthplace" and GPA value are stored there. An important feature of AO induction is that it can generalize the values of the tuples in a relation to higher level concepts, and subsequently merge those tuples that have become identical into generalized tuples. An important observation is that each one of these resulted generalized tuples certainly reflects some common characteristics of the group of tuples in the original relation from which it is generated. Basic AO induction has been implemented in the mining system DBMiner, (whose prototype was called DBLearn) [16, 17, 19]. Besides AO induction, DBMiner also has incorporated many interesting mining techniques including mining multiple-level knowledge, meta-rule guided mining, and can discover many different types of rules, patterns, trends and deviations [17]. AO induction, not only can be applied to relation databases, but also performed on unconventional databases such as spatial, object-oriented, and deductive databases [19]. The engine for concept generalization in basic AO induction is the concept ascension. It relies on a concept tree or lattice to represent the background knowledge for generalization. [22]. However, concept tree and lattice have their limitation in terms of background knowledge representation. In order to further enhance the capability of AO induction, there is a need to replace them by a more general concept hierarchy. In this paper, the work has been focused on the development of a rule-based concept hierarchy to support a more general concept generalization. Rule-based concept generalization was first studied in [7]. In general, concepts in a concept tree or lattice are generalized to higher level concepts unconditionally. For example, in a concept tree defined for the attribute GPA of a student database, a 3.6 GPA (in a 4 points system) can be generalized to a higher level concept, perhaps to the concept excellent. This generalization depends only on the GPA value but not any other information (or attribute) of a student. However, in some institutions, they may want to apply different rules to different types of students. The same 3.6 GPA may only deserve a good, if the student is a graduate; and it may be excellent, if the student is an undergraduate. This suggests that a more general concept generalization scheme should be conditional or rule-based. In a Rule-Based Concept Graph, a concept can be generalized to more than one higher level concept, and rules are used to determine which generalization path should be taken. To support AO induction on a rule-based concept graph, we have extended the basic AO induction to Rule-Based Attribute-Oriented Induction. However, if the technique of induction on a concept tree is applied directly to a concept graph, a problem of induction anomaly would occur. In [7], a "backtracking" technique was proposed to solve this problem. It is designed based on generalized relation which is proposed originally in the basic AO induction. The backtracking algorithm has an O(n log n) complexity, where n is the number of tuples in the induction domain. In [23], a more efficient technique for induction has been proposed. In this paper, we apply the technique to rule-based concept graph, and propose an algorithm for rule-based induction whose complexity is improved to O(n). The algorithm has avoided to use the data structure of generalized relation. Instead, it uses a multi-dimensional data cube [20] or a generalized-attribute tree depending on the sparseness of the data distribution. Extensive performance studies on the algorithm have been done and the results show that it is more efficient than the backtracking algorithm based on generalization relation. poor excellent strong Figure 1: Concept tree table entries for a university student database The paper is organized as follows. The primitives of knowledge discovery and the principle of basic AO induction are briefly reviewed in Section 2. The general notions of rule-based concept generalization and rule-based concept hierarchy are discussed in Section 3. The model of rule-based AO induction is defined in Section 4. A new technique of using path relation and data cube instead of generalized relation to facilitate rule-based AO induction is discussed in Section 5. An efficient rule-based AO induction algorithm is presented in Section 6. Section 7 is the performance studies. Discussions and conclusions are in Sections 8 and 9. Basic Attribute-Oriented Induction The purpose of AO induction is to discover rules from relations. The primary technique used is concept generalization. In a database such as a university student relation with the schema the values of attributes like Status, Age, Birthplace, GPA can be generalized according to a concept hierarchy. For example, GPA between 0.0 and 1.99 can be generalized to "poor", those between 2.0 and 2.99 to "average", other values to "good" or "excellent". After this process, many records would have the same values except on those un-generalizable attributes such as Name. By merging the records which have the same generalized values, important characteristics of the data can be captured in the generalized tuples and rules can be generated from them. Basic AO induction is proposed along this approach. Task-relevant data, background knowledge, and expected representation of learning results are the three primitives that specify a learning task in basic AO induction [16, 19]. 2.1 Primitives in AO Induction The first primitive is the task-relevant data. A database usually stores a large amount of data, of which only a portion may be relevant to a specific induction task. A query that specifies an induction task can be used to collect the task-relevant set of data from a database as the domain of an induction. In AO induction, the retrieved task-relevant tuples are stored in a table called initial relation. The second primitive is the background knowledge. In AO induction, background knowledge is necessary to support generalization, and it is represented by concept hierarchies. Concept hierarchies could be supplied by domain experts. As have been pointed out, generalization is the key engine in induction. M.A. Ph.D. sophomore senior junior M.S. Undergraduate Graduate freshman ANY Figure 2: A concept tree for Status Therefore, the structure and representation power of the concept hierarchies is an important issue in AO induction. Concepts hierarchies in a particular domain are often organized as a multi-level taxonomy in which concepts are partially ordered according to a general-to-specific ordering. The most general concept is the null description (described by a reserved word "ANY"), and the most specific concepts corresponds to the low level data values in the database [22]. The simplest hierarchy is concept tree in which a node can only be generalized to one higher level node at each step. Example 1 Consider a a typical university student database with a schema Part of the corresponding concept tree table is shown in Figure 5, where A ! B indicates that B is a generalization of the members of A. An example of a concept tree on the attribute status is shown in Figure 6. 2 Student records in the university relation can be generalized following the paths in the above concept trees. For example, the status value of all students can be generalized to "undergraduate" or "graduate". Note that the generalization should be performed iterative from lower levels to higher levels together with the merging of tuples which have the same generalized values. The generalization should stop once the generalized tuples resulted from the merging have reached a reasonable level in the concept trees. Otherwise, the resulted tuples could be over generalized and the rules generated subsequently would have no practical usage. The third primitive is the representation of learning results. The generalized tuples at the end will be used to generate rules by converting them to logic formula. This follows the fact that a tuple in a relation can always be viewed as a logic formula in conjunctive normal form, and a relation can be characterized by a large set of disjunctions of such conjunctive forms [13, 26]. Thus, both the data for learning and the rules discovered can be represented in either relational form (tuples) or first-order predicate calculus. For exam- ple, if one of the generalized tuples resulted from the generalization and merging of the computer science student records in the university database is (graduate(Status), NorthAmerican(Birthplace), good(GPA)), then the following rule in predicate calculus can be generated: Note that the above rule is only one of the many rules that would be generated, and it is not quantified yet. We will explain how quantification of the rules is done in AO induction. Many kinds of rules, such as characteristic rules, and discriminant rules can be discovered by induction processes [15]. A characteristic rule is an assertion that characterizes a concept satisfied by all or most of the examples in the class targeted by a learning process. For example, the symptoms of a specific disease can be summarized by a characteristic rule. A discriminant rule is an assertion which discriminates a concept of the class being learned from other classes. For example, to distinguish one disease from others, a discriminant rule should summarize the symptoms that discriminate this disease from others. 2.2 Concept Generalization The most important mechanisms in AO Induction are concept generalization and rule creation. Generalization is performed on all the tuples in the initial relation with respect to the concept hierarchy. All values of an attribute in the relation are generalized to the same higher level. A selected set of attributes of the tuples in the relation are generalized to possibly different higher levels synchronously and redundant tuples are merged to become generalized tuples. The resulted relation containing these generalized tuples is called a generalized relation, and it is smaller than the initial relation. In other words, a generalized relation is a relation which consists of a set of generalized attributes storing generalized values of the corresponding attributes in the original relation. Even though, a generalized relation is smaller than the initial relation; however it may still contain too many tuples, and is not practical to convert them to rules. Therefore, some principles are required to guide the generalization to do further reduction. An attribute in a generalized relation is at a desirable level if it contains only a small number of distinct values in the relation. A user of the mining system can specify a small integer as a desirable attribute threshold to control the number of distinct values of an attribute. An attribute is at the minimum desirable level if it would contain more distinct values than a defined desirable attribute threshold when generalized to a level lower than the current one [16]. The minimum desirable level for an attribute can also be specified explicitly by users or experts. A special generalized relation R 0 of an original relation R is the prime relation [19] of R if every attribute in R 0 is at the minimum desirable level. The first step of AO induction is to generalize the tuples in the initial relation to proper concept levels such that the resulted relation becomes the prime relation. The prime relation has a useful characteristic of containing minimal distinct values for each attribute. However, it may still has many tuples, and would not be suitable for rule generation. Therefore, AO induction will generalize and reduce the prime relation further until the final relation can satisfy the user's expectation in terms of rule generation. This generalization can be done repetitively in order to generate rules at different concept levels so that a user can find out the most suitable levels and rules. This is the technique of progressive generalization (roll-up) [19]. If the rules discovered in a level is found to be too general, then re-generalization to some lower levels can be performed and this technique is called progressive specialization [19]. DBMiner has implemented the roll-up and drill-down techniques to support user to explore different generalization paths until the resulted relation and rules so created can satisfy his expectation. A discovery system can also quantify a rule generated from a generalized tuple by registering the number of tuples from the initial relation, which are generalized to the generalized tuple, as a special attribute count in the final relation. The attribute count carries database statistics to higher level concept rules, supports pruning scattered data and searching for substantially weighted rules. A set of basic principles for AO induction related to the above discussion has been proposed in [15, 16]. 3 Rule-Based Concept Generalization In Basic AO Induction, the key component to facilitate concept generalization is the concept tree. Its generalization is unconditional and has limited generality. From this point on, we will focus our investigation on the rule-based concept generalization which is a more general scheme. Concepts are partially ordered in a concept hierarchy by levels from specific (lower level) to general (higher level). Generalization is achieved by ascending concepts along the paths of a concept hierarchy. In general, a concept can ascend via more than one path. A generalization rule can be assigned to a path in a concept hierarchy to determine whether a concept can be generalized along that path. For example, the generalization of GPA in Figure 5 could depend not only on the GPA of a student but also his status. A GPA could be a good GPA for an undergrad but a poor one for a graduate. For example, the rules to categorize a GPA in (2.0 - 2.49) may be defined by the following two conditional generalization rules: If a student's GPA is in the range (2.0 - 2.49) and he is an undergrad, then it is an average GPA. If a student's GPA is in the range (2.0 - 2.49) and he is a grad, then it is a poor GPA. Concept hierarchies whose paths have associated generalization rules are called rule-based concept hier- archies. Concept hierarchies can be balanced or unbalanced. Unbalanced hierarchy can always be converted to a balanced one. For easy of discussion, we will assume all hierarchies are balanced. Also, similar to concept tree, we will assume that the concepts in a hierarchy are partially ordered into levels such that lower level concepts will be generalized to the next higher level concepts and the concepts converge to the null concept "ANY" in the top level (root). (The two notions of "concept" on the concept hierarchy and "generalized attribute value" are equivalent, depending on the context, we will use the two notions interchangeably.) In the following, three types of concept generalization, their corresponding generalization rules and concept hierarchies are classified and discussed. 3.1 Unconditional Concept Generalization This is the simplest type of concept generalization. The rules associated with these hierarchies are the unconditional IS-A type rules. A concept is generalized to a higher level concept because of the subsumption relationship indicated in the concept hierarchy. This type of hierarchies support concept climbing generalization. The most popular unconditional concept generalization are performed on concept tree and lattice. The hierarchies represented in both Figure 5 and Figure 6 belong to this type. 3.2 Deductive Rule Generalization In this type of generalization, the rule associated with a generalization path is a deduction rule. For example, the deduction rule: if a student's GPA is in the range (2.0 - 2.49) and he is a grad, then it is a poor GPA, can be associated with the path from the concept GPA 2 (2:0 \Gamma 2:49) to the concept poor in the GPA hierarchy. This type of rules is conditional and can only be applied to generalize a concept if the corresponding condition can be satisfied. A deduction generalization rule has the following form: For a tuple x, concept (attribute value) A can be generalized to concept C if condition B can be satisfied by x. The condition B(x) can be a simple predicate or a general logic formula. In the simplest case, it can be a predicate involving a single attribute. A concept hierarchy associated with deduction generalization rules is called deduction-rule-based concept graph. This structure is suitable for induction in a database that supports deduction. 3.3 Computational Rule Generalization The rules for this type of generalization are computational rules. Each rule is represented by a condition which is value-based and can be evaluated against an attribute or a tuple or the database by performing some computation. The truth value of the condition would then determine whether a concept can be generalized via the path. For example, in the concept hierarchy for a spatial database, there may be three generalization paths from regional spatial data to the concepts of small region, medium size region and large region. Conditions like "region size - SMALL REGION SIZE", "region size ? SMALL REGION SIZE - region size ! LARGE REGION SIZE", and "region size - LARGE REGION SIZE" can be assigned to these paths respectively. The conditions depend on the computation of the value of region size from the regional spatial data. In general, computation rules may involve sophisticated algorithms or methods which are difficult to be represented as deduction rules. The hierarchy with associated computation rules is called computation-based concept graph. This type of hierarchy is suitable for induction in databases that involve a lot of numerical data, e.g., spatial databases and statistical databases. 3.4 Hybrid Rule-Based Concept Generalization A hierarchy can have paths associated with all the above three different types of rules. This type of hierarchy is called hybrid rule-based concept graph. It has a powerful representation capability and is suitable for many kinds of applications. For example, in many spatial databases, some generalization paths are computation bound and are controlled by computation rules. While some symbolic attributes can be generalized by deduction rules. And many simple attributes can be generalized by unconditional IS-A rules. In the scope of database induction, the same technique can be used on all the different types of rule-based hierarchies. Therefore, in the rest of this paper, we will use deduction-rule-based concept graph as a typical concept hierarchy. Rule-Based Attribute-Oriented Induction In order to discuss the technique of AO induction in the rule-based case. We first define a general model for rule-based AO induction. A rule-based AO induction system is defined by five components (DB, CH, DS, KR, t a ). DB is the underlying extensional database. CH is a set of rule-based concept hierarchies associated with the attributes in DB. We assume these hierarchies are deduction-rule-based concept graphs. DS is a deduction system supporting the concept generalization. The generalization rules in CH, together with some other deduction rules form the core of DS. In the simple case, DS may consist of only the rules in CH. KR is a knowledge representation scheme for the learned result. It can be any one of the popular schemes predicate calculus, frames, semantic nets, production rules, etc. Following the approach in basic AO induction, we assume KR in the induction system is first-order predicate calculus. The last component t a is the desirable attribute threshold defined in the basic induction. Note that all these five components are input to the rule-based deduction system. Output of the system is the rules discovered from the database. The generalization and rule creation processes in rule-based induction is fundamentally the same as that in the basic induction. However, an attribute value could be generalized to different higher level concepts depending on the concept graph. As a consequence, the techniques in basic induction have to be modified to solve the induction problem in this case. We will describe in this section the frame work of the rule-based induction, and explain the induction anomaly problem which occurs in this case. Same as the basic induction, the first step of rule-based induction is to generalize and reduce the initial relation to the prime relation. The minimum desirable level can be found in a scan of the initial relation. Once the minimum desirable levels have been determined, the initial relation can be generalized to these levels in a second scan, and the result would be the prime relation. This step of inducing the prime relation is basically the same as that of the basic induction, except that the induction is performed on a general concept graph rather than a restricted concept tree. Once the prime relation is found, selected attributes are to be generalized further and the generalization- comparison-merge process will be repeated to perform roll-up or drill-down. (Some selected attributes can also be removed before the generalization starts.) In the basic induction, attributes in a prime relation can always be generalized further by concept tree ascension because the generalization is based only on the current generalized attribute values. However, this may not be the case in the rule-based induction because the application of a rule on the prime relation may require additional information not available in the prime relation. This phenomenon is called induction anomaly. Following are some cases that will cause this anomaly to happen. (1) A rule may depend on an attribute which has been removed. (2) A rule may depend on an attribute whose concept level in the prime relation has been generalized too high to match the condition of the rule. (3) A rule may depend on a condition which can only be evaluated against the initial relation, e.g., the number-of-tuples in the relation. In the following, an example will be used to illustrate the rule-based induction on a concept graph and the associated induction anomaly. Example 2 This example is based on the induction in Example 1. We will enhance the concept tree there to a rule-based concept graph to explain the rule-based induction. The database DB is the same student database in Example 1, and the mining task is the same which is to discover the characteristic rules for CS students. The modification is in the concept hierarchy CH. The unconditional rules for the attribute GPA in Figure 5 are replaced by the set of deduction rules in Figure 7. And the concept tree for GPA is enhanced to the rule-based concept graph in Figure 8 which has been labeled with the corresponding deduction rules in Figure 7. For example, the GPA in the range (3:5 \Gamma 3:79) poor poor R 4 R 5 excellent excellent strong strong strong. Figure 3: Conditional Generalization Rules for GPA strong average good excellent poor ANY R6 R7 R8 Figure 4: A rule-based concept graph for GPA would no longer be generalized to "excellent" only, as would be the case if the concept tree in Figure 5 is being followed. Instead, it will be checked against the two rules R 6 and R 7 in Figure 7. If it is the GPA of a graduate student, then it will be generalized to "good"; otherwise, it must be an undergraduate, and will be generalized to "excellent". Suppose the tuples in the initial relation in Example 1 has been generalized according to the rule-based concept graph in Figure 8. After comparison and merging, the prime relation resulted is the one in Table 1 1 . In performing further generalization in Table 1, some rules which reference no other information than the generalized attribute values, such as R 9 , R 12 and R 13 in Figure 7, can be applied directly to the prime relation to further generalize the GPA attribute. For example, the GPA value "good" in the second tuple can be generalized to "strong" according to R 12 , and the value "poor" in the fifth row to "weak" according to R 9 . However, for the GPA value "average" in the first tuple, it cannot be decided with the information in the prime relation which one of the two rules R 10 and R 11 should be applied. If the student is either a senior or graduate, then R 10 should be used to generalize the GPA to "weak"; otherwise, it should be generalized to "strong". However, the status information (freshman=sophomore=junior=senior) has been lost during the previous generalization, and is not available in the prime relation. In fact, the 40 1 The comparison and merging technique used here follows that proposed in [15] Status Sex Age GPA Count undergrad M undergrad M 16 25 good 20 undergrad F Table 1: A Prime Relation from the Rule-Based Generalization. student tuples in the initial relation which are generalized and merged into the first tuple may have all the different status. Therefore, if further generalization is performed, its value "average" will be generalized to both "weak" and "strong", and the tuple will be split into two generalized tuples. 2 It is clear from Example 2 that further generalization from a prime relation may have difficulty for rule-based induction. Therefore, the generalization technique has to be modified to suit the rule-based case. In Section 5, we will describe a new method of using path relation instead of generalized relation to solve the induction anomaly problem. 5 Path Relation Since any generalization could introduce induction anomaly into the generalized relation, any further generalization in the rule-based case has to be started again from the initial relation which has the full information. However, re-applying the deduction rules all over again on the initial relation is costly and wasteful. All the deduction that have been done previously in the generation of the prime relation is wasted, and has to be redo. In order to solve this problem, we propose to use a path relation to capture the generalization result from one application of the rules on the initial relation such that the result can be reused in all subsequent generalization. An attribute value may be generalized to the root via multiple possible paths on a concept graph. However, for the attribute value of a given tuple in the initial relation, it can only be generalized via a unique path to the root. Each one of the multiple paths on which an attribute value can be generalized is a generalization path. Since the concepts on the graph are partially ordered, there are only a finite number of distinct generalization paths from the bottom level. In general, the number of generalization paths of an attribute should be small. Before an induction starts, a preprocessing is used to identify and label the generalization paths of all the attributes. For example, the generalization paths of the concept graph of GPA in Figure 8 are identified and labeled in Figure 9. For every attribute value of a tuple in the initial relation, its generalization path can be identified by generalizing the tuple to the root. Therefore, each tuple in the initial relation is associated with a tuple of generalization paths. In a scan of the initial relation, every tuple can be transformed into a tuple of ids of the associated generalization paths. The result of the transformation is the path relation of the initial relation. It is important to observe that the path relation has captured completely the generalization result of the Path 4 Path 5 Figure 5: Generalization Paths for GPA initial relation at all levels. In other words, given the generalization paths of a tuple, the generalized values of a tuple can be determined easily from the concept graph without redoing any deduction. Furthermore, the set of generalization paths on which some attribute values in the initial relation are generalized to the root can be determined during the generation of the path relation. By checking the number of distinct attribute values (concepts) on each level in the concept graph through which some paths found above have traversed, the minimum desirable level can be found. It can be concluded at this point that the path relation is an effective structure for capturing the generalization result in the rule-based case. By using it, the repetitive generalization required by roll-up and drill-down can be done in an efficient way without the problem introduced by the induction anomaly. Name Status Sex Age GPA J. Wong freshman M C. Chan freshman F 19 2.8 D. Zhang senior M 21 2.7 A. Deng senior F 21 3.3 C. Ma M.A. M 28 2.3 A. Chan sophomore M 19 2.4 Table 2: Initial Relation from the student database. Example 3 Assume that the initial relation of the induction in Example 2 is the one in Table 2. Its path relation can be generated in one scan by using the generalization rules in Figure 7 and the path ids specified in Figure 9. (Figure 7 only has the rules for GPA, the rules of the other attributes are simple.) Table 3 is the path relation of Table 2. 2 Another issue in rule-based generalization is the cyclic dependency. A generalization rule may introduce dependency between attributes. The generalization of an attribute value may depend on that of another Status path id Sex path id Age path id GPA path id Table 3: Path Relation from the Initial Relation. Status GPA Age Sex Figure Generalization Dependency Graph attribute. If the dependency is cyclic, it could introduce deadlock in the generalization process. In order to prevent cyclic dependency, rule-based induction creates a generalization dependency graph from the generalization rules and prevents deadlock by ensuring the graph is acyclic. The nodes in the generalization dependency graph are the levels of each attributes in the concept graph. (In the rest of the paper, we have adopted the convention of numbering the top level (root) of a concept graph level 0, and to increase the numbering from top to bottom. In other word, the bottom level would have the highest level number.) We use L ij to denote the node associated with level j of an attribute A i . In the dependency graph, there is an edge from every lower level node L ik to the next higher level node L i(k\Gamma1) for each attribute A i . Also, if the generalization of an attribute A i from L ik to level L i(k\Gamma1) depends on another attribute A j in level L jm , (i 6= j), then there is an edge from L jm to L i(k\Gamma1) . In Figure 10, we have the generalization dependency graph of the generalization rules in Example 2. For example, the edges from L 22 to L 21 , from L 12 to L 21 , and from L 11 to L 21 are introduced by the rule R 10 . If the dependency graph is acyclic, a generalization order of the concepts can be derived from a partial ordering of the nodes. Following this order, every attribute values of a tuple can be generalized to any level in the concept graphs. For example, a generalization order of the graph in Figure 10 is (L 12 Moreover, any tuple in the initial relation in Table 2 can be generalization to the root following this order. 5.1 Data Structure for Generalization poor good excellent average senior freshman sophomore junior 0_15 GPA Status Age Figure 7: A multi-dimensional data cube In the prototype of the DBMiner system (called DBLearn), the data structure of generalization relation is used to store the intermediate results. Both generalized tuples and their associated aggregate values such as "count" are stored in relation tables. However, it is discovered that generalized relation is not the most efficient structure to support insertion of new generalized tuples and comparison of identical generalized tuples. Furthermore, the prime relation may not have enough information to support further generalization in the rule-based case, which makes it an inappropriate choice for storing intermediate results. To facilitate rule-based induction, we propose to use either a multi-dimensional data cube or a generalized-attribute tree. A data cube is a multi-dimensional array, as shown in Figure 11, in which each dimension represents a generalized attribute and each cell stores the value of some aggregate attributes, such as "count" or "sum". For example, the data cube in Figure 11 can store the generalization result of the initial relation in Table 2 to levels 2,2,1 of the attributes Status, GPA and Age. (Please refer to the generalized attributes of the levels in Figure 10.) Let v be a vector of desirable levels of a set of attributes, and the initial relation is required to be generalized to the levels in v. During the generalization, for every tuple p in the initial relation, the generalized attribute values of p with respect to the levels in v can be derived from its path ids in the path relation and the concept graphs. Let p g be the tuple of these generalized attribute values. In order to record the count of p and update the aggregate attribute values, p g is used as an index to a cell in the data cube. The count and the aggregate attribute values of p are recorded in this cell. For example, the count of the tuple (J.Wong, freshman, M, 18, 3.2) in the initial relation in Table 2 would be recorded in the cell whose index is the generalized tuple (undergraduate, M, Many works have been published on how to build data cubes [1, 6, 14]. In particular, how to compute data cubes storing aggregated values efficiently from raw data in a database. In our case, we only need to use a data cube as a data structure to store the "counts", i.e., number of tuples that have been generalized into a higher level tuple. Therefore, the details of how to compute a cube on aggregations from a base cube has no relevancy. For the AO induction algorithm, the cube is practically a multi-dimensional array. In [17], the data cube has been compared with the generalized relation. It costs less to produce a data cube, and its performance is better except when the data is extremely sparse. In that case, the data cube may waste some storage. A more space efficient option is to use a b-tree type data structure. We propose to use a b-tree called generalized-attribute tree to store the count and aggregate attribute values. In this approach, the generalized tuple p g will be used as an index to a node in the generalized-attribute tree and the count and aggregate values will be stored in the corresponding node. According to the experience in the DBMiner system, data cube is very efficient as long as the percentage of occupied cells is reasonably dense. Therefore, in rule-based induction, data cube is the favorable data structure; and the generalized-attribute tree should be used only in case the sparseness is extremely high. 6 An Efficient Rule-Based AO Induction Algorithm We have shown that in the case of rule-based induction, it is more efficient to capture the generalization results in the path relation, and to use the data cube to store the intermediate results. In the following, we present the path relation algorithm for rule-based induction. Algorithm 1 Path Relation Algorithm for Rule-Based Induction task specification is input into a Rule-Based AO Induction System (DB, CH, DS, KR, t a ) /* the initial relation R, whose attributes are A i , (1 - i - n), is retrieved from DB */ Step One: Inducing the prime relation Rpm from R 1: transform R into the path relation R 2: compute the minimum desirable level L i for each A i (1 - i - n); 3: create a data cube C with respect to the levels L i 4: scan R p ; for each tuple p 2 R p , compute the generalized tuple p g with respect to the levels L i , update the count and aggregate values in the cell indexed by p 5: convert C into the prime relation Rpm ; Step Two: Perform progressive generalization to create rules 1: Select a set of attributes A j and corresponding desirable levels L j for further generalization; 2: create a data cube C for the attributes A j with respect to the levels L 3: scan the path relation R p ; compute the generalized tuple p g for each tuple p 2 R p with respect to the desirable levels update the count and aggregate values in the cell indexed by p 4: convert all non-empty cells of C into rules. Two will be repeated until the rules generated satisfy the discovery goal 2 . 2 Explanation of the algorithm: In the first step, the path relation R p is first created from the initial relation. After that, the minimum desirable levels L i , (1 - i - n), are computed by scanning R p once, and a data cube C is created for the generalized attributes at levels L i . In 4, R p is scanned again and every tuple in R p is generalized to levels L i . For every p in R p , its generalized tuple p g is used as an index to 2 The meaning of "discovery goal" follows those defined in [15], and will be discussed in the following paragraph. locate a cell in C in which the count and other aggregate values are updated. At the end of the first step, the non-empty cells in C are converted to tuples of the prime relation Rpm . The second step is the progressive generalization, it is repeated until the rules generated satisfy the discovery goal. There are two ways to define the discovery goal. In [15], a threshold was defined to control the number of rules generated. Once the number of rules generated is reduced beyond the given threshold, then the goal has been reached. Another way is to allow the generalization to go through an interactive and iterative process until the user is satisfied with the rules generated. In other words, no pre-defined threshold would be given, but the goal is reached when the user is comfortable with the rules generated. This is compatible with the roll-up and drill-down approach used in many data mining systems. The details of step two in the algorithm is the following. In the beginning of each iteration, a set of attributes A j and levels L j are selected for generalization, and a data cube C is created with respect to the levels L j . Following that, the tuples in R p are generalized to the levels L j in the same way as that in the first step. After all corresponding cells in the data cube have been updated, the non-empty cells are converted into rules. This can be repeated until the number of rules reached a pre-defined threshold or the user is satisfied with the rules generated. In the above algorithm, if the rules discovered are too general and in a level which is too high, then the generalization can be redone to a lower level. Hence, it can perform not only progressive generalization, but also progressive specialization. Example 4 We extend Example 2 here to provide a complete walk through of Algorithm 1. The task is to discover the characteristic rules from the computer science students in a university database. The initial relation has been extracted from the database, and it is presented in Table 2. In the following, we will describe the execution of Algorithm 1 on Table 2 in detail. The input are the same as those described in Example 2. The database D is the database of computer science students. The concept hierarchy CH is the one described in Figure 7. The initial relation R is the one in Table 2. Step One: Inducing the prime relation Rpm from R 1: Scan R and generalize each tuple in R to the root with respect to the concept graph (Figure 7) to identify the associated path ids. For example, the 2.8 GPA of the second student in Table 2 can be generalized to "average" by R 4 , and then to "strong" by R 11 Figure 7). (Note that R 11 is applied instead of R 10 , because the student is a "freshman".) Therefore, the path for the GPA attribute associated with this tuple is Path 6 Figure 9), and the associated path id tuple is the second one in Table 3. Following this mechanism, every tuple in R is transformed into a tuple of path ids, and R is transformed into the path relation R p in Table 3. 2: Compute the minimum desirable levels for each attribute in Table 2 by checking the number of concepts at each level through which some generalization paths identified in the previous step have traversed. 3: Create a data cube C with respect to the minimum desirable levels found. 4: Scan the path relation R p (Table 3). For each tuple p 2 R p , by using the corresponding path ids, find out the generalized values p g of p on the concept graphs with respect to the minimum levels. For example, assume the minimum level for GPA is found to be level 2, since the path id for Status Age GPA Count undergrad undergrad grad grad Table 4: A Final Relation from the Rule-Based Generalization. GPA of the first tuple in R p (Table 3) is "7", it can be identified from Path 7 (Figure that the generalized value at level 2 is "good". Once p g is found, update the count and aggregate values in the cell indexed by p g . For example, the first tuple of Table 3 is generalized to (undergrad, average), the count in the corresponding cell is updated. 5: For every nonempty cell in C, a corresponding tuple can be created in a generalized relation, and the result is the prime relation Rpm in Table 1. For example, the count in the cell in C indexed by (undergrad, M, 16 25, average) has count equal to 40 and it is converted to the first tuple in Table 1). Step Two: Perform progressive generalization to create rules 1: In order to perform further generalization on the prime relation Rpm (Table 1), the attributes Sex and Birthplace are removed and GPA is generalized one level higher from level 2 to level 1. 2: A data cube C is created for the remaining attributes Status, Age and GPA with respect to the new levels. (In fact, only GPA is moved one level higher.) 3: Scan the path relation R p to compute the generalized tuple p g for each tuple p 2 R p with respect to the new levels, and update the count and aggregate values in the cell. 4: Convert all non-empty cells of C into a generalized relation, and the result is the final relation in Table 4. If the final relation can satisfy the user's expectation, it is then converted to the us analyze the complexity of the path relation algorithm. The cost of the algorithm can be decomposed into the cost of induction and that of deduction. The deduction portion is the one time cost of generalizing the attribute values to the root in building the path relation. The induction portion covers those spent on inducing the prime and the final relations. The induction portion of the algorithm is very efficient, which will be discussed in the following theorem. However, the cost of the deduction portion will depend on the complexity of the rules and the efficiency of the deduction system DS. A general deductive database system may consist of complex rules involving multiple levels of deduction, recursion, negation, aggregation, etc. and thus may not exist an efficient algorithm to evaluate such rules [26]. However, the deduction rules in the algorithm are the conditional rules associated with a concept graph, which in most cases, are very simple conditional rules. Therefore, it is practical to assume each generalization in the deduction process is bounded by a constant in the analysis of the algorithm. When the concept graphs involve more complex deduction rules, the complexity of the algorithm will depend on the complexity of the deduction system. The following theorem shows the complexity of the path relation algorithm under the assumption of the bounded cost of the deduction processes. Theorem 1 If the cost of generalizing an attribute to any level is bounded, the complexity of the path relation algorithm for rule-based induction is O(n), where n is the number of tuples in the initial data relation. Proof. In the first step, the initial relation and the path relation are scanned once in steps 1 and 4. The time to access a cell in the data cube is constant, therefore, the complexity of this step is O(n). In the subsequent progressive generalization, assume that the number of rounds of generalization is k, which is much smaller than n. In each round, the path relation will be scanned once only. Therefore, the complexity is bounded by n \Theta k. Adding the costs of the two steps together, the complexity of the entire induction process is O(n). 2 7 Performance Study Our analysis in Section 6 has shown that the complexity of the relation path algorithm is O(n), which is as good as that of the algorithms proposed for the more restricted non-rule-based case. Moreover, the path relation algorithm proposed here is more efficient than a previously proposed backtracking algorithm [7] which has a complexity of O(n log n). To confirm this analysis, an experiment has been conducted to compare the performance between the path relation algorithm and the backtracking algorithm. There are two main differences between the two algorithms. (1) The generalization in the backtracking algorithm uses generalized relation as the data structure. (2) All further generalization after the prime relation has been generated is based on the information in the prime relation. As has been explained, the prime relation will introduce the induction anomaly in rule-based induction. Because of that, the generalized tuples in the prime relation have to be backtracked to the initial relation and split according to the multiple possible generalization paths in the further generalization. The backtracking and splitting has to be performed in every round of progressive generalization. This impacts the performance of the backtracking algorithm when comparing with the path relation algorithm. The path relation algorithm is more efficient because the path relation has captured all the necessary induction information in the path ids and its tuples can be generalized to any level in the rule-based case. For comparison purpose, both algorithms were implemented and executed on a synthesized student database similar to the one in Example 1. The records in the database have the following attributes f Name, Status, Sex, Age, GPAg. The records are generated such that values in each attribute are random within the range of possible values and satisfy some conditions. The following conditions are observed so that the data will contain some interesting patterns rather than being completely random. 1. Graduate students are at least 22 years old. 2. Ph.D. students are at least 25 years old. 3. Graduate students' GPAs are at least 3.00. For each attribute, there is a corresponding rule-based concept graph same as that in Example 1. In order to compare the two algorithms, we assume the attributes Status, Sex, Age, GPA are generalized to level 1,1,1,3 respectively in the prime relation, (the levels are those indicated in Figure 10). Following that, GPA is further generalized to level 2 and then level 1. Hence the the generalization order is: Number of No. of tuples in Path Backtracking Backtracking of records final relation relation Alg Alg. (case 1) Alg. (case 2) 10000 6 192 4028 24033 10000 12 191 4028 24033 10000 1000 6 21 395 2396 1000 12 20 395 2396 1000 Figure 8: Number of pages read Since the cost in the algorithms is dominated by the scanning in the database, the metric in the comparison is the number of pages required to read and write. For the path relation algorithm, the initial relation is first transformed into the path relation, and the algorithm use the path relation to generalize the records. For each generalization step, we use a data cube to store the counts and aggregate values of the generalized tuples. The number of pages written is much smaller than the number of pages read as the path relation contains only the path id for the data of each attribute and is much smaller than the initial relation. For the backtracking algorithm, the initial relation is used to form the prime relation and a virtual attribute is added to the initial relation to record the linkages between the tuples in the initial and prime relations. (Please see [7] for details). In the generalization of the attribute GPA from level 3 to level 2, the induction anomaly occurs. The initial relation with the virtual attribute is read and the selected attribute GPA is generalized and form the enhanced-prime relation by merging the tuples. The enhanced- prime relation has the virtual attribute values to support splitting of the prime relation for the further generalization. Two cases are considered for the implementation of the enhanced-prime relation. In the first case the enhanced-prime relation is stored in the main memory. In the second case, the enhanced-prime relation is stored in the secondary storage and the pages read and written are counted accordingly. The results of the comparison are shown in Figures 12 and 13. It can be seen from the two figures that the number of pages read and written by the path relation algorithm is much smaller. And the reduction ratio between the two algorithms is comparable to the ratio between n and n log n, where n is the number of records, which matches our analysis in the previous section. This has clearly demonstrated that the path relation algorithm is more efficient. As an example, the final relation for generalizing 10000 records with the final relation threshold set to 8 is shown in Figure 14. Number of No. of tuples in Path Backtracking Backtracking of records final relation relation Alg Alg. (case 1) Alg. (case 2) 1000 6 4 26 28 1000 12 4 26 28 1000 Figure 9: Number of pages written Status Sex GPA Count Undergraduate M Weak 1545 Undergraduate M Strong 1326 Undergraduate F Weak 1574 Undergraduate F Strong 1266 Graduate M Strong 2167 Graduate F Strong 2122 Figure Final relation The path relation has an important role in the rule-based induction proposed above. If the initial relation is very large, it may not be cost effective to create another large table as the path relation. A better alternative would be to encode and store the path ids in the initial relation. Also, it is useful to note that it is not necessary to encode all the paths in a concept graph together. For example, in Figure 8, for a given GPA value, only one bit at each level is sufficient to record the generalization path of the GPA value. Therefore, in order not to create another large relation (path relation), generalization paths can be tagged to tuples in the initial relation with a minimal amount of space. On the other hand, if it is feasible to create a path relation, then another interesting observation is that the path relation can be reduced without lossing of any information. In the generation of the path relation, the path id tuples generated are stored in the same order as that in the initial relation and redundant path id tuples are not compared and merged. Hence, the size of both relations are equal. If the initial relation has m relevant attributes A i , (1 - i - m), then the number of distinct tuples in the path relation would be less than m), is the number of distinct generalization paths in the concept graph of attribute A i . In some cases, ae would be much smaller than the size of the initial relation, and it would be beneficial to merge the redundant path id tuples in the path relation and record their counts there. By using this compact path relation, the cost of induction will be reduced significantly in the progressive generalization. In [17], it is suggested that mining characterization rules should be multiple-level. And the techniques of progressive generalization and specialization should be used. The techniques rely on a minimally generalized relation in which concepts are generalized to some minimal levels such that it can be used to perform drill-down and roll-up from any level. The compact path relation discussed above has all necessary information to support any generalization and its size is smaller than the initial relation. Therefore, it can be a good candidate of minimally generalized relation for performing rule-based progressive generalization and specialization. In fact, because of the induction anomaly problem, it would not be efficient to use other generalized relation as a minimally generalized relation. Many large organizations have started to use data warehouse to store valuable data from different information sources. In some cases, the data could come from 20 to 30 legacy sources. In this type of warehouse, the number of attributes of an object could be in the order of hundreds. One such example is the data warehouses in large international banking corporations. In order to perform rule-based induction on the data in large warehouses, the path relation algorithm is an excellent choice because many warehousing systems have already implemented the data cube technique for on-line analytical processing (OLAP) and decision support. 9 Conclusion A rule-based attribute-oriented induction technique is proposed and studied in this paper. The technique extends the previously developed basic attribute-oriented induction method and make the method more generally applicable to databases containing both data and rules. Also, the given concept hierarchies may contain unconditional and conditional rules, which enlarges the application domain and handles more sophisticated situations. Rule-based concept graph extends substantially the representation power of the background knowledge in an induction system. It also has converted the discovery system into an integrated deduction system and may eventually lead to the integration of a discovery system into an intelligent and cooperative information system [5]. This study has developed a general model of rule-based database induction and extended the basic AO Induction to the rule-based AO induction which handles induction on a rule-based concept graph. A new technique of using path relation and data cube to replace the initial relation and the generalized relations has been proposed to facilitate a more efficient induction. The proposed path relation algorithm has an improved complexity of O(n) which is faster than the previously proposed rule-based induction algorithms. The performance studies have also confirmed that the path relation algorithm is more efficient than the "backtracking" algorithm. A data mining system DBMiner has been developed for interactive mining of multiple-level knowledge in large databases [17]. The system has implemented a wide spectrum of data mining techniques, including generalization, characterization, association, classification, and prediction. Our proposed rule-based induction can extend the mining capability of DBMiner. In particular, a rule-based progressive generalization and specialization could be very useful in mining multiple-level knowledge. Furthermore, unconditional and rule-based induction, including both generalization and specialization, in multidatabases or data warehouses with large number of views are among the interesting topics in data mining for future research. --R On the computation of multidimensional aggregates. An interval classifier for database mining applications. Database mining: A performance perspective. Fast algorithms for mining association rules. On Intelligent and Cooperative Information Systems: A Workshop Summary. An Overview of Data Warehousing and OLAP Technology. Knowledge discovery in databases: A rule-based attribute-oriented approach Maintenance of Discovered Association Rules in Large Databases: An Incremental Updating Technique. A Fast Distributed Algorithm for Mining Association Rules. Advances in Knowledge Discovery and Data Mining. Improving inference through conceptual clustering. Knowledge discovery in databases: An overview. Logic and databases: A deductive approach. Data cube: A relational aggregation operator generalizing group-by Knowledge discovery in databases: An attribute-oriented approach A System for Mining Knowledge in Large Relational Databases. Discovery of multiple-level association rules from large databases Exploration of the power of attribute-oriented induction in data mining Implementing Data Cubes Efficiently. Mining for knowledge in databases: Goals and general description of the INLEN system. Learning conjuctive concepts in structural domains. Efficient Algorithms for Attribute-Oriented Induction A theory and methodology of inductive learning. Database Achievements and Opportunities Into the 21st Century. Principles of Database and Knowledge-Base Systems Research Problems in Data Warehousing. Interactive mining of regularities in databases. --TR Principles of database and knowledge-base systems, Vol. I Research problems in data warehousing Implementing data cubes efficiently An overview of data warehousing and OLAP technology Advances in knowledge discovery and data mining Attribute-oriented induction in data mining Logic and Databases: A Deductive Approach A fast distributed algorithm for mining association rules Data-Driven Discovery of Quantitative Rules in Relational Databases Database Mining Maintenance of Discovered Association Rules in Large Databases Data Cube An Interval Classifier for Database Mining Applications Knowledge Discovery in Databases Fast Algorithms for Mining Association Rules in Large Databases Discovery of Multiple-Level Association Rules from Large Databases On the Computation of Multidimensional Aggregates A Case-Based Reasoning Approach for Associative Query Answering --CTR Klaus Julisch, Clustering intrusion detection alarms to support root cause analysis, ACM Transactions on Information and System Security (TISSEC), v.6 n.4, p.443-471, November Jeffrey Hsu, Critical and future trends in data mining: a review of key data mining technologies/applications, Data mining: opportunities and challenges, Idea Group Publishing, Hershey, PA,

data mining;rule-based concept hierarchy;rule-based concept generalization;inductive learning;learning and adaptive systems;attribute-oriented induction;knowledge discovery in databases

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The FERET Evaluation Methodology for Face-Recognition Algorithms.

AbstractTwo of the most critical requirements in support of producing reliable face-recognition systems are a large database of facial images and a testing procedure to evaluate systems. The Face Recognition Technology (FERET) program has addressed both issues through the FERET database of facial images and the establishment of the FERET tests. To date, 14,126 images from 1,199 individuals are included in the FERET database, which is divided into development and sequestered portions of the database. In September 1996, the FERET program administered the third in a series of FERET face-recognition tests. The primary objectives of the third test were to 1) assess the state of the art, 2) identify future areas of research, and

Introduction Over the last decade, face recognition has become an active area of research in computer vision, neuroscience, and psychology. Progress has advanced to the point that face-recognition systems are being demonstrated in real-world settings [5]. The rapid development of face recognition is due to a combination of factors: active development of The work reported here is part of the Face Recognition Technology (FERET) program, which is sponsored by the U.S. Department of Defense Counterdrug Technology Development Program. Portions of this work was done while Jonathon Phillips was at the U.S. Army Research Laboratory (ARL). Jonathon Phillips acknowledges the support of the National Institute of Justice. algorithms, the availability of a large database of facial images, and a method for evaluating the performance of face-recognition algorithms. The FERET database and evaluation methodology address the latter two points and are de facto standards. There have been three FERET evaluations with the most recent being the Sep96 FERET test. The Sep96 FERET test provides a comprehensive picture of the state-of-the-art in face recognition from still images. This was accomplished by evaluating algorithms' ability on different scenarios, categories of images, and versions of algorithms. Performance was computed for identification and verification scenarios. In an identification application, an algorithm is presented with a face that it must identify the face; whereas, in a verification application, an algorithm is presented with a face and a claimed identity, and the algorithm must accept or reject the claim. In this paper, we describe the FERET database, the evaluation protocol, and present identification results. Verification results are presented in Rizvi et al. [8]. To obtain a robust assessment of performance, algorithms are evaluated against different categories of images. The categories are broken out by lighting changes, people wearing glasses, and the time between the acquisition date of the database image and the image presented to the algorithm. By breaking out performance into these categories, a better understanding of the face recognition field in general as well as the strengths and weakness of individual algorithms is obtained. This detailed analysis helps to assess which applications can be successfully addressed. All face recognition algorithms known to the authors consist of two parts: (1) face detection and normalization and (2) face identification. Algorithms that consist of both parts are referred to as fully automatic algorithms, and those that consist of only the second part are partially automatic algorithms. The Sep96 test evaluated both fully and partially automatic algorithms. Partially automatic algorithms are given a facial image and the coordinates of the center of the eyes. Fully automatic algorithms are only given facial images. The availability of the FERET database and evaluation methodology has made a significant difference in the progress of development of face-recognition algorithms. Before the FERET database was created, a large number of papers reported outstanding recognition results (usually ? 95% correct recognition) on limited-size databases (usually (In fact, this is still true.) Only a few of these algorithms reported results on images utilizing a common database, let alone met the desirable goal of being evaluated on a standard testing protocol that included separate training and testing sets. As a consequence, there was no method to make informed comparisons among various algorithms. The FERET database has made it possible for researchers to develop algorithms on a common database and to report results in the literature using this database. Results reported in the literature do not provide a direct comparison among algorithms because each researcher reported results using different assumptions, scoring methods, and images. The independently administered FERET test allows for a direct quantitative assessment of the relative strengths and weaknesses of different approaches. More importantly, the FERET database and tests clarify the current state of the art in face recognition and point out general directions for future research. The FERET tests allow the computer vision community to assess overall strengths and weaknesses in the field, not only on the basis of the performance of an individual algorithm, but in addition on the aggregate performance of all algorithms tested. Through this type of assessment, the community learns in an unbiased and open manner of the important technical problems to be addressed, and how the community is progressing toward solving these problems. Background The first FERET tests took place in August 1994 and March 1995 (for details of these tests and the FERET database and program, see Phillips et al [5, 6] and Rauss et al [7]); The FERET database collection began in September 1993 along with the FERET program. The August 1994 test established, for the first time, a performance baseline for face-recognition algorithms. This test was designed to measure performance on algorithms that could automatically locate, normalize, and identify faces from a database. The test consisted of three subtests, each with a different gallery and probe set. The gallery contains the set of known individuals. An image of an unknown face presented to the algorithm is called a probe, and the collection of probes is called the probe set. The first subtest examined the ability of algorithms to recognize faces from a gallery of 316 individuals. The second was the false-alarm test, which measured how well an algorithm rejects faces not in the gallery. The third baselined the effects of pose changes on performance. The second FERET test, that took place in March 1995, measured progress since August 1994 and evaluated algorithms on larger galleries. The March 1995 evaluation consisted of a single test with a gallery of 817 known individuals. One emphasis of the test was on probe sets that contained duplicate images. A duplicate is defined as an image of a person whose corresponding gallery image was taken on a different date. The FERET database is designed to advance the state of the art in face recognition, with the images collected directly supporting both algorithm development and the FERET evaluation tests. The database is divided into a development set, provided to researchers, and a set of sequestered images for testing. The images in the development set are representative of the sequestered images. The facial images were collected in 15 sessions between August 1993 and July 1996. Collection sessions lasted one or two days. In an effort to maintain a degree of consistency throughout the database, the same physical setup and location was used in each photography session. However, because the equipment had to be reassembled for each session, there was variation from session to session (figure 1). Images of an individual were acquired in sets of 5 to 11 images, collected under relatively unconstrained conditions. Two frontal views were taken (fa and fb); a different facial expression was requested for the second frontal image. For 200 sets of images, a third frontal image was taken with a different camera and different lighting (this is referred to as the fc image). The remaining images were collected at various aspects between right and left profile. To add simple variations to the database, photographers sometimes took a second set of images, for which the subjects were asked to put on their glasses and/or pull their hair back. Sometimes a second set of images of a person was taken on a later date; such a set of images is referred to as a duplicate set. Such duplicates sets result in variations in scale, pose, expression, and illumination of the face. By July 1996, 1564 sets of images were in the database, consisting of 14,126 total images. The database contains 1199 individuals and 365 duplicate sets of images. For duplicate I fc duplicate II Figure 1: Examples of different categories of probes (image). The duplicate I image was taken within one year of the fa image and the duplicate II and fa images were taken at least one year apart. some people, over two years elapsed between their first and most recent sittings, with some subjects being photographed multiple times (figure 1). The development portion of the database consisted of 503 sets of images, and was released to researchers. The remaining images were sequestered by the Government. 3 Test Design 3.1 Test Design Principles The FERET Sep96 evaluation protocol was designed to assess the state of the art, advance the state of the art, and point to future directions of research. To succeed at this, the test design must solve the three bears problem. The test cannot be neither too hard nor too easy. If the test is too easy, the testing process becomes an exercise in "tuning" existing algorithms. If the test is too hard, the test is beyond the ability of existing algorithmic techniques. The results from the test are poor and do not allow for an accurate assessment of algorithmic capabilities. The solution to the three bears problem is through the selection of images in the test set and the testing protocol. Tests are administered using a testing protocol that states the mechanics of the tests and the manner in which the test will be scored. In face recognition, the protocol states the number of images of each person in the test, how the output from the algorithm is recorded, and how the performance results are reported. The characteristics and quality of the images are major factors in determining the difficulty of the problem being evaluated. For example, if faces are in a predetermined position in the images, the problem is different from that for images in which the faces can be located anywhere in the image. In the FERET database, variability was introduced by the inclusion of images taken at different dates and locations (see section 2). This resulted in changes in lighting, scale, and background. The testing protocol is based on a set of design principles. Stating the design principle allows one to assess how appropriate the FERET test is for a particular face recognition algorithm. Also, design principles assist in determining if an evaluation methodology for testing algorithm(s) for a particular application is appropriate. Before discussing the design principles, we state the evaluation protocol. In the testing protocol, an algorithm is given two sets of images: the target set and the query set. We introduce this terminology to distinguish these sets from the gallery and probe sets that are used in computing performance statistics. The target set is given to the algorithm as the set of known facial images. The images in the query set consists of unknown facial images to be identified. For each image q i in the query set Q, an algorithm reports a similarity s i (k) between q i and each image t k in the target set T . The testing protocol is designed so that each algorithm can use a different similarity measure and we do not compare similarity measures from different algorithms. The key property of the new protocol, which allows for greater flexibility in scoring, is that for any two images q i and t k , we know s i (k). This flexibility allows the evaluation methodology to be robust and comprehensive; it is achieved by computing scores for virtual galleries and probe sets. A gallery G is a virtual gallery if G is a subset of the target set, i.e., G ae T . Similarly, P is a virtual probe set if P ae Q. For a given gallery G and probe set P, the performance scores are computed by examination of similarity measures s i (k) such that q G. The virtual gallery and probe set technique allows us to characterize algorithm performance by different categories of images. The different categories include (1) rotated images, (2) duplicates taken within a week of the gallery image, (3) duplicates where the time between the images is at least one year, (4) galleries containing one image per person, and (5) galleries containing duplicate images of the same person. We can create a gallery of 100 people and estimate an algorithm's performance by recognizing people in this gallery. Using this as a starting point, we can then create virtual galleries of 200; people and determine how performance changes as the size of the gallery increases. Another avenue of investigation is to create n different galleries of size 100, and calculate the variation in algorithm performance with the different galleries. To take full advantage of virtual galleries and probe sets, we selected multiple images of the same person and placed them into the target and query sets. If such images were marked as the same person, the algorithms being tested could use the information in the evaluation process. To prevent this from happenning, we require that each image in the target set be treated as an unique face. (In practice, this condition is enforced by giving every image in the target and query set a unique random identification.) This is the first design principle. The second design principle is that training is completed prior to the start of the test. This forces each algorithm to have a general representation for faces, not a representation tuned to a specific gallery. Without this condition, virtual galleries would not be possible. For algorithms to have a general representation for faces, they must be gallery (class) insensitive. Examples are algorithms based on normalized correlation or principal component analysis (PCA). An algorithm is class sensitive if the representation is tuned to a specific gallery. Examples are straight forward implementation of Fisher discriminant analysis [1, 9]. Fisher discriminant algorithms were adapted to class insensitive testing methodologies by Zhao et al [13, 14], with performance results of these extensions being reported in this paper. The third design rule is that all algorithms tested compute a similarity measure between two facial images; this similarity measure was computed for all pairs of images between the target and query sets. Knowing the similarity score between all pairs of Face Recognition Algorithm (run at Testee's site) Output File Scoring Code Government (run at Results Probe Image Name Gallery Image Name (One Image/Person) Gallery images List of Probes Figure 2: Schematic of the FERET testing procedure images from the target and query sets allows for the construction of virtual galleries and probe sets. 3.2 Test Details In the Sep96 FERET test, the target set contained 3323 images and the query set 3816 images. All the images in the target set were frontal images. The query set consisted of all the images in the target set plus rotated images and digitally modified images. We designed the digitally modified images to test the effects of illumination and scale. (Results from the rotated and digitally modified images are not reported here.) For each query image q i , an algorithm outputs the similarity measure s i (k) for all images t k in the target set. For a given query image q i , the target images t k are sorted by the similarity scores s i (\Delta). Since the target set is a subset of the query set, the test output contains the similarity score between all images in the target set. There were two versions of the Sep96 test. The target and query sets were the same for each version. The first version tested partially automatic algorithms by providing them with a list of images in the target and query sets, and the coordinates of the center of the eyes for images in the target and query sets. In the second version of the test, the coordinates of the eyes were not provided. By comparing the performance between the two versions, we estimate performance of the face-locating portion of a fully automatic algorithm at the system level. The test was administered at each group's site under the supervision of one of the au- thors. Each group had three days to complete the test on less than 10 UNIX workstations (this limit was not reached). We did not record the time or number of workstations because execution times can vary according to the type of machines used, machine and network configuration, and the amount of time that the developers spent optimizing their code (we wanted to encourage algorithm development, not code optimization). (We imposed the time limit to encourage the development of algorithms that could be incorporated into operational, fieldable systems.) The images contained in the gallery and probe sets consisted of images from both the developmental and sequestered portions of the FERET database. Only images from the FERET database were included in the test; however, algorithm developers were not prohibited from using images outside the FERET database to develop or tune parameters in their algorithms. The FERET test is designed to measure laboratory performance. The test is not concerned with speed of the implementation, real-time implementation issues, and speed and accuracy trade-offs. These issues and others, need to be addressed in an operational, fielded system, were beyond the scope of the Sep96 FERET test. Figure presents a schematic of the testing procedure. To ensure that matching was not done by file name, we gave the images random names. The nominal pose of each face was provided to the testee. 4 Decision Theory and Performance Evaluation The basic models for evaluating the performance of an algorithm are the closed and open universes. In the closed universe, every probe is in the gallery. In an open universe, some probes are not in the gallery. Both models reflect different and important aspects of face-recognition algorithms and report different performance statistics. The open universe models verification applications. The FERET scoring procedures for verification is given in Rizvi et al [8]. The closed-universe model allows one to ask how good an algorithm is at identifying a probe image; the question is not always "is the top match correct?" but "is the correct answer in the top n matches?" This lets one know how many images have to be examined to get a desired level of performance. The performance statistics are reported as cumulative match scores. The rank is plotted along the horizontal axis, and the vertical axis is the percentage of correct matches. The cumulative match score can be calculated for any subset of the probe set. We calculated this score to evaluate an algorithm's performance on different categories of probes, i.e., rotated or scaled probes. The computation of an identification score is quite simple. Let P be a probe set and jPj the size of P. We score probe set P against gallery G, where and by comparing the similarity scores s i (\Delta) such that G. For each probe image G. We assume that a smaller similarity score implies a closer match. If g k and p i are the same image, then The function id(i) gives the index of the gallery image of the person in probe is an image of the person in g id(i) . A probe p i is correctly identified if s i (id(i)) is the smallest scores for G. A probe p i is in the top k if s i (id(i)) is one of the k-th smallest score s i (\Delta) for gallery G. Let R k denote the number of probes in the top k. We reported R k =jPj, the fraction of probes in the top k. As an example, let and 100. Based on the formula, the performance score for R 5 is In reporting identification performance results, we state the size of the gallery and the number of probes scored. The size of the gallery is the number of different faces (people) contained in the images that are in the gallery. For all results that we report, there is one image per person in the gallery, thus, the size of the gallery is also the number of images in the gallery. The number of probes scored (also, size of the probe set) is jPj. The probe set may contain more than one image of a person and the probe set may not contain an image of everyone in the gallery. Every image in the probe set has a corresponding image in the gallery. 5 Latest Test Results The Sep96 FERET test was designed to measure algorithm performance for identification and verification tasks. Both tasks are evaluated on the same sets of images. We report the results for 12 algorithms that includes 10 partially automatic algorithms and 2 fully automatic algorithms. The test was administered in September 1996 and March 1997 (see table 1 for details of when the test was administered to which groups and which version of the test was taken). Two of these algorithms were developed at the MIT Media Laboratory. The first was the same algorithm that was tested in March 1995. This algorithm was retested so that improvement since March 1995 could be measured. The second algorithm was based on more recent work [2, 3]. Algorithms were also tested from Excalibur Corp. (Carlsbad, CA), Michigan State University (MSU) [9, 14], Rutgers University [11], University of Southern California (USC) [12], and two from University of Maryland (UMD) [1, 13, 14]. The first algorithm from UMD was tested in September 1996 and a second version of the algorithm was tested in March 1997. For the fully automatic version of test, algorithms from MIT and USC were evaluated. The final two algorithms were our implementation of normalized correlation and a principal components analysis (PCA) based algorithm [4, 10]. These algorithms provide a performance baseline. In our implementation of the PCA-based algorithm, all images were (1) translated, rotated, and scaled so that the center of the eyes were placed on specific pixels, (2) faces were masked to remove background and hair, and (3) the non-masked facial pixels were processed by a histogram equalization algorithm. The training set consisted of 500 faces. Faces were represented by their projection onto the first 200 eigenvectors and were identified by a nearest neighbor classifier using the L 1 metric. For normalized correlation, the images were (1) translated, rotated, and scaled so that the center of the eyes were placed on specific pixels and (2) faces were masked to remove background and hair. 5.1 Partially automatic algorithms We report identification scores for four categories of probes. The first probe category was the FB probes (fig 3). For each set of images, there were two frontal images. One of the images was randomly placed in the gallery, and the other image was placed in the FB probe set. (This category is denoted by FB to differentiate it from the fb images in the FERET database.) The second probe category contained all duplicate frontal images in the FERET database for the gallery images. We refer to this category as the duplicate I probes. The third category was the fc (images taken the same day, but with a different camera and lighting). The fourth consisted of duplicates where there is at least one year between the acquisition of the probe image and corresponding gallery image. We refer to this category as the duplicate II probes. For this category, the gallery images were acquired before January 1995 and the probe images were acquired after January 1996. Table 1: List of groups that took the Sept96 test broken out by versions taken and dates administered. (The 2 by MIT indicates that two algorithms were tested.) Test September March Version of test Group 1996 1997 Baseline Fully Automatic MIT Media Lab [2, 3] ffl U. of So. California (USC) [12] ffl Eye Coordinates Given Baseline PCA [4, 10] ffl Baseline Correlation ffl Excalibur Michigan State U. [9, 14] ffl Rutgers U. [11] ffl U Maryland [1, 13, 14] ffl ffl The gallery for the FB, duplicate I, and fc probes was the same and consisted of 1196 frontal images with one image person in the gallery (thus the gallery contained 1196 individuals). Also, none of the faces in the gallery images wore glasses. The gallery for duplicate II probes was a subset of 864 images from the gallery for the other categories. The results for identification are reported as cumulative match scores. Table 2 shows the categories corresponding to the figures presenting the results, type of results, and size of the gallery and probe sets (figs 3 to 6). In figures 7 and 8, we compare the difficulty of different probe sets. Whereas, figure 4 reports identification performance for each algorithm, figure 7 shows a single curve that is an average of the identification performance of all algorithms for each probe category. For example, the first ranked score for duplicate I probe sets is computed from an average of the first ranked score for all algorithms in figure 4. In figure 8, we presented current upper bound for performance on partially automatic algorithms for each probe category. For each category of probe, figure 8 plots the algorithm with the highest top rank score (R 1 ). Figures 7 and 8 reports performance of four categories of probes, FB, duplicate I, fc, duplicate II. Table 2: Figures reporting results for partially automatic algorithms. Performance is broken out by probe category. Figure no. Probe Category Gallery size Probe set size 4 duplicate I 1196 722 6 duplicate II 864 234 Cumulative match score Rank MSU UMD 96 MIT 95 Baseline Baseline EF Excalibur Rutgers (a)0.80.91 Cumulative match score Rank UMD 97 USC UMD 96 Baseline Baseline EF (b) Figure 3: performance against FB probes. (a) Partially automatic algo-0.95 Cumulative match score Rank Excalibur Baseline EF Baseline MIT 95 MSU UMD 96 Rutgers (a)0.40.60.81 Cumulative match score Rank USC UMD 97 Baseline EF Baseline UMD 96 (b) Figure 4: performance against all duplicate I probes. (a) Partially automatic Cumulative match score Rank UMD 96 MSU Excalibur Baseline EF Rutgers MIT 95 Baseline (a)0.20.61 Cumulative match score Rank USC UMD 97 UMD 96 Baseline EF Baseline (b) Figure 5: performance against fc probes. (a) Partially automatic algorithms Cumulative match score Rank Baseline EF Excalibur Rutgers MIT 95 Baseline UMD 96 MSU (a)0.20.61 Cumulative match score Rank USC Baseline EF UMD 97 Baseline UMD 96 (b) Figure performance against duplicate II probes. (a) Partially automatic Cumulative Match Score Rank FB probes Duplicate I probes fc probes Duplicate II probes Figure 7: Average identification performance of partially automatic algorithms on each probe category.0.20.61 Cumulative Match Score Rank FB probes fc probes Duplicate I probes Duplicate II probes Figure 8: Current upper bound identification performance of partially automatic algorithm for each probe category. Cumulative Match Score Rank USC partially automatic MIT partially automatic USC fully automatic MIT fully automatic Figure 9: Identification performance of fully automatic algorithms against partially automatic algorithms for FB Cumulative Match Score Rank USC partially automatic USC fully automatic MIT partially automatic MIT fully automatic Figure 10: Identification performance of fully automatic algorithms against partially automatic algorithms for duplicate I probes. 5.2 Fully Automatic Performance In this subsection, we report performance for the fully automatic algorithms of the MIT Media Lab and USC. To allow for a comparison between the partially and fully automatic algorithms, we plot the results for the partially and fully automatic algorithms. Figure 9 shows performance for FB probes and figure 10 shows performance for duplicate I probes. (The gallery and probe sets are the same as in subsection 5.1.) 5.3 Variation in Performance From a statistical point of view, a face-recognition algorithm estimates the identity of a face. Consistent with this view, we can ask about the variance in performance of an algorithm: "For a given category of images, how does performance change if the algorithm is given a different gallery and probe set?" In tables 3 and 4, we show how algorithm performance varies if the people in the galleries change. For this experiment, we constructed six galleries of approximately 200 individuals, in which an individual was in only one gallery (the number of people contained within each gallery versus the number of probes scored is given in tables 3 and 4). Results are reported for the partially automatic algorithms. For the results in this section, we order algorithms by their top rank score on each gallery; for example, in table 3, the UMD Mar97 algorithm scored highest on gallery 1 and the baseline PCA and correlation tied for 9th place. Also included in this table is average performance for all algorithms. Table 3 reports results for FB probes. Table 4 is organized in the same manner as table 3, except that duplicate I probes are scored. Tables 3 and 4 report results for the same gallery. The galleries were constructed by placing images within the galleries by chronological order in which the images were collected (the first gallery contains the first images collected and the 6th gallery contains the most recent images collected). In table 4, mean age refers to the average time between collection of images contained in the gallery and the corresponding duplicate probes. No scores are reported in table 4 for gallery 6 because there are no duplicates for this gallery. 6 Discussion and Conclusion In this paper we presented the Sep96 FERET evaluation protocol for face recognition algorithms. The protocol makes it possible to independently evaluate algorithms. The protocol was designed to evaluate algorithms on different galleries and probe sets for different scenarios. Using this protocol, we computed performance on identification and verification tasks. The verification results are presented in Rizvi et al. [8], and all verification results mentioned in this section are from that paper. In this paper we presented detailed identification results. Because of the Sep96 FERET evaluation protocol's ability to test algorithms performance on different tasks for multiple galleries and probe sets, it is the de facto standard for measuring performance of face recognition algorithms. These results show that factors effecting performance include scenario, date tested, and probe category. The Sep96 test was the latest FERET test (the others were the Aug94 and Mar95 tests [6]). One of the main goals of the FERET tests has been to improve the performance of face recognition algorithms, and is seen in the Sep96 FERET test. The first case is the improvement in performance of the MIT Media Lab September 1996 algorithm over Table 3: Variations in identification performance on six different galleries on FB probes. Images in each gallery do not overlap. Ranks range from 1-10. Algorithm Ranking by Top Match Gallery Size / Scored Probes 200/200 200/200 200/200 200/200 200/199 196/196 Algorithm gallery 1 gallery 2 gallery 3 gallery 4 gallery 5 gallery 6 Baseline Baseline correlation 9 9 9 Excalibur Michigan State Univ. 3 4 5 8 4 4 Rutgers Univ. 7 8 9 6 7 9 Average Score 0.935 0.857 0.904 0.918 0.843 0.804 Table 4: Variations in identification performance on five different galleries on duplicate probes. Images in each of the gallery does not overlap. Ranks range from 1-10. Algorithm Ranking by Top Match Gallery Size / Scored Probes 200/143 200/64 200/194 200/277 200/44 Mean Age of Probes (months) 9.87 3.56 5.40 10.70 3.45 Algorithm gallery 1 gallery 2 gallery 3 gallery 4 gallery 5 Baseline Baseline correlation Excalibur Michigan State Univ. 9 Rutgers Univ. Average Score 0.238 0.620 0.645 0.523 0.687 the March 1995 algorithm; the second is the improvement of the UMD algorithm between September 1996 and March 1997. By looking at progress over the series of FERET tests, one sees that substantial progress has been made in face recognition. The most direct method is to compare the performance of fully automatic algorithms on fb probes (the two earlier FERET tests only evaluated fully automatic algorithms. The best top rank score for fb probes on the Aug94 test was 78% on a gallery of 317 individuals, and for Mar95, the top score was 93% on a gallery of 831 individuals [6]. This compares to 87% in September 1996 and 95% in March 1997 (gallery of 1196 individuals). This method shows that over the course of the FERET tests, the absolute scores increased as the size of the database increased. The March 1995 score was from one of the MIT Media Lab algorithms, and represents an increase from 76% in March 1995. On duplicate I probes, MIT Media Lab improved from 39% (March 1995) to 51% (September 1996); USC's performance remained approximately the same at 57-58% between March 1995 and March 1997. This improvement in performance was achieved while the gallery size increased and the number of duplicate I probes increased from 463 to 722. While increasing the number of probes does not necessarily increase the difficulty of identification tasks, we argue that the Sep96 duplicate I probe set was more difficult to process then the Mar95 set. The Sep96 duplicate I probe set contained the duplicate II probes and the Mar95 duplicate I probe set did not contain a similar class of probes. Overall, the duplicate II probe set was the most difficult probe set. Another goal of the FERET tests is to identify areas of strengths and weaknesses in the field of face recognition. We addressed this issue by computing algorithm performance for multiple galleries and probe sets. From this evaluation, we concluded that algorithm performance is dependent on the gallery and probe sets. We observed variation in performance due to changing the gallery and probe set within a probe category, and by changing probe categories. The effect of changing the gallery while keeping the probe category constant is shown in tables 3 and 4. For fb probes, the range for performance is 80% to 94%; for duplicate I probes, the range is 24% to 69%. Equally important, tables 3 and 4 shows the variability in relative performance levels. For example, in table 4, UMD duplicate performance varies between number three and nine. Similar results were found in Moon and Phillips [4] in their study of principal component analysis-based face recognition algorithms. This shows that an area of future research could measure the effect of changing galleries and probe sets, and statistical measures that characterize these variations. Figures 7 and 8 shows probe categories characterized by difficulty. These figures show that fb probes are the easiest and duplicate II probes are the most difficult. On average, duplicate I probes are easier to identify than fc probes. However, the best performance on fc probes is significantly better than the best performance on duplicate I and II probes. This comparative analysis shows that future areas of research could address processing of duplicate II probes and developing methods to compensate for changes in illumination. The scenario being tested contributes to algorithm performance. For identification, the MIT Media Lab algorithm was clearly the best algorithm tested in September 1996. However, for verification, there was not an algorithm that was a top performer for all probe categories. Also, for the algorithms tested in March 1997, the USC algorithm performed overall better than the UMD algorithm for identification; however, for verification, UMD overall performed better. This shows that performance on one task is not predictive of performance on another task. The September 1996 FERET test shows that definite progress is being made in face recognition, and that the upper bound in performance has not been reached. The improvement in performance documented in this paper shows directly that the FERET series of tests have made a significant contribution to face recognition. This conclusion is indirectly supported by (1) the improvement in performance between the algorithms tested in September 1996 and March 1997, (2) the number of papers that use FERET images and report experimental results using FERET images, and (3) the number of groups that participated in the Sep96 test. --R Discriminant analysis for recognition of human face images. Bayesian face recognition using deformable intensity surfaces. Probabilistic visual learning for object detection. Analysis of PCA-based face recognition algorithms The face recognition technology (FERET) program. 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face recognition;algorithm evaluation;FERET database

354183

Completion Energies and Scale.

AbstractThe detection of smooth curves in images and their completion over gaps are two important problems in perceptual grouping. In this study, we examine the notion of completion energy of curve elements, showing, and exploiting its intrinsic dependence on length and width scales. We introduce a fast method for computing the most likely completion between two elements, by developing novel analytic approximations and a fast numerical procedure for computing the curve of least energy. We then use our newly developed energies to find the most likely completions in images through a generalized summation of induction fields. This is done through multiscale procedures, i.e., separate processing at different scales with some interscale interactions. Such procedures allow the summation of all induction fields to be done in a total of only $O(N \log N)$ operations, where $N$ is the number of pixels in the image. More important, such procedures yield a more realistic dependence of the induction field on the length and width scales: The field of a long element is very different from the sum of the fields of its composing short segments.

Introduction The smooth completion of fragmented curve segments is a skill of the human visual system that has been demonstrated through many compelling examples. Due to this skill people often are able to perceive the boundaries of objects even in the lack of sufficient contrast or in the presence of oc- clusions. A number of computational studies have addressed the problem of curve completion in an attempt to both provide a computational theory of the problem and as part of a process of extracting the smooth curves from images. These studies commonly obtain two or more edge elements (also referred to as edgels) and find either the most likely completions that connect the elements or the smoothest curves trav- Research supported in part by Israel Ministry of Science Grant 4135- 1-93 and by the Gauss Minerva Center for Scientific Computation. y Research supported in part by the Unites States-Israel Binational Science Foundation, Grant No. 94-00100. eling through them. The methods proposed for this problem generally require massive computations, and their results strongly depend on the energy function used to evaluate the curves in the image. It is therefore important to develop methods which simplify the computation involved in these methods while providing results competitive with the existing approaches. Below we present such a method that directly relates to a number of recent studies of completion and curve salience [9, 18, 5, 19, 12, 7] (see also [2, 6, 8, 14, 15]). Along with simplifying the computations proposed in these studies our method also takes into account the size of edge elements, allowing for a proper computation of completion and saliency at different scales. A number of studies have addressed the problem of determining the smoothest completion between pairs of edge elements These studies seek to define a functional that, given two edge elements defined by their location and orientation in the image, selects the smoothest curve that connects the two as its minimizing curve. The most common functional is based on the notion of elas- tica, that is, minimizing the total squared curvature of the curve [9]. Scale invariant variations of this functional were introduced in [18, 5]. While the definition of scale-invariant elastica is intuitive, there exists no simple analytic expression to calculate its shape or its energy, and existing numerical computations are orders-of-magnitude too expensive, as will be shown below. In the first part of this paper we revisit the problem of determining the smoothest completion between pairs of edges and introduce two new analytic approximations to the curve of least energy. The first approximation is obtained by assuming that the deviation of the two input edgels from the straight line connecting them is relatively small. This assumption is valid in most of the examples used to demonstrate perceptual completions in humans and monkeys [10, 11]. We show that under this simplifying assumption the Hermite spline (see, e.g., [13]) provides a good approximation to the curve of least energy and a very good approximation to the least energy itself. We further develop a second expression, which directly involves the angles formed by the edgels and the straight line connecting them. The second expression is shown to give extremely accurate approximations to the curve of least energy even when the input edgels deviate significantly from the line connecting them. We then introduce a new, fast numerical method to compute the curve of least energy and show that our analytic approximations are obtained at early stages of this numerical computation. Several recent studies view the problems of curve completion and salience as follows. Given M edge elements, the space of all curves connecting pairs of elements is examined in an attempt to determine which of these completions is most likely using smoothness and length considerations. For this purpose [7, 19] define an affinity measure between two edge elements that grows with the likelihood of these elements being connected by a curve. By fixing one of the elements and allowing the other element to vary over the entire image an induction field representing the affinity values induced by the fixed element on the rest of the image is obtained. The system finds the most likely completions for the M elements by applying a process that includes a summation of the induction fields for all M elements. In the second part of this paper we use our newly developed completion energies to define an affinity measure that encourages smoothness and penalizes for gap length. We then use the induction fields defined by this affinity measure to solve the problem of finding the most likely completions for M elements. Since in practice edge elements are never dimensionless, because they are usually obtained by applying filters of a certain width and length to the image, we adjust our affinity measure to take these parameters into ac- count. We do so by relating the scale of these filters to the range of curvatures which can be detected by them and to the orientational resolution needed. Finally, we show that our affinity measure is asymptotically smooth, and so can be implemented using multigrid methods and run efficiently in time complexity O(nm) (where n is the number of pixels and m is the number of discrete orientations at every pixel). The paper is divided as follows. In Section 2 we review the notion of elastica and its scale invariant variation. In Section 3 we introduce the two analytic approximations to the curve of least energy. Then, in Section 4 we develop a fast numerical method to compute the curve of least energy and compare it to our analytic approximations. Finally, in Section 5 we construct an affinity measure taking into account the length and width of the edge filters applied to the image. We then discuss a multiscale (multigrid) method for fast summation of induction fields. 2. Elastica Consider two edge elements positioned at R 2 with directed orientations Y 1 and Y 2 respectively measured from the right-hand side of the line passing through Below we shall confine ourselves to the case that may conveniently assume that This is illustrated in Fig. 1(a). Let C 12 denote the set of curves through e 1 and e 2 . Denote such a curve by its orientation representation Y(s), where 0 - s - L is the arclength along the curve. That is, R s s and R s s. Also denote the curvature of the curve at s by dY(s)=ds. r (a) Y y F F F 1- r (b) Figure 1. (a) The planar relation between two edge ele- ments, This relation is governed by are measured from the line (b) The more general relation between F i and Y i . The most common functional used to determine the smoothest curve traveling through P 1 and P 2 with respective orientations Y 1 and Y 2 is the elastica functional. Namely, the smoothest curve through e 1 and e 2 is the curve Y(s) which minimizes the functional G el (Y) def R L Elastica was already introduced by Euler. It was first applied to completion by Ullman [17], and its properties were further investigated by Horn [9]. One of the problems with the classical elastica model is that it changes its behavior with a uniform scaling of the image. In fact, according to this model if we increase r, the distance between the two input elements, the energy of the curve connecting them proportionately decreases, as can be easily seen by rescaling s (cf. [1]). This is somewhat counter-intuitive since psychophysical and neurobiological evidence suggests that the affinity between a pair of straight elements drops rapidly with the distance between them [11]. Also, the classical elastica does not yield circular arcs to complete cocircular elements. To solve these problems Weiss [18, 5] proposed to modify the elastica model to make it scale invariant. His functional is defined as G inv (Y) R L We believe that a proper adjustment of the completion energy to scale must take into account not only the length of the curve (or equivalently the distance between the input elements), but also the dimensions of the input edge elements. Both the elastica functional and its scale invariant version assume that the input elements have no dimensions. In practice, however, edge elements are frequently obtained by convolving the image with filters of some specified width and length. A proper adjustment of the completion energy as a result of scaling the distance between the elements should also consider whether a corresponding scaling in the width and length of the elements has taken place. Below we first develop useful approximations to the scale invariant functional. (These approximations can also readily be used with slight modifications to the classical elastica measure.) Later, in Section 5, we develop an affinity measure between elements that also takes into account both the distance between the elements and their dimensions. 3. Analytic simplification of G inv Although the definition of both the classical and the scale invariant elastica functionals is fairly intuitive, there is no simple closed-form expression that specifies the energy or the curve shape obtained with these functionals. In this section we introduce two simple, closed-form approximations to these functionals. Our first approximation is valid when the sum of angles jF relatively small. This assumption represents the intuition that in most psychophysical demonstrations gap completion is perceived when the orientations of the curve portions to be completed are nearly collinear. With this assumption we may also restrict for now the range of applicable orientations to Y The second approximation will only assume that jF is small, i.e., that the curve portions to be completed are nearly cocircular. Since the curve of least energy is supposed to be very smooth, it is reasonable to assume that within the chosen range of Y the smoothest curve will not wind much. Consequently, it can be described as a function in Fig. 1(a). Expressing the curvature in terms of x and y we obtain that G inv R L For we get that L ! r, and that the variation of unimportant for the comparison of G inv (Y) over different curves Y 2 C 12 , so that G inv (Y) ' r R r Hence G inv (Y) ' r min The minimizing curve is the appropriate cubic Hermite spline (see [1]) r so that Evidently, this simple approximation to E inv is scale- independent. This leads us to define the scale-invariant spline completion energy as: E spln Although the spline energy provides a good approximation to the scale invariant elastica measure for small values of jF the measure diverges for large values. An alternative approximation to E inv can be constructed by noticing that for such small values tan F 1 ' F 1 and tan F 2 ' F 2 . Thus, we may define: We refer to this functional as the scale-invariant angular completion energy. This measure does not diverge for large values of jF In fact, when F In Section 4 below we show that this angular energy is obtained in an early stage of the numeric computation of E inv , and that it provides extremely accurate approximations to the scale invariant least energy functional even for relatively large values of jF 1 j+jF 2 j, especially for small jF for the range of nearly cocircular elements. Using the numeric computation we can also derive the smoothest curve according to The angular completion energy can be generalized as follows: where Eq. (4) is identical to Eq. (6) with 2. That is, the angular completion energy is made of an equal sum of two penalties. One is for the squared difference between F 1 and F 2 , and the other is for the growth in each of them. This suggests a possible generalization of E ang to other weights a - 0 and b - 0. One can also think of using energies such as E circ more elaborate study of these types of energies and their properties is presented in [1]. Finally, we note that the new approximations at small angles can also be used to approximate the classical elastica energy, since el G el (Y) ' 1 4. Computation of E inv We use the scale-invariance property of G inv in order to reformulate the minimization problem into minimizing over all C 12 curves of length Applying Euler-Lagrange equations (see, e.g., [13]) we get that a necessary condition for - s) to be an extremal curve is that it should satisfy for some -: Y s:t: (8) Considering the very nature of the original minimization problem, and also by repeatedly differentiating both sides of the ODE equation, it can be shown that its solution must be very smooth. Hence, we can well approximate the solution by a polynomial of the form where n is small. (By comparison, the discretization of the same problem presented in [5] is far less efficient, since it does not exploit the infinite smoothness of the solution on the full interval (0,1). As a result the accuracy in [5] is only second order, while here it is "1-order", i.e., the error decreases exponentially in the number of discrete variables :::; an from Eq. (8) and Fixing n, as well as two other integers - n and p, we will build the following system of n+2 equations for the n+2 unknowns a an ; and - collocating the ODE, and n) are the weights of a p-order numerical integration. Generally, we increase n gradually and increase - n and p as functions of n in such a way that the discretization error will not be governed by the discretization error of the integration. The nonlinear system of equations is solved by Newton iterations (also called Newton-Raphson; see, e.g. [13].) We start the Newton iterations from a solution previously obtained for a system with a lower n. Actually, only one Newton iteration is needed for each value of n if n is not incremented too fast. In this way convergence is extremely fast. At each step, in just several dozen computer operations, the error in solving the differential equation can be squared. In fact, due to the smoothness of the solution for the ODE, already for the simple 2)-system and the Simpson integration rule very good approximation to the accurate solution and also to E inv is obtained, as can be seen in [1]. The good approximations obtained already for small values of (n, - n) suggest that E inv can be well approximated by simple analytic expressions, as indeed we show in [1] by comparingbetween several simple approximations to E inv . Fig. 2 illustrates some of the completions obtained using inv and the two analytic approximations E ang and E spln . It can be seen that the differences between the three curves is barely noticeable, except in large angles where E spln diverges. Notice especially the close agreement between the curve obtained with the angular energy (Eq. (5)) and that obtained with the scale-invariant elastica measure even in large angles and when the angles deviate significantly from cocircularity. Note that although the spline curve does not approximate the scale invariant elastica curve for large angles jF 1 j and jF 2 j it still produces a reasonable completion for the elements. In fact, when the two elements deviate from co- circularity the elastica accumulates high curvature at one of its ends, whereas in the spline curve continues to roughly follow the tangent to the two elements at both ends (see, e.g, Fig. 2(b)) . This behavior is desirable especially when the elements segments (see Section 5.2). 5. Completion field summation Until now we have considered the problem of finding the smoothest completion between pairs of edge elements. A natural generalization of this problem is, given an image from which M edge elements are extracted, find the most likely completions connecting pairs of elements in the image and rank them according to their likelihoods. This problem has recently been investigated in [7, 19]. In these studies affinity measures relating pairs of elements were defined. The measures encourage proximity and smoothness of com- pletion. Using the affinity measures the affinities induced by an element over all other elements in the image (referred to as the induction field of the element) are derived. The likelihoods of all possible completions are then computed simultaneously by a process which includes summation of the induction fields for all M elements. An important issue that was overlooked in previous ap- proaches, however, is the issue of size of the edge elements. Most studies of curve completion assume that the edge elements are dimensionless. In practice, however, edge elements are usually obtained by convolving the image with filters of certain width and length. A proper handling of scale must take these parameters into account. Thus, for example, one may expect that scaling the distance between two elements would not result in a change in the affinity of the two elements if the elements themselves are scaled by the same proportion. Below we first present the general type of non-scaled induction underlying previous works. We then modify that induction to properly account for the width and length of the edge elements. Finally, the process of summing the induction fields may be computationally intensive. Nevertheless, in the third part of this section we show that the summation kernel obtained with our method is very smooth. Thus, the summation of our induction fields can be speeded up considerably using a multigrid algorithm. This result also applies to the summation kernels in [19, 16, 7], and so an efficient implementation of these methods can be obtained with a similar multigrid algorithm. 5.1. Non-scaled induction In [12, 19] a model for computing the likelihoods of curve completions, referred to as Stochastic Completion Fields, Figure 2. Completion curves: elastica in solid line, - (Eq. (5)) in dotted line, and the cubic Hermite spline (Eq. (2)) in dashed line. (a) was proposed. According to this model, the edge elements in the image emit particles which follow the trajectories of a Brownian motion. It was shown that the most likely path that a particle may take between a source element and a sink element is the curve of least energy according to the Elastica energy function 1 . To compute the stochastic completion fields a process of summing the affinity measures representing the source and sink fields was used. In [1] we show, by further analyzing the results in [16], that the affinity measure used for the induction in [19, 16] is of the general type: A(e 1 Fig. 1(b) ), where r 0 and oe 0 are strictly positive a-priori set parameters. These parameters need to be adjusted properly according to the scale involved (see Sec. 5.2). Note that for small values of (jF 1 j,jF 2 el . Hence, el =oe 0 . Another method which uses summation of induction fields to compute the salience of curves was presented in [7]. In their method the affinity between two edge elements which are cocircular has the form: e \Gammafl r e \Gammaffi- , where fl and ffi are strictly positive constants, - is the curvature of the circle connecting e 1 and e 2 , and r is the distance between e 1 and e 2 . A reasonable and straightforward definition in that spirit is - serves as an approximation according to Eq. (3). Fig. 3 shows an example of computing the "stochastic completion field," suggested by Williams and Jacobs in [19], while replacing their affinity measure with the simple expression - It can be verified by comparing the fields obtained with our affinity measure with the fields presented in [19] that the results are very similar although a much simpler affinity measure was employed. 5.2. Induction and scale Given an image, an edge element is producedby selecting a filter of a certain length l and width w (e.g., rectangular filters) and convolving the filter with the image at a certain position and orientation. The result of this convolution is a scalar value, referred to as the response of the filter. An edge filter may, for example, measure the contrast along its primary axis, in which case its response represents the "edgeness level", or the likelihood of the relevant subarea of the image to contain an edge of (l; w) scale. Similarly, 1 Actually, the path minimizes the energy functional R L for some predetermined constant -. (a) (b) Figure 3. Stochastic completion fields (128 \Theta 128 pixels, 36 orientations) with the induction e \Gamma2r e \Gamma20E spln . (a) F . The results closely resemble those obtained in [19]. a filter may indicate the existence of fiber-like shapes in the image, in which case its response represents the "fiberness level" of the relevant subarea of the image. Below we use the term "straight responses" to refer to the responses obtained by convolving the image with an edge or a fiber filter. Consider now the edge elements obtained by convolving the image with a filter of some fixed length l and width w. Every edge element now is positioned at a certain pixel P and is oriented in two opposite directed orientations Y and -. The number of edge elements required to faithfully represent the image at this scale depends on l and Thus, long and thin elements require finer resolution in orientation than square elements. In fact, the orientational resolution required to sample significantly different orientations increases linearly with l=w (see [4]). Similarly, elements of larger size require less spatial resolution than elements of smaller size. Brandt and Dym ([4]) use these observations in order to introduce a very efficient computation (O(N log N), where N is the number of pixels in the image) of all significantly different edge elements. Given a particular scale determined by the length l and width w of edge elements, we would like to compute a completion field for this scale. Note that only curves within a relevant range of curvature radii can arouse significant responses for our l \Theta w elements. Denote the smallest curvature radius that will arouse a still significant response by curvature radii will arouse significant responses also in larger l=w scales, implying there for a farther-reaching and more orientation-specific continua- tion.) By Fig. 4(a) we see that Consequently, we have (l=2ae) 2 - implying that ae - l 2 =8w. Next, consider a pair of straight responses. Assuming these elements are roughly cocircular, then, using the relations defined in Fig. 4(b) , the differential relation Y 0 1=ae(s) can be approximated by (Y so that W - r=ae. Hence, for completion at a particular scale (l; w), it is reasonable to define for every pair of points P 1 and P 2 a scale for the turning angle W given by r=ae(l; w). That is, in the scale (l,w) we define the completion energy between the pair of straight responses so as to depend on the scaled turning angle Wae=r. Since is straightforward to show that 0:5W - reasonable definition for the scaled angular energy, therefore, is a monotonically decreasing function of (ae=r) a (a) rcosa r Y F 1Y F r (b) Y r Figure 4. (a) The relation between l,w, and the curvature radius ae. (b) The turn W that a moving particle takes in its way between two straight responses, characterized each by a planar location and an orientation. Obviously, in any given scale of straight responses,(l; w), for every F 1 and F 2 , the induction of P 1 upon P 2 should decrease with an increase of r=ae. Hence, we define the field induced by an element e 1 of length l and width w at location directed orientation Y 1 on a similar element e 2 at where u 1 denotes the strength of response at e 1 , some appropriate function of this response, and ae r F d and F t (the distance and turning attenuation functions, are smoothly decreasing dimensionless functions that should be determined by further considerations and experience. Thus, our summation kernel is a product of the orientational and the spatial components involved in completing a curve between e 1 and e 2 . As we shall see below, this definition has many computational advantages. Let fu i g denote the set of straight responses for a given scale (l; w), where each u i is associated with two directed edges -). The total field induced at any element e is expressed by \Theta - The total field induced at - is given by G (l;w) \Theta \Theta Since in general the responses obtained by convolving the image with edge filters are bi-directional we may want to combine these two fields into one. This can be done in various ways. The simplest way is to take the sum fv as the completion field. Another possibility, in the spirit of [19], is to take the product fv j - as the completion field. Note that the field of a long straight response should be very different (farther-reaching and more orientation- specific) than the sum of the fields of shorter elements composing it, and should strongly depend on its width (see Fig. 5). This suggests that for a comprehensive completion process one must practice a multiscale process, performing a separate completion within each scale. The scaled induction field (10)-(11), avoids a fundamental difficulty of non-scaled fields like [7, 19, 16]. The latter exhibit so weak a completion for far elements, that it would be completely masked out by local noise and foreign local features.103050(a)103050(b) Figure 5. Induction fields (200 \Theta 200 pixels) in different scales using F d (a) The induction field of one long directed orientations. (b) The sum of induction fields of the three shorter elements composing this long element, each consist of: directed orientations. 5.3. Fast multigrid summation of induction-fields be the number of sites (P ), and the number of orientations (Y) at each site, that are required in order to describe all the l \Theta w straight responses that are significantly different from each other. It can be shown (see [4]) that if l and w are measured in pixel units then, for any N-pixel picture, so the total number of l \Theta w elements is O(N=w). Hence, for any geometric sequence of scales (e.g., l=1,2,4,., and w=1,3,9,.) the total number of straight elements is O(N log N). It has been shown (in [4]) that all the responses at all these elements can be calculated in only O(N log N) operations, using a multiscale algorithm that constructs longer-element responses from shorter ones. At any given scale l \Theta w, it seems that the summations (Eqs. (12) and (13)), summing over for each value of would require a total of operations (even though some of them can be performed in parallel to each other, as in [20]). However, using the smoothness properties of the particular kernel (11), the summation can be reorganized in a multiscale algorithm that totals only O(nm) operations (and the number of unparallelizable steps grows only logarithmically in nm). Indeed, the functions in (11) would usually take on the typical form F d . For such choices of the functions, and practically for any other reasonable choice, the kernel has the property of "asymptotic smoothness." By this we mean that any q- order derivative of G with respect to any of its six arguments decays fast with r \Theta and the higher q is the faster the decay is. Also, for any fixed r ij (even the smallest, i.e., r O(l)), G is a very smooth function of Y i and of Y j . Due to the asymptotic smoothness, the total contribution to v j (and - elements far from P j is a smooth function of need not be computed separately for each j, but can be interpolated (q-order interpolation, with as small an error as desired by using sufficiently high q) from its values at a few representative points. For this and similar reasons, multiscale algorithms, which split the summations into various scales of farness (see details in [3]) can perform all the summations in merely O(nm) operations. 6. Conclusion Important problems in perceptual grouping are the detection of smooth curves in images and their completion over gaps. In this paper we have simplified the computation involved in the process of completion, exploiting the smoothness of the solution to the problem, and have defined affinity measures for completion that take into a proper account the scale of edge elements. In particular, we have introduced new, closed-form approximations for the elas- tica energy functional and presented a fast numeric method to compute the curve of least energy. In this method the error decreases exponentially with the number of discrete elements. We then have used our approximations to define an affinity measure which takes into account the width and length of the edge elements by considering the range of curvatures that can be detected with corresponding filters of the same scale. Finally, we have shown that solutions to the problem of finding the most likely completions in an image can be implemented using a multigrid algorithm in time that is linear in the number of discrete edge elements in the image. This last observation applies also to recent methods for completion and salience [7, 19]. In the future we intend to use the multigrid algorithm to simultaneously detect completions at different scales in order to combine these completions into a single saliency map. --R "Completion energies and scale," "Shape Encoding and Subjective Contours," "Multilevel computations of integral transforms and particle interactions with oscillatory kernels," "Fast computation of multiple line integrals," "On Minimal Energy Trajec- tories," "The Role of Illusory Contours in Visual Segmentation," "Inferring Global Perceptual Contours from Local Features," "A Computational Model of Neural Contour Processing: Figure-Ground Segregation and Illusory Contours," "The Curve of Least Energy," "Organization in Vision," "Improve- ment in visual sensitivity by changes in local context: Parallel studies in human observers and in V1 of alert monkeys," "Elastica and Computer Vision," "Handbook of applied mathematics - Second Edition," "Shape Completion," "Structural Saliency: The Detection of Globally Salient Structures Using a Locally Connected Network," "Analytic Solution of Stochastic Completion Fields," "Filling-In the Gaps: The Shape of Subjective Contours and a Model for Their Generation," "3D Shape Representation by Contours," "Stochastic Completion Fields: A Neural Model of Illusory Contour Shape and Salience," "Local Parallel Computation of Stochastic Completion Fields," --TR --CTR Washington Mio , Anuj Srivastava , Xiuwen Liu, Contour Inferences for Image Understanding, International Journal of Computer Vision, v.69 n.1, p.137-144, August 2006 Sylvain Fischer , Pierre Bayerl , Heiko Neumann , Rafael Redondo , Gabriel Cristbal, Iterated tensor voting and curvature improvement, Signal Processing, v.87 n.11, p.2503-2515, November, 2007 Marie Rochery , Ian H. Jermyn , Josiane Zerubia, Higher-Order Active Contour Energies for Gap Closure, Journal of Mathematical Imaging and Vision, v.29 n.1, p.1-20, September 2007 Dan Kushnir , Meirav Galun , Achi Brandt, Fast multiscale clustering and manifold identification, Pattern Recognition, v.39 n.10, p.1876-1891, October, 2006 Fragment-based image completion, ACM Transactions on Graphics (TOG), v.22 n.3, July Benjamin B. Kimia , Ilana Frankel , Ana-Maria Popescu, Euler Spiral for Shape Completion, International Journal of Computer Vision, v.54 n.1-3, p.157-180, August-September Song Wang , Joachim S. Stahl , Adam Bailey , Michael Dropps, Global Detection of Salient Convex Boundaries, International Journal of Computer Vision, v.71 n.3, p.337-359, March 2007 Jonas August , Steven W. Zucker, Sketches with Curvature: The Curve Indicator Random Field and Markov Processes, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.25 n.4, p.387-400, April Xiaofeng Ren , Charless C. Fowlkes , Jitendra Malik, Learning Probabilistic Models for Contour Completion in Natural Images, International Journal of Computer Vision, v.77 n.1-3, p.47-63, May 2008 Song Wang , Toshiro Kubota , Jeffrey Mark Siskind , Jun Wang, Salient Closed Boundary Extraction with Ratio Contour, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.27 n.4, p.546-561, April 2005

elastica curve;multiscale;induction field;completion field;perceptual grouping;fast summation;scale;least-energy curve;curve completion;curve saliency

354191

Self-Calibration of a 1D Projective Camera and Its Application to the Self-Calibration of a 2D Projective Camera.

AbstractWe introduce the concept of self-calibration of a 1D projective camera from point correspondences, and describe a method for uniquely determining the two internal parameters of a 1D camera, based on the trifocal tensor of three 1D images. The method requires the estimation of the trifocal tensor which can be achieved linearly with no approximation unlike the trifocal tensor of 2D images and solving for the roots of a cubic polynomial in one variable. Interestingly enough, we prove that a 2D camera undergoing planar motion reduces to a 1D camera. From this observation, we deduce a new method for self-calibrating a 2D camera using planar motions. Both the self-calibration method for a 1D camera and its applications for 2D camera calibration are demonstrated on real image sequences.

Introduction A CCD camera is commonly modeled as a 2D projective device that projects a point in P 3 (the projective space of dimension 3) to a point in P 2 . By analogy, we can consider what we call a 1D projective camera which projects a point in P 2 to a point in P 1 . This 1D projective camera may seem very abstract, but many imaging systems using laser beams, infra-red or ultra-sound acting only on a source plane can be modeled this way. What is less obvious, but more interesting for our purpose, is that in some situations, the usual 2D camera model is also closely related to this 1D camera model. One first example might be the case of the 2D affine camera model operating on line segments: The direction vectors of lines in 3D space and in the image correspond to each other via this 1D projective camera model [21]. Other cases will be discussed later. In this paper, we first introduce the concept of self-calibration of a 1D projective camera by analogy to that of a 2D projective camera which is a very active topic [17, 12, 7, 13, 1, 29, 20] since the pioneering work of [18]. It turns out that the theory of self-calibration of 1D camera is considerably simpler than the corresponding one in 2D. It is essentially determined in a unique way by a linear algorithm using the trifocal tensor of 1D cameras. After establishing this result, we further investigate the relationship between the usual 2D camera and the 1D camera. It turns out that a 2D camera undergoing a planar motion can be reduced to a 1D camera on the trifocal line of the 2D cameras. This remarkable relationship allows us to calibrate a real 2D projective camera using the theory of self-calibration of a 1D camera. The advantage of doing so is evident. Instead of solving complicated Kruppa equations for 2D camera self-calibration, exact linear algorithm can be used for 1D camera self-calibration. The only constraint is that the motion of the 2D camera should be restricted to planar motions. The other applications, including 2D affine camera calibration, are also briefly discussed. Part of this work was also presented in [10]. The paper is organised as follows. In Section 2, we review the 1D projective camera and its trifocal tensor. Then, an efficient estimation of the trifocal tensor is discussed in Section 3. The theory of self-calibration of a 1D camera is introduced and developed in Section 4. After pointing out some direct applications of the theory in Section 5, we develop in Section 6 a new method of 2D camera self-calibration by converting a 2D camera undergoing planar motions into a 1D camera. The experimental results on both simulated and real image sequences are presented in Section 7. Finally, some concluding remarks and future directions are given in Section 8. Throughout the paper, vectors are denoted in lower case boldface, matrices and tensors in upper case boldface. Some basic tensor notation is used: Covariant indices as subscripts, contravariant indices as superscripts and the implicit summation convention. projective camera and its trifocal tensor We will first review the one-dimensional camera which was abstracted from the study of the geometry of lines under affine cameras [21]. We can also introduce it directly by analogy to a 2D projective camera. A 1D projective camera projects a point (projective plane) to a point (projective line). This projection may be described by a 2 \Theta 3 homogeneous matrix M as x: We now examine the geometric constraints available for points seen in multiple views similar to the 2D camera case [23, 24, 13, 28, 9]. There is a constraint only in the case of 3 views, as there is no any constraint for 2 views (two projective lines always intersect in a point in a projective plane). Let the three views of the same point x be given as (1) These can be rewritten in matrix form as 0: The vector cannot be zero, so 0: (2) The expansion of this determinant produces a trifocal constraint for the three views where T ijk is a 2 \Theta 2 \Theta 2 homogeneous tensor whose components T ijk are 3 \Theta 3 minors (involving all three views) of the following 6 \Theta 3 joint projection matrix: The components of the tensor can be made explicit as T where the bracket denotes the 3 \Theta 3 minor of i-th,j 0 -th and k 00 -th row vector of the above joint projection matrix and bar "-" in - i, - j and - k denotes the mapping (1; 2) 7! (2; \Gamma1): It can be easily seen that any constraint obtained by adding further views reduces to a trilinearity. This proves the uniqueness of the trilinear constraint. Moreover, the 2 \Theta 2 \Theta 2 homogeneous tensor has so it is a minimal parametrization of three views in the uncalibrated setting since three views have exactly 3 \Theta (2 \Theta 3 \Gamma to a projective transformation in P 2 . This result for the one-dimensional projective camera is very interesting. The trifocal tensor encapsulates exactly the information needed for projective reconstruction in P 2 . Namely, it is the unique matching constraint, it minimally parametrizes the three views and it can be estimated linearly. Contrast this to the 2D image case in which the multilinear constraints are algebraically redundant and the linear estimation is only an approximation based on over-parametrization. 3 Estimation of the trifocal tensor of a 1D camera Each point correspondence in 3 views u yields one homogeneous linear equation for the 8 tensor components T ijk for 2: With at least 7 point correspondences, we can solve for the tensor components linearly. A careful normalisation of the measurement matrix is nevertheless necessary just like that stressed in [11] for the linear estimation of the fundamental matrix. The points at each image are first translated so that the centroid of the points is the origin of the image coordinates, then scaled so that the average distance of the points from the origin is 1. This is achieved by an affine transformation of the image coordinates in each image: - With these normalised image coordinates, the normalised tensor components - T ijk are linearly estimated by SVD from - The original tensor components T ijk are recovered by de-scaling the normalised tensor - T ijk as a c 4 Self-calibration of a 1D camera from 3 views The concept of camera self-calibration using only point correspondences became popular in computer vision community following Maybank and Faugeras [18] by solving the so-called Kruppa equations. The basic assumption is that the internal parameters of the camera remain invariant. In the case of the 2D projective camera, the internal calibration (the determination of the 5 internal parameters) is equivalent to the determination of the image ! of the absolute conic in P 3 . 4.1 The internal parameters of a 1D camera and the circular points For a 1D camera represented by a 2 \Theta 3 projection matrix M 2\Theta3 , this projection matrix can always be decomposed into R 2\Theta2 where K ff represents the two internal parameters: ff the focal length in pixels and u 0 the position of the principal point; the external parameters are represented by a 2 \Theta 2 rotation matrix R 2\Theta2 cos ' sin ' sin ' cos ' and the translation vector t 2\Theta1 The object space for a 1D camera is a projective plane, and any rigid motion on the plane leaves the two circular points I and J invariant (a pair of complex conjugate points on the line at infinity) of the plane. Similar to the 2D camera case where the knowledge of the internal parameters is equivalent to that of the image of the absolute conic, the knowledge of the internal parameters of a 1D camera is equivalent to that of the image points i and j of the circular points in P 2 . The relationship between the image of the circular points and the internal parameters of the 1D camera follows directly by projecting one of the circular points I = (i; \Gamma1, by the camera M 2\Theta3 ff R 2\Theta2 @ It clearly appears that the real part of the ratio of the projective coordinates of the image of the circular point i is the position of the principal point u and the imaginary part is the focal length ff. 4.2 Determination of the images of the circular points Our next task is to locate the circular points in the images. Let us consider one of the circular points, say I . This circular point is projected onto i, i 0 and i 00 in the three views. As they should be invariant because of our assumption that the internal parameters of the camera are constant, we have: where The triplet of corresponding points i satisfies the trilinear constraint (3) as all corresponding points do, therefore, T ijk 0: This yields the following cubic equation in the unknown A cubic polynomial in one unknown with real coefficients has in general either three real roots or one real root and a pair of complex conjugate roots. The latter case of one real and a pair of complex conjugates is obviously the case of interest here. In fact, Equation (4) characterizes all the points of the projective plane which have the same coordinates in three views. This is reminiscent of the 3D case where one is interested in the locus of all points in space that project onto the same point in two views (see Section 6). The result that we have just obtained is that in the case where the internal parameters of the camera are constant, there are in general three such points: the two circular points which are complex conjugate, and a real point with the following geometric interpretation. Consider first the case of two views and let us ask the question, what is the set of points such that their images in the two views are the same? This set of points can be called the 2D horopter (h) of the set of two 1D views. Since the two cameras have the same internal parameters we can ignore them and assume that we work with the calibrated pixel coordinates. In that case, a camera can be identified to an orthonormal system of coordinates centered at the optical center, one axis is parallel to the retina, the other one is the optical axis. The two views correspond to each other via a rotation followed by a translation. This can always be described in general as a pure rotation around a point A whose coordinates can easily be computed from the cameras' projection matrices. A simple computation then shows that the horopter (h) is the circle going through the two optical centers and A, as illustrated in Figure 1.a. In fact it is the circle minus the two optical centers. Note that since all circles go through the circular points (hence their name), they also belong to the horopter curve, as expected. In the case of three views, the real point, when it exists, must be at the intersection of the horopter of the first two views and the horopter (h 23 ) of the last two views. The first one is a circle going through the optical centers C 1 and C 2 , the second one is a circle going through the optical centers C 2 and C 3 . Those two circles intersect in general at a second point C which is the real point we were discussing, and the third circle (h 13 corresponding to the first and third views must also go through the real point C, see Figure 1.b. a. First camera Second camera center of rotation b.C Second camera C2First camera Third camera Figure 1: a. The two dimensional horopter which is the set of points having the same coordinates in the 2 views (see text). b. The geometric interpretation of the real point C which has the same images in all three views (see text). We have therefore established the interesting result that the internal parameters of a 1D camera can be uniquely determined through at least 7 point correspondences in 3 views: the seven points yield the trifocal tensor and Equation (4) yields the internal parameters. Applications The theory of self-calibration of 1D camera is considerably simpler than the corresponding one in 2D [18] and can be directly used whenever a 1D projective camera model occurs, for instance: self-calibration of some active systems using laser beams, infra-red [3] or ultra-sound whose imaging system is basically reduced to a 1D camera on the source plane; and partial/full self-calibration of 2D projective camera using planar motions. The first type of applications is straightforward. The interesting observation is that the 1D calibration procedure can also be used for self-calibrating a real 2D projective camera if the camera motion is restricted to planar motions. This is discussed in detail in the remaining of this paper. 6 Calibrating a 2D projective camera using planar motions A planar motion consists of a translation in a plane and a rotation about an axis perpendicular to that plane. Planar motion is often performed by a vehicle moving on the ground, and has been used for camera self-calibration by Beardsley and Zisserman [4] and by Armstrong et al. [1]. Recall that the self-calibration of a 2D projective camera [8, 18] consists of determining the 5 unchanging internal parameters of a 2D camera, represented by a 3 \Theta 3 upper triangular matrix This is mathematically equivalent to the determination of the image of the absolute conic !, which is a plane conic described by x Given the image of the absolute conic x T the calibration matrix K can be found from C using the Choleski decomposition. Converting 2D images into 1D images For a given planar motion, the trifocal plane-the plane through the camera centers-of the camera is coincident with the motion plane as the camera is moving on it. Therefore the image location of the motion plane is the same as the trifocal line which could be determined from fundamental matrices. The determination of the image location of the motion plane has been reported in [1, 4]. Obviously, if restricting the working space to the trifocal plane, we have a perfect 1D projective camera model which projects the points of the trifocal plane onto the trifocal line in the 2D image plane, as the trifocal line is the image of the trifocal plane. In practice, very few or no any points at all really lie on the trifocal plane. However, we may virtually project any 3D points onto the trifocal plane, therefore comes the central idea of our method: the 2D images of a camera undergoing any planar motion reduce to 1D images by projecting the 2D image points onto the trifocal line. This can be achieved in at least two ways. First, if the vanishing point v of the rotation axis is well-defined. This vanishing point of the rotation axis being the direction perpendicular to the common plane of motion can be determined from fundamental matrices by noticing that the image of the horopter for planar motion degenerates to two lines [1], one of which goes through the vanishing point of the rotation axis, we may refer to [1] for more details. Given a 3D point M with image m, we mentally project it to - M in the plane of motion, the projection being parallel to the direction of rotation. The image - m of this virtual point can be obtained in the image as the intersection of the line v \Theta m with the trifocal line t, i.e. - (v \Theta m): Since the vanishing point v of the rotation axis and the trifocal line t are well defined, this construction illustrated in Figure 2 is a well-defined geometric operation. Note this is also a projective projection from P 2 (image plane) to P 1 (trifocal line): m 7! - m; as is illustrated in Figure 3. Alternatively, if the vanishing point is not available, we can nonetheless create the virtual points in the trifocal plane. Given two points M and M 0 with images m and m 0 , the line (M; M 0 ) intersects the plane of motion in - M . The image - m of this virtual point can be obtained in the image as the intersection of the line (m; m 0 ) with the trifocal line t, see Figure 4. Another important consequence of this construction is that 2D image line segments can also be converted into 1D image points! The construction is even simpler, as the resulting 1D image point is just the intersection of the line segment with the trifocal line. Direction of the axis of rotation Figure 2: Creating a 1D image from a 2D image from the vanishing point of the rotation axis and the trifocal line (see text). trifocal line vanishing point of the rotation axis 2D image point 1D image point Figure 3: Converting 2D image points into 1D image points in the image plane is equivalent to a projective projection from the image plane to the trifocal line with the vanishing point of the rotation axis as the projection center. Figure 4: Creating a 1D image from any pairs of points or any line segments (see text). 1D Self-calibration At this point, we have obtained the interesting result that a 1D projective camera model is obtained by considering only the re-projected points on the trifocal line for a planar motion. The 1D self-calibration method just described in Section 4 will allow us to locate the image of the circular points common to all planes parallel to the motion plane. Estimation of the image of the absolute conic for the 2D camera Each planar motion generally gives us two points on the absolute conic, together with the vanishing point of the rotation axes as the pole of the trifocal line w.r.t. the absolute conic. The pole/polar relation between the vanishing point of the rotation axes and the trifocal line was introduced in [1]. As a whole, this provides 4 constraints on the absolute conic. Since a conic has 5 d.o.f., at least different planar motions, yielding 8 linear constraints on the absolute conic, will be sufficient to determine the full set of 5 internal parameters of a general 2D camera by fitting a general conic of If we assume a 4-parameter model for camera calibration with no image skew (i.e. planar motion yielding 4 constraints is generally sufficient to determine the 4 internal parameters of the 2D camera. However this is not true for the very common planar motions such as purely horizontal or vertical motions with the image plane perpendicular to the motion plane It can be easily proven that there are only 3 instead of 4 independent constraints on the absolute conic in these configurations. We need at least 2 different planar motions for determining the 4 internal parameters. This also suggests that even if the planar motion is not purely horizontal or vertical, but close, the vanishing point of the rotation axes only constrains loosely the absolute conic. Using only the circular points located on the absolute conic is preferable and numerically stable, but we may need at least 3 planar motions to determine the 5 internal parameters of the 2D camera. Note that the numerical instability of the vanishing point for nearly horizontal trifocal line was already reported by Armstrong in [2]. Obviously, if we work with a 3-parameter model with known aspect ratio and without skew, one planar motion is sufficient [1]. As we have mentioned at the beginning of this section the method described in this section is related to the work of Armstrong et al. [1], but there are some important differences which we explain now. ffl First, our approach gives an elegant insight of the intricate relationship between 2D and 1D cameras for a special kind of motion called planar motion. ffl Second, it allows us to use only the fundamental matrices of the 2D images and the trifocal tensor of 1D images to self-calibrate the camera instead of the trifocal tensor of 2D images. It is now well known that fundamental matrices can be very efficiently and robustly estimated [31, 27]. The same is true of the estimation of the 1D trifocal tensor [21] which is a linear process. Armstrong et al., on the other hand, use the trifocal tensor of 2D images which so far has been hard to estimate due to complicated algebraic constraints to our knowledge. Also, the trifocal tensor of 2D images takes a special form in the planar motion case [1] and the new constraints have to be included in the estimation process. It may be worth mentioning that in the case of interest here, planar motion of the cameras, the Kruppa equations become degenerate [30] and the recover the internal parameters is impossible from the Kruppa equations. Since it is known that the trifocal tensor of 2D images is algebraically equivalent to the three fundamental matrices plus the restriction of the trifocal tensor to the trifocal plane [14, 15, 9], our method can be seen as an inexpensive way of estimating the full trifocal tensor of 2D images: first estimate the three fundamental matrices (nonlinear but simple and well understood), then estimate the trifocal tensor in the trifocal plane (linear). Although it looks superficially that both the 1D and 2D trifocal tensors can be estimated linearly with at least 7 image correspondences, this is misleading since the estimation of the 1D trifocal tensor is exactly linear for 7 d.o.f. whereas the linear estimation of the 2D trifocal tensor is only a rough approximation based on a set of 26 auxiliary parameters for its d.o.f. and obtained by neglecting 8 complicated algebraic constraints. ffl Third, but this is a minor point, our method may not require the estimation of the vanishing point of the rotation axes. 7 Experimental results The theoretical results for 1D camera self-calibration and its applications to 2D camera calibration have been implemented and experimented on synthetic and real images. Due to space limitation, we are not to present the results on synthetic data, the algorithms generally perform very well. We only show some real examples. Here we consider a scenario of a real camera mounted on a robot's arm. Two sequences of images are acquired by the camera moving in two different planes. The first sequence contains 7 (indexed from 16 to 22) images (cf. Figure 5) and the second contains 8 (indexed from 8 to 15). The calibration grid was used to have the ground truth for the internal camera parameters which have been measured as ff using the standard calibration method [6]. Figure 5: Three images of the first planar motion. We take triplets of images from the first sequence and for each triplet we estimate the trifocal line and the vanishing point of the rotation axes by the 3 fundamental matrices of the triplet. The 1D self-calibration is applied for estimating the images of the circular points along the trifocal lines. To evaluate the accuracy of the estimation, the images of the circular points of the trifocal plane are re-computed in the image plane from the known internal parameters by intersecting the image of the absolute conic with the trifocal line. Table 1 shows the results for different triplets of images of the first sequence. Image triplet Fixed point Circular points by self-calibration Circular points by calibration Table 1: Table of the estimated positions of the images of the circular points by self-calibration with different triplets of images of the first sequence. The quantities are expressed in the first image pixel coordinate system. The location of circular points by calibration vary as the trifocal line location varies. Since we have more than 3 images for the same planar motion of the camera, we could also estimate the trifocal line and the vanishing point of the rotation axes by using all available fundamental matrices of the 7 images of the sequence. The results using redundant images are presented for different triplets in Table 2. We note the slight improvement of the results compared with those presented in Table 1. Image triplet Circular points Fixed point known position by calibration 262:1 \Sigma i2590:6 Table 2: Table of the estimated positions of the image of circular points with different triplets of images. These quantities vary because the 1D trifocal tensor varies. The trifocal line and the vanishing point of the rotation axes are estimated using 7 images of the sequence instead of the minimum of 3 images. The same experiment was carried out for the other sequence of images where the camera underwent a different planar motion. Similar results to the first image sequence are obtained, we give only the result for one triplet of images in Table 3 for this sequence. Image triplet Fixed point Circular points by self-calibration Circular points by calibration Table 3: Table of the estimated position of the image of circular points with one triplet of second image sequence. Now two sequences of images each corresponding to a different planar motion yield four distinct imaginary points on the image plane which must be on the image ! of the absolute conic. Assuming that there is no camera skew, we could fit to those four points an imaginary ellipse using standard techniques and compute the resulting internal parameters. Note that we did not use the pole/polar constraint of the vanishing point of the ratation axes on the absolute conic as it was discussed in Section 6 that this constraint is not numerically reliable. To have an intuitive idea of the planar motions, the two trifocal lines together with one image are shown in Figure 6. -1000 -500 500 1000 150050015002500 Figure The image of the motion planes of the two planar motions. The ultimate goal of self-calibration is to get 3D metric reconstruction. 3D reconstruction from two images of the sequence is performed by using the estimated internal parameters as illustrated in Figure 7. To evaluate the reconstruction quality, we did the same reconstruction using the known internal parameters. Two such reconstructions differ merely by a 3D similarity transformation which could be easily estimated. The resulting relative error for normalised 3D coordinates by similarity between the reconstruction from self-calibration and off-line calibration is 3:4 percent. -0.4 -0.3 -0.2 -0.10.10.3 -0.23.43.84.4 Figure 7: Two views of the resulting 3D reconstruction by self-calibration. 8 Conclusions and other applications We have first established that the 2 internal parameters of 1D camera can be uniquely determined through the trifocal tensor of three 1D images. Since the trifocal tensor can be estimated linearly from at least 7 points in three 1D images, the method of the 1D self-calibration is a real linear method (modulo the fact that we have to find the roots of a third degree polynomial in 1 variable), no over- parameterisation was introduced. Secondly, we have proven that if a 2D camera undergoes a planar motion, the 2D camera reduces to a 1D camera in the plane of motion. The reduction of a 2D image to a 1D image can be efficiently performed by using only the fundamental matrices of 2D images. Based on this relation between 2D and 1D images, the self-calibration of 1D camera can be applied for self-calibrating a 2D camera. Our experimental results based on real image sequences show the very large stability of the solutions yielded by the 1D self-calibration method and the accurate 3D metric reconstruction that can be obtained from the internal parameters of the 2D camera estimated by the 1D self-calibration method. The camera motions that may defeat the self-calibration method developed in Section 4 are described in [26]. --R Invariancy Methods for Points Affine calibration of mobile vehicles. The twisted cubic and camera calibration. Camera Calibration for 3D Computer Vision Stratification of three-dimensional vision: Projective Motion from point matches: Multiplicity of solutions. About the correspondences of points between n images. In defence of the 8-point algorithm Euclidean reconstruction from uncalibrated views. A linear method for reconstruction from lines and points. Geometry and Algebra of Multiple Projective Transformations. Algebraic Properties of Multilinear Constraints. A Common Framework for Multiple-View Tensors A theory of self calibration of a moving camera. Affine structure from line correspondences with uncalibrated affine cameras. Algebraic Projective Geometry. Algebraic functions for recognition. A unified theory of structure from motion. Critical Motion Sequences for Monocular Self-Calibration and Uncalibrated Euclidean Recon- struction Vision 3D non calibr-ee : contributions - a la reconstruction projective et - etude des mouvements critiques pour l'auto-calibrage Performance characterization of fundamental matrix estimation under image degradation. Matching constraints and the joint image. Autocalibration and the Absolute Quadric. Camera self-calibration from video sequences: the Kruppa equations revisited A robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry. --TR --CTR Mandun Zhang , Jian Yao , Bin Ding , Yangsheng Wang, Fast individual face modeling and animation, Proceedings of the second Australasian conference on Interactive entertainment, p.235-239, November 23-25, 2005, Sydney, Australia A. C. Murillo , C. Sags , J. J. Guerrero , T. Goedem , T. Tuytelaars , L. Van Gool, From omnidirectional images to hierarchical localization, Robotics and Autonomous Systems, v.55 n.5, p.372-382, May, 2007

self-calibration;camera model;vision geometry;planar motion;1D camera

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Look-Ahead Procedures for Lanczos-Type Product Methods Based on Three-Term Lanczos Recurrences.

Lanczos-type product methods for the solution of large sparse non-Hermitian linear systems either square the Lanczos process or combine it with a local minimization of the residual. They inherit from the underlying Lanczos process the danger of breakdown. For various Lanczos-type product methods that are based on the Lanczos three-term recurrence, look-ahead versions are presented which avoid such breakdowns or near-breakdowns at the cost of a small computational overhead. Different look-ahead strategies are discussed and their efficiency is demonstrated by several numerical examples.

Introduction . Lanczos-type product methods (LTPMs) like (Bi)CGS [42], BiCGStab [44], BiCGStab2 [22], and BiCGStab(') [38], [41] are among the most efficient methods for solving large systems of linear equations sparse system matrix A 2 C N \ThetaN . Compared to the biconjugate gradient method they have the advantage of converging roughly twice as fast and of not requiring a routine for applying the adjoint system matrix A H to a vector. Nev- ertheless, they inherit from BiCG the short recurrence formulas for generating the approximations x k and the corresponding residuals r k := b \Gamma Ax k . As for BiCG, where the convergence can be smoothed by applying the quasi- minimal residual (QMR) method [16] or the local minimum residual process (MR smoothing) [37], [47], product methods can be combined with the same techniques [14], [47] to avoid the likely "erratic" convergence behavior [11], [36]; the benefit of these smoothing techniques is disputed, however. A well-known problem of all methods that make implicit use of the Lanczos polynomials generated by a non-Hermitian matrix is the danger of breakdown. Although exact breakdowns are very rare in practice, it has been observed that near-breakdowns can slow down or even prevent convergence [15]. Look-ahead techniques for the Lanczos process [15], [21], [23], [34], [35], [43] allow us to avoid this problem when we use variants of the BiCG method with or without the mentioned smoothing tech- niques. However, the general look-ahead procedures that have so far been proposed for (Bi)CGS or other LTPMs are either limited to exact breakdowns [8], [9] or are based on different look-ahead recursions: in fact, the look-ahead steps proposed by Brezinski and Redivo Zaglia [5] to avoid near-breakdowns in CGS and those in [6], which are applicable to all LTPMs, are based on the so-called BSMRZ algorithm, which is itself based on a generalization of coupled recurrences (different from those of the standard BiOMin and BiODir versions of the BiCG method) for implementing the Lanczos method. For a theoretical comparison of the two approaches we refer to [25, x 19]. Recently, a number of further "look-ahead-like" algorithms for Lanczos- type solvers have been proposed by Ayachour [1], Brezinski et al. [7], Graves-Morris Applied Mathematics, ETH Zurich, ETH-Zentrum HG, CH-8092 Zurich, Switzerland (mhg@sam.math.ethz.ch). Formerly at the Swiss Center for Scientific Computing (CSCS/SCSC). y German Aerospace Research Establishment (DLR), German Remote Sensing Data Center (DFD), D-82234 Oberpfaffenhofen, Germany (kjr@dfd.dlr.de). Formerly also at CSCS/SCSC. [19], and Ziegler [48]. Their discussion is beyond the scope of this paper, but it seems that [1, 7, 48] are restricted to exact breakdowns. In this paper, we start from the approach in [15] and [23, x 9] and derive an alternative look-ahead procedure for LTPM algorithms that make use of the Lanczos three-term recurrences. Compared to the standard coupled two-term recurrences they have the advantage of being simpler to handle with regard to look-ahead, since they are only affected by one type of breakdown. In contrast to the first version of this work [26], we also capitalize upon an enhancement for the look-ahead Lanczos algorithm pointed out by Hochbruck recently (see [28], [25]), which is here adapted to LTPMs. Other improvements help to further reduce the overhead and stabilize the process. Starting with an initial approximation x 0 and a corresponding initial residual steps a basis of the 2n-dimensional Krylov space in such a way that the even indexed basis vectors are of the form where ae n is the nth Lanczos polynomial (see below) and n is another suitably chosen polynomial of exact degree n. In the algorithms we discuss, these vectors will be either the residual of the nth approximation xn or a scalar multiple of it. By allowing them to be multiples of the residuals, that is, by considering so-called unnormalized [20] or inconsistent [25] Krylov space solvers, we avoid the occurrence of pivot (or ghost) breakdowns. (For the various types of breakdowns and their connection to the block structure of the Pad'e table see [20, 24, 30, 31]. In the setting of [5, 6], they have been addressed particularly in [4].) More generally, we define a doubly indexed sequence of product vectors w l n by The aim is to find in the (n 1)th step of an LTPM an improved approximation xn+1 by computing a new product vector w n+1 n+1 from previously determined ones in a stable way. To visualize the progression of an algorithm and the recurrences it uses we arrange the product vectors w l n in a w-table 1 . Its n-axis points downwards and its l-axis to the right. We describe then how an algorithm moves in this table from the upper left corner downwards to the right. This paper is organized as follows. In Section 2 we review the look-ahead Lanczos process. Versions based on the Lanczos three-term recurrences of various LTPMs are introduced in Section 3. In Section 4 we present for these LTPMs look-ahead procedures and analyze their computational overhead, and in Section 5 several look-ahead strategies are discussed. Our preferred way of applying the look-ahead procedures to obtain the solution of a linear system is presented in Section 6. In Section 7 the efficiency of the proposed algorithms is demonstrated by numerical examples, and in Section 8 we draw some conclusions. 2. The Look-Ahead Lanczos Process. The primary aim of the Lanczos process [32] is the construction of a pair of biorthogonal bases for two nested sequences of Krylov spaces. Given A 2 C N \ThetaN and a pair of starting vectors, (ey Biorthogonalization (BiO) Algorithm generates a pair of finite sequences, fey n g 1 The w-table is different from the scheme introduced by Sleijpen and Fokkema [38] for BiCGstab('). fyng n=0 , of left and right Lanczos vectors, such that e and ae or, equivalently, Here, h\Delta; \Deltai denotes the inner product in C N , which we choose to be linear in the second argument, and ? indicates the corresponding orthogonality. The sequence of pairs of Lanczos vectors can be constructed by the three-term recursions with coefficients ff n and fi n that are determined from the orthogonality condition (2.2) and nonvanishing scale factors fl n that can be chosen arbitrarily. Choosing would allow us to consider the right Lanczos vectors yn as residuals and to update the iterates xn particularly simply. But this choice also introduces the possibility of a breakdown due to ff n Therefore, we suggest in Section 6 a different way of defining iterates. From (2.4) it follows directly that the Lanczos vectors can be written in the form e yn=ae n where ae n denotes the n-th Lanczos polynomial. Since we aim here at LTPMs, we consider for the Krylov spaces e Kn more general basis vectors of the form e with arbitrary, but suitably chosen polynomials n of exact degree n and e z 0 := e y 0 . In general, e z n+1 ? Kn+1 will no longer hold, but e Kn+1 ? yn+1 can still be attained by enforcing e z means choosing the coefficients ff n and in (2.4) in the following way: if n ? 0 we need but since hez When Similarly, from the orthogonality condition hez we obtain Equations (2.7)-(2.9) and the first recurrence of (2.4) specify what one might call the one-sided Lanczos process, formulated in [36] to derive LTPMs. Clearly, this recursive process terminates with y or it breaks down with ffi The look-ahead Lanczos process [23, x 9], [15] overcomes such a breakdown if curable, i.e., if for some k, hez ; A k y i 6= 0. However, its role is not restricted to treating such exact breakdowns with ffi 0: it allows us to continue the biorthogonalization process whenever for stability reasons we choose to enforce the orthogonality condition (2.2) only partially for a couple of steps. We will come back later to the conditions that make us start such a look-ahead phase. In the look-ahead Lanczos process the price we have to pay is that the Gramian matrix D := (hez m ; yn i) becomes block-lower triangular instead of triangular (as in the generic one-sided Lanczos algorithm) or diagonal (as in the generic two-sided Lanczos algorithm). In other words, we replace the conditions e by e and nonsingular is the subsequence of indices of (well-condi- tioned) regular Lanczos vectors yn j ; for the other indices the vectors are called inner vectors. Likewise, we refer to n as a regular index if while n is called an inner index otherwise. Note that there is considerable freedom in choosing the subsequence fn j g J example, we might request that the smallest singular value oe min (D j ) is sufficiently large. The sequence fyn g is determined by the condition e hence it does not depend directly on the sequence fez n g but only on the spaces e that are spanned by the first n j elements of fez n g. On the other hand, the smallest singular value of each D j depends on the basis chosen, a fact that one should take into account. Forming the blocks \Theta \Theta e z we can express relation (2.10) as e ae The look-ahead step size is in the following denoted by h j := n In the look-ahead case the Lanczos vectors fyng can be generated by the recursion [23, x 9], [15]: \Theta denotes the not yet fully completed j-block if is an inner index, while b regular, that is, if . The coefficient vector fi n is determined by the condition thus, e The coefficient vector ff n can be chosen arbitrarily if n+1 is an inner index. Obviously, the choice ff yields the cheapest recursion, but we may gain numerical stability by choosing ff n 6= 0. On the other hand, if results from the condition that is e Recently, it was pointed out by Hochbruck [28] that the recursion (2.13) can be simplified since the contribution of the older block Y j \Gamma1 can be represented by multiples of a single auxiliary vector y 0 . This is due to the fact (noticed in [23]) that the matrix made up of the coefficient vectors fi n j is of rank one; see also [25, x 19]. This simplification has also been capitalized upon in the look-ahead Hankel solver of Freund and Zha [17], which is closely related to the look-ahead Lanczos algorithm. In particular, due to (2.10) or (2.12) we have Therefore, (2.14) simplifies to e Introducing the auxiliary vector we have e z H so that the recurrence (2.13) becomes and (2.15) changes to e (2. 3. Lanczos-Type Product Methods (LTPMs). LTPMs are based on two ideas. The first one is to derive for a certain sequence of product vectors w l+1 recursion formulas that involve only previously computed product vectors, so that there is no need of explicitly computing the vectors e z l and yn . By multiplying the three-term recursion (2.4) for the right Lanczos vectors yn with l (A) we obtain Aw l This recursion can be applied to move forward in the vertical direction in the w-table. To obtain a formula for proceeding in the horizontal direction, the recursion for the chosen polynomials l (i) is capitalized upon in an analogous way. The second basic idea is to rewrite the inner products that appear in the Lanczos process in terms of product vectors: For the coefficients ff n and fi n of (2.8) and (2.9) this results in LTPMs still have short recurrence formulas if the polynomials l have a short one. In addition, they have two advantages over BiCG: first, multiplications with the adjoint system matrix A H are avoided; second, for an appropriate choice of the polynomials l (i) smaller new residuals r l := w l (A)y l can be expected because of a further reduction of y l by the operator l (A). Different choices of the polynomials l (i) lead to different LTPMs. In the following we briefly review some possible choices for these polynomials. We start with the general class where they satisfy a three-term recursion. Since then both the 'left' polynomials l and the `right' polynomials ae n fulfill a three-term recursion, we say that this is the class of (3; 3)-type LTPMs. In the following we briefly review some possible choices for these polynomials. For a more detailed discussion and a pseudocode for the resulting algorithms we refer to [36]. 3.1. LTPMs based on a three-term recursion for f l g: BiOxCheb and BiOxMR2. Assume the polynomials l (i) are generated by a three-term recursion of the form [22] l. Note that, by induction, l has exact degree l and l l. Hence, the polynomials qualify as residual polynomials of a Krylov space method. Multiplying yn by l+1 (A) from the left and applying (3.3) yields the horizontal recurrence By applying (3.1) and (3.4) the following loop produces a new product vector w n+1 from previously calculated ones, namely w l 1), and the product Aw also evaluated before. At the end of the loop, besides n+1 , which will be needed in the next run through the loop, are available. Loop 3.1. (General (3; 3)-type LTPM) 1. Compute n and determine fi n and ff n . 2. Use (3.1) to compute w n+1 . 3. Compute n+1 and determine n and jn . 4. Use (3.4) to compute w n+1 n and w n+1 n+1 . Next we discuss two ways of choosing the polynomials l , that is, of specifying the recurrence coefficients l and j l . One possibility is to combine the Lanczos process with the Chebyshev method [13], [45] by choosing l as a suitably shifted and scaled Chebyshev polynomial. This combination was suggested in [3, 22, 44]. Let us call the resulting (3; 3)-type LTPM BiOxCheb. It is well known that these polynomials satisfy a recurrence of the form (3.3). After acquiring some information about the spectrum of the matrix A, for example, by performing a few iterations with another Krylov space method, one can determine the recursion for the scaled and shifted Chebyshev polynomials that correspond to an ellipse surrounding the estimated spectrum [33]. However, the necessity to provide spectral information is often seen as a drawback of this method. Another idea is to use the coefficients l and j l of (3.3) for locally minimizing the norm of the new residual w l+1 l+1 . This idea is borrowed from the BiCGStab and methods [44], [22] reviewed below: l and j l are determined by solving the two-dimensional minimization problem min Introducing the N \Theta 2 matrix B l+1 := \Theta (w l we can write (3.5) as the least-squares problem min Therefore, l and j l can be computed as the solution of the normal equations l In view of the two-dimensional local residual norm minimization performed at every step (except the first one) we call this the BiOxMR2 method 2 . A version based on coupled two-term Lanczos recurrences of this method was introduced by the first author in a talk in Oberwolfach (April 1994). Independently it was as well proposed by Cao [10] and by Zhang [46], whose Technical Report is dated April 1993. Zhang also considered two-term formulas for l and presented very favorable numerical results. 2 The letter 'x' in the name BiOxMR2 reflects the fact that the residual polynomials of this method are the products of the Lanczos polynomials generated by the BiO process with polynomials obtained from a successive two-dimensional minimization of the residual (MR2). Similarly, BiOxCheb means a combination of the BiO process with a Chebyshev process. 3.2. LTPMs based on a two-term recursion for f l g: BiOStab. In Van der Vorst's BiCGStab [44] the polynomials l (i) are built up successively as products of polynomials of degree 1: Inserting here for i the system matrix A and multiplying by yn from the right yields The coefficient l is determined by minimizing the norm of w l+1 that is by solving the one-dimensional minimization problem min 2C which leads to hAw l hAw l Recurrences (3.1) and (3.8) can be used to compute a new product vector w n+1 from w n n and w n as described in Loop 3.1 with is calculated from (3.10). However, since (3.8) is only a two-term recurrence, there is no need now to compute w n+1 in substep 2 of the loop. We call this algorithm BiOStab since it is a version of BiCGStab that is based on the three-term recurrences of the Lanczos biorthogonalization process 3 . 3.3. BiOStab2. BiOStab2 is the version of BiCGStab2 [22] based on the three-term Lanczos recurrences. The polynomials l (i) satisfy the recursions l is even; l is odd: Here l may be obtained by solving the one-dimensional minimization problem (3.9), so l is given by (3.10). However, if j l j is small, this choice is dangerous since the vector component needed to enlarge the Krylov space becomes negligible [38]. Then some other value of l should be chosen. Except for roundoff, the choice has no effect on later steps, because l and j l are determined by solving the two-dimensional minimization problem (3.5). Multiplying yn by l+1 (A) and applying (3.11) leads to ae l is even Aw l l is odd: 3.1 applies with is even, while n and jn are chosen as indicated above if n is odd. If n is even, there is no need to compute w n+1 in substep 2 of the loop. 3 Eijkhout [12] also proposed such a variant of BiCGStab; however, his way of computing the Lanczos coefficients is much too complicated. 3.4. BiO-Squared (BiOS). BiOS is obtained by "squaring" the three-term Lanczos process: among the basis vectors generated are those Krylov space vectors that correspond to the squared Lanczos polynomials. By complementing BiOS with a recursion for Galerkin iterates we will obtain BiOResS, a (3; 3)-type version of Sonneveld's (2; 2)-type conjugate gradient squared (CGS ) method [42]. The method fits into the framework of LTPMs if we identify l The vectors e yn are then exactly the left Lanczos vectors so that now yn ? e Kn as well as e z n ? Kn is fulfilled. Thus, the coefficient ff n in (3.2) simplifies to and the w-table becomes symmetric since Consequently, we have in analogy to (3.1) Aw l These two recursions lead to the following loop: Loop 3.2. (BiOS) 1. Compute n and determine fi n and ff n . 2. Use (3.1) to compute w n+1 . 3. Compute n+1 . 4. Use (3.16) to compute w n+1 n+1 . Note that we exploit in substep 2 the symmetry of the w-table: the product vector which is needed for the calculation of w n n+1 by (3.1), is equal to w n and need not be stored. We also point out that (3.16) is not of the form (3.4); in particular, the coefficients of w n and w l n need not sum up to 1. 4. Look-Ahead Procedures for LTPMs. Look-ahead steps in an LTPM serve to stabilize the Lanczos process, the vertical movement in the w-table. Except for BiOS, the recursion formulas for the horizontal movement remain the same. The vertical movement is now in general based on the recurrence formula (2.21) for the Lanczos vectors, but we need to replace these vectors by product vectors w l We introduce the blocks of product vectors c l \Theta \Theta and the auxiliary product vector w 0 l defined by Again c l is a regular index, while c W l denotes the not yet completed jth block if n+ 1 is an inner index. Now, multiplying (2.21) by l (A) from the left, we obtain Aw l As in Section 3, the coefficient vectors fi 0 n and, in the regular case, ff n can be expressed in terms of the product vectors by rewriting all inner products in such a way that the part l z l on the left side is transferred to the right side of the inner product. For the diagonal blocks D j of the Gramian this yields \Theta and fi 0 n of (2.18) becomes Likewise, ff n of (2.22) turns into For advancing in the horizontal direction of the w-table we will still use (3.4), (3.8), the combination (3.12), or (3.16). Substituting Aw n in (4.4) according to these formulas and letting i be the (n n)-element of D (4.4) simplifies to We remark that by using (4.6) instead of hez we avoid to compute and store the complete block. Now only few elements of this block are needed, see Figure 4.1 below. In this figure we display those entries w l n and products Aw l n in the w-table that are needed to compute by (4.2) a new vector w l n+1 marked by ' '. inner case d d d d d d v 0 . V A regular case d d d d d d ff 0 ff . D D V D . D D V D A A A A Fig. 4.1. Entries in the w-table needed for the construction of a new inner or regular vector marked by ' ' by recurrence formula (4.2). Here, `v 0 ' indicates entries needed for w 0 l those for c l , 'd' those for those additionally required for D j . Moreover, 'ff 0 ' marks auxiliary vectors in the formula (4.5) for ff n , and 'A' stands for a matrix-vector product (MV) with A in this formula. One such product also appears explicitly in (4.2). Note that the horizontal recursions are valid for the auxiliary product vectors example, (3.4) and (4.1) imply while (3.8) and (4.1) provide Thus, the auxiliary vector can be updated in the horizontal direction at the cost of one matrix-vector product (MV) per step. Combining the recursions for the vertical movement with those for the horizontal movement leads to the look-ahead version of an LTPM. (Actually, we will also need to compute approximations to the solution A \Gamma1 b of the linear system; but we defer this till section 6). Because of the additional vectors involved in the above recurrence formulas, it seems that such a look-ahead LTPM requires much more computational work and storage than its unstable, standard version. However, the number of look-ahead steps as well as the size of their blocks is small in practice, so that the overhead is moderate. Moreover, we describe in the following for various choices of polynomials how MVs that are needed can be computed indirectly by applying the recurrence formulas. In the same way also the values of inner products can be obtained indirectly at nearly no cost. 4.1. Look-ahead LTPMs based on a three-term recursion for f l g: LA- BiOxCheb and LA-BiOxMR2. For methods incorporating a horizontal three-term recurrence, such as BiOxCheb and BiOxMR2, applying the above principles for look-ahead LTPMs leads to the general Loop 4.1. Note that the first four substeps are identical in both cases. We will see in Section 5 that the decision between a regular and an inner loop is made during substep 5. In Figure 4.2 we display the action of this loop in the w-table, but now we use a different format than before, which, on the one hand specifies what has been known before the current sweep through the loop and what is being computed in this sweep. In particular, the following symbols and indicators are used: ffl 'V' indicates that the corresponding product vector is already known; ffl 'A' as `denominator' indicates that the product of the vector represented by 'V' with the matrix A was needed; ffl a solid box around a 'fraction' means that this product by A required (or requires) an MV; ffl no box around a 'fraction' means that this product can be obtained by applying a recurrence formula; ffl a number as entry specifies in which substep of the current sweep this entry is calculated; ffl a prime indicates that the vector is an auxiliary one, as defined in (4.1) (these vectors are displayed in the last row of the corresponding block of the indicates that the product of the vector represented by with the matrix A was needed; ffl double primes will indicate that the vector is an auxiliary one of the type defined in (4.11) used in LA-BiCGS (these vectors are displayed in the lower right corner of the corresponding block of the w-table); ffl 'S' will indicate that the vector or the MV is obtained for free due to the symmetry of the w-table of LA-BiCGS. Loop 4.1. (Look-ahead for (3; 3)-type LTPM.) Let := min fn Inner Loop: (n 1. Compute n . 2. If n ? use (4.2) to compute indirectly Aw n . 3. If n n . 4. If n n j +3, use (3.4) to compute indirectly Aw +1 n . 5. Use (4.2) to compute w n n+1 . 6. Compute n+1 and n ; jn . 7. Use (3.4) to compute w n+1 n+1 . 8. Compute use (4.7) to compute w 0 n+1 Regular Loop: (n 1. Compute n . 2. If n ? use (4.2) to compute indirectly Aw n . 3. If n n . 4. If n n j +3, use (3.4) to compute indirectly Aw +1 n . 5. Use (4.2) to compute w n n+1 and n+1 . 6. Compute n+1 and n ; jn . 7. Use (3.4) to compute w n+1 n+1 . 8. Compute according to definition (4.1). Inner Loop A A A A A A Regular Loop A A A A A Inner Loop A A A A A A A A A A A A Regular Loop A A A A A A A A A A A A A A A Fig. 4.2. Action of Loop 4.1 (Look-ahead for (3; 3)-type LTPM) in the w-table. For blocks of different sizes an inner step (at left) and the following regular step (at right) are shown. In the first two w-tables of Figure 4.2 we display what happens in a first inner step (at left) and in a regular step (at right) that follows directly a regular step (so that 1). In the second pair of w-tables of Figure 4.2 we consider an inner and a regular step in case of a large block size. Note that the substeps 2-4 of the loop do not appear in the first pair, and that substep 4 would still not be active in the regular loop at the end of a look-ahead step of length h active. For such a look-ahead step of length h j , the cost in terms of MVs is 4h MVs if h j ? 1. In fact, in the first substep of the h inner and the one regular loops h j MVs are consumed; another h are needed in substep 3, further in substep 6, and the remaining h are used in substep 6 of the inner loops. If no look-ahead is needed, that is if h are required, both in our procedure and in the standard one that does not allow for look-ahead. Hence, in a step of length 2, we have 25% overhead, in a step of length 3 there is 50% overhead, and for even longer steps, which are very rare in practice, the overhead grows gradually towards 100%. 4.2. Look-ahead for BiOStab and BiOStab2. Since the horizontal recurrence (3.8) for BiOStab is only a two-term one, there is no need to compute elements of the second subdiagonal of the w-table as long as we are not in a look-ahead step. In case of a look-ahead step, this remains true for those of these elements that lie in a subdiagonal block, but, of course, not for those in a diagonal block. For Loop 4.1 this means that simplifies to := n j and that in substep 5 of a regular loop there is no need to compute w . All the other changes refer to equation numbers or the coefficients n and jn . In summary, we obtain Loop 4.2. Again the first four substeps are the same in both cases, and the choice between them will be made in substep 5. In Figure 4.3 we display for this loop the two sections of w-tables that correspond to the second pair in Figure 4.2. Since those product vectors from Loop 4.1 that are no longer needed in Loop 4.2 were found without an extra MV before, the overhead in terms of MVs remains the same here. Look-ahead for BiOStab2 could be defined along the same lines, by alternating between steps of Loop 4.1 and Loop 4.2. At this point it also becomes clear how to obtain a look-ahead version of BiOStab('), an algorithm analogous to BiCGStab(') of Sleijpen and Fokkema [38], but based on the three-term Lanczos process instead of coupled two-term BiCG formulas. 4.3. Look-ahead (bi)conjugate gradient squared: LA-BiOS. For BiOS, which will be the underlying process for our BiOResS version of BiCGS, the horizontal recurrence (3.4) has to be substituted by the Lanczos recurrence given in (3.16), which may need to be replaced by the look-ahead formula that is analogous to (4.2) with l and n exchanged and with suitably defined auxiliary vectors and blocks of vectors. But since the w-table is symmetric, we can build it up by vertical recursions only and reflections at the diagonal. We only formulate the recursion for w 0 l as a horizontal one that replaces (4.7): where now \Theta Recurrence (4.9) follows from the horizontal Lanczos look-ahead recurrences for computing derived from (2.13) instead of (2.21), which can be gathered Loop 4.2. (Look-ahead for BiOStab.) Inner Loop: (n 1. Compute n . 2. If n ? use (4.2) to compute indirectly Aw n . 3. If n n . 4. If n n j +3, use (3.8) to compute indirectly Aw n j +1 n . 5. Use (4.2) to compute w n n+1 . 6. Compute n+1 and n . 7. Use (3.8) to compute w n+1 n+1 . 8. Compute use (4.8) to compute w 0 n+1 Regular Loop: (n 1. Compute n . 2. If n ? use (4.2) to compute indirectly Aw n . 3. If n n . 4. If n n j +3, use (3.8) to compute indirectly Aw n j +1 n . 5. Use (4.2) to compute w n n+1 . 6. Compute n+1 and n . 7. Use (3.8) to compute w n+1 n+1 . 8. Compute according to definition (4.1). Inner Loop A A A A A A A A A A A Regular Loop A A A A A A A A A A A A A A Fig. 4.3. Action of Loop 4.2 (Look-ahead for BiOStab) in the w-table. An inner step (at left) and the following regular step (at right) are shown. into one recurrence for these columns of W l+1 post-multiplied by as in (4.1). Again, (4.9) can be simplified: if we define for block and the needed values of j the auxiliary vector and for each l with n k l ! n k+1 the same coefficient fi 0 l := ffi nk l nk \Gamma1 as in (4.6), then in view of (2.17) and (4.6), Loop 4.3. (Look-ahead for BiOS.) Inner Loop: (n 1. Compute n . 2. If n n j +2, use (4.2) to compute indirectly Aw n n\Gamma2 . 3. Use (4.2) to compute w n n+1 . 4. Compute n+1 and Aw 0 n 5. Use (4.13) to compute w 0 n+1 6. Use (4.2) to compute w n+1 n+1 . Regular Loop: (n 1. Compute n . 2. If n n j +2, use (4.2) to compute indirectly Aw n n\Gamma2 . 3. Use (4.2) to compute w n n+1 . 4. Compute n+1 and Aw 0 n 5. Use (4.13) to compute w 0 n+1 6. Use definition (4.1) to compute j and use definition (4.11) to compute w 00 j . 7. Use (4.2) to compute w n+1 n+1 . 8. Use definition (4.1) to compute . Inner Loop A A A A A A A A VS Regular Loop A A A A A A A A A A A A A A A A Fig. 4.4. Action of Loop 4.3 (Look-ahead for BiOS) in the w-table. An inner loop (at the top) and the following regular loop (at the bottom) are shown. Consequently, (4.9) simplifies to l A step of the resulting LA-BiOS algorithm is summarized as Loop 4.3. Here, the decision between a regular and an inner loop is made in substep 3 (which corresponds to substep 5 in Loops 4.1 and 4.2). Now even the first five substeps of the two versions of the loop are identical. In Figure 4.4 we display for LA-BiOS the same pair of loops as we did in Figure 4.3 LA-BiOStab. For a look-ahead step of length h j , the cost in terms of MVs is now 3h j MVs if while it is only 3h As the standard, non-look- ahead algorithm requires 2 MVs per step, this means that the overhead is at most 50%. Here, in the first substep of the h inner and the one regular loop h j MVs are consumed, and another 2h j MVs are needed in substep 4. 4.4. Overhead for Look-Ahead. In this subsection we further discuss the overhead of the look-ahead process in terms of MVs, inner products (IPs) and the necessary storage of N-vectors. First we note that all IPs required in the look-ahead algorithms have the fixed initial vector e z 0 as their first argument. Therefore, if the second argument of an IP was computed by a recurrence formula, then also the IP of this vector with e z 0 can be computed indirectly by applying the same recurrence formula. Thus, only IPs of the n i, for which Aw l n is computed directly, need to be computed explicitly. This means that in all algorithms the number of required IPs is equal to the number of required MVs. We must admit, however, that such recursively computed inner products, as well as the recursively computed matrix-vector products, may be the source of additional roundoff, which may cause instability. Actually, for understanding how to compute all the inner products needed, the reader may want to introduce a ffi-table and a oe-table with entries ffi l and oe l Our loops and figures about generating the w- table then hold as well for these two tables. Note that the ffi-table just contains the transposed of the matrix D. Table Cost and overhead of look-ahead LTPMs when constructing a look-ahead step of length h j ? 1. The construction of iterates is not yet included. By further capitalizing upon storage locations that become available during a look-ahead step, the storage overhead could be reduced by roughly 50 %. method total cost cost overhead storage overhead (MVs, IPs) (MVs, IPs) relativ (N-vectors) In terms of MVs, the cost of a look-ahead step of length h j ? 1 has been specified in the previous subsection. By subtracting the cost for h j non-look-ahead steps, that is 2h j MVs, we obtain the overhead summarized in Table 4.1. We stress that when our algorithm has no overhead except for the necessary test of regularity, which, if it fails, would reveal an upcoming instability and initiate a look-ahead step. The table lists additionally the overhead in storage of N-vectors in a straightforward implementation that is not optimized with respect to memory usage. For comparison, we cite from page 60 of [5] or page 180 of [6] that the CGS look-ahead procedure of Brezinski and Redivo Zaglia requires 6h (the typical case) and 5h j +n (which means that a relatively large look-ahead step is needed in one of the first few iterations). Therefore, compared to our numbers, the overhead in terms of MVs is about four times (if h j is large, but five times (if h larger than in our LA-BiOS (assuming as the basis a standard CGS implementation requiring 2MVs per ordinary iteration). However, we also note that, according to the above numbers, for a step without look-ahead 1), the methods of [5] and [6] need 3MVs instead of 2MVs. 5. Look-Ahead Strategies. In this section we address the delicate issue of when to perform a look-ahead step, which means in an LTPM to decide whether the new vertical index n+1 is a regular index or an inner index, i.e., whether the required product vectors w l n+1 in the (n 1)th row of the w-table should be computed as regular or as inner vectors. Therefore, the look-ahead procedure in LTPMs serves to stabilize the underlying Lanczos method, the vertical movement of the product method in the w-table. Consequently, the criterion, when to carry out a look-ahead step in an LTPM can be based on the criterion given in [15] for the Lanczos algorithm. However, since the Lanczos vectors e yn and yn are not computed explicitly in a product method, we need to rewrite the conditions of this criterion in terms of the product vectors w l . Let us first motivate these conditions. In the case of an exact breakdown, where e z n 6= 0; yn 6= 0, a division by zero would occur in the next Lanczos step. The first task of the look-ahead process is to circumvent these exact breakdowns without the necessity of restarting the Lanczos process and loosing its superlinear convergence. In finite precision arithmetic, exact breakdowns are very unlikely. However, near breakdowns, where jffi very small, may occur and cause large relative roundoff errors in the Lanczos coefficients ff n and fi n given by (3.2). To be more precise, we recall that the relative roundoff error in the computation of the inner product ffi is bounded by [18, p. 64] " denotes the roundoff unit. Thus, a small value of jffi leads in finite precision arithmetic to a big relative roundoff error in the computation of the inner product n , which also causes a perturbation of the Lanczos coefficients ff n and fi n , since they depend on ffi respectively. The second task of the look-ahead process is therefore, to avoid a convergence deterioration due to perturbed Lanczos coefficients. Of course, similar roundoff effects may come up in the numerators of the formulas (3.2) for ff n and fi n , but large relative errors in those will only be harmful if the denominators are small too. We would like to point out that in an LTPM the inner products ffi n can be enlarged to a certain extent by an appropriate adaptive choice of the polynomials n [39, 40], as long as h example, considering the BiOStab case, where we obtain, since hez Thus, minimizing the relative roundoff error in the calculation of ffi n is equivalent to choosing n\Gamma1 such that it minimizes to OR n i, which corresponds to the use of orthogonal residual polynomials of degree 1 instead of minimal residual polynomials of degree 1 in the recursive definition of l (i). Therefore, minimizing the relative roundoff error in the inner product ffi conflicts often with the objective of avoiding large intermediate residuals in order to prevent the recursive residual to drift apart from the true residual [40]. Performing a look-ahead step is then the only possible remedy. We need now to find a criterion for deciding when a look-ahead step should be performed so that both the above objectives can be attained. In view of the recursion (4.2) for the product vectors in the case of look-ahead, it follows that a block can be closed and a new product vector can be computed as regular vector only if the diagonal blocks D j \Gamma1 and D j of the Gramian are numerically nonsingular. Thus, the first condition that needs to be fulfilled in order to compute a new product vector n+1 as regular vector (n oe min (D j ) "; where oe min (D j ) denotes the minimal singular value of the block D j . Note that this does not mean that D j is well conditioned; it just guarantees numerical nonsingu- larity. In practice, (5.1) can be replaced by some other condition that implies this nonsingularity, for example by one implemented in a linear solver used for computing in (2.19) and ff n in (2.22). Freund et al. [15] use a second condition to guarantee that the Krylov space is stably extended in the next Lanczos step in the sense that the basis of Lanczos vectors is sufficiently well conditioned. In terms of product vectors, this second condition amounts to computing, for any l, the new product vector w l n+1 as regular vector if in addition to (5.1) the following conditions for the coefficients ff n and fi n are fulfilled: Here k\Deltak 1 denotes the ' 1 -norm and n(A) is an estimate for kAk which is updated dynamically to ensure that the blocks W n j do not become larger than a user specified maximal size [15]. The motivation for (5.2) was to ensure that in the new regular vector (obtained from (4.2) with c since n+ is regular) the component in the new direction Aw n n is sufficiently large, which will be the case if n and tol 2 is a chosen tolerance. Compared to (5.2), condition (5.3) costs additional two inner products and the calculation of w t , which only in the regular case can be reused for the computation of the new product vector w n n+1 . Since we have we could replace (5.3) by the less expensive condition However, in an LTPM it is not possible to normalize all product vectors w l (see Section 6); so C is not equal to 1. Moreover, (5.5) is less strict than (5.3) and (5.2). Since a look-ahead step is more expensive than regular steps providing the same increase of the Krylov space dimension, a tight look-ahead criterion can save overall computational cost. Therefore, it is reasonable to spend extra effort for it. For this reason we favor criterion (5.3). A drawback of (5.3) is that it does not take the angle between Aw n n and w t into account. If C c should be chosen larger than in the case where it is nearly 1. This motivates the choice with suitably chosen constants C depending on the roundoff unit ". This criterion requires an extra inner product and an appropriate choice for . For many small problems C worked well, but for larger problems we observed that C c decays dramatically with the block size. Therefore, the probability that (5.3) with tol 2 as defined in (5.6) will be fulfilled decreases with the block length and leads very often (especially in BiOS) to situations where the maximal user specified block size was reached. Further investigations are needed to see if this problem can be solved by a better choice of C 1 and C 2 or by a more appropriate selection for ff n in the inner case, instead of using, as in [15], 2. 6. Obtaining the solution of b. So far we have only introduced various algorithms for constructing a sequence of product vectors w l n that provide a basis for Km . However, our goal is to solve the linear system b. We now describe how to accomplish this with these algorithms. There are several basic approaches to constructing approximate solutions of linear systems from a Krylov space basis. Ours is related to the Galerkin method, but avoids the difficulty that arises when the Galerkin solution does not exist. (This difficulty causes, for example, the so-called pivot break-down of the biconjugate gradient method.) Our approach is a natural generalization of the one that lead to "unnormalized BiORes" introduced in [20] and renamed "in- consistent BiORes" in [25]. In contrast to the BiOMin, BiODir, and BiORes versions of the BiCG method, the inconsistent BiORes variant is not endangered by pivot breakdowns. An alternative would be to construct approximate solutions based on the quasi-minimal residual (QMR) approach [16]. For a combination of this approach with LTPMs we refer the reader to [36]. Let the doubly indexed sequence of scalars ae l n be given by ae l We define a doubly indexed sequence of product iterates x l n as follows. Starting with an arbitrary x 0 N , we choose the initial product vector w 0 0 so that ae 0 Here, ae 0 1. For product iterates are now implicitly defined by ae l ae l n ae l n if ae l Of course, x l n will be constructed only when w l is. If ae l it follows from (6.3) that x l ae l n can be considered as an approximate solution of corresponding residual is w l ae l n . In order to derive recursions for the scalars ae l n and the product iterates x l n , we introduce the blocks \Theta x l \Theta x l \Theta ae l ae l \Theta ae l ae l as well as the auxiliary product iterates x 0l auxiliary scalars ae 0l defined by x 0l ae 0l Again, b Then, by (6.3), (4.1), and (6.4), ae 0l Next, using (4.2) we conclude that ae l Aw l l Aw l ae 0l ae 0l This shows that x l ae l ae 0l If we arrange the product iterates x l n and the scalars ae l n in two tables analogous to the w-table (with the n-axis pointing downwards and the l-axis to the right), these two recursions can be used to proceed in vertical direction. To obtain recursions for a horizontal movement, we assume first that the polynomials are given by the normalized three-term recurrence (3.3). This covers all algorithms described in this paper except look-ahead BiOS. Using (3.4) and (6.3) we see that the product iterates satisfy For the scalars ae l n the recursion ae l+1 ae l ae are valid, but since the polynomials l are normalized (that is, l all l), the scalars ae l n do not change with the index l, and we have simply ae l ae 0 In look-ahead BiOS only one horizontal movement is explicitly computed per step, namely in substep 5 of Loop 4.3 based on the recurrence (4.13). If we define in analogy to (4.10) and (4.11) \Theta x 0nk \Theta x 0n \Theta ae 0n k ae 0l \Theta ae 0n ae and x ae recurrences for the auxiliary iterates and the corresponding scalars are given by l ae 0l+1 ae l Using for the scaling parameter fl n in (4.2) the special choice n\Gamman with 1m := [1 also the Lanczos polynomials ae n could be normalized (ae n 1), so that ae l However, as we mentioned before, some fl n might turn out to be zero, which would lead to a so-called pivot breakdown. Moreover, to avoid overflow or underflow, in the Lanczos process the scaling parameter fl n is often used to normalize the Lanczos vectors yn . But since the Lanczos vectors yn are not explicitly computed in an LTPM, we cannot base the choice of fl n here on their norm. However, independent of the size of the blocks generated by the look-ahead process, it is always necessary to compute the product vectors w n n+1 in an LTPM. Therefore, we chose here fl n to normalize w n n+1 , that is, 7. Numerical Examples. In this section we demonstrate the practical performance of our look-ahead versions of LTPMs in numerical examples. The tests are restricted to BiOStab, BiOxMR2 and BiOS. The look-ahead versions of these LTPMs are denoted by LABiOStab, LABiOxMR2, and LABiOS, respectively. For all tests the initial iterate x used, and the iteration is terminated when the norm of the recursive residual is less than ", the square root of the roundoff unit. The test programs were written in FORTRAN90/95 and run on workstations with 64-bit IEEE arithmetic. We start with small, artificially constructed model problems and move gradually to large real-world problems. Example 1. The following small test example was proposed by Joubert [29] and also used by Brezinski and Redivo-Zaglia [6]: . The Lanczos process, and hence, BiOStab, BiOxMR2, and BiOS without look-ahead break down at step 2. On the contrary, all look-ahead versions avoid this breakdown and converge after 4 iterations as shown in our plots of the true residual norms kb \Gamma Ax l ae l n k in Figure 7.1. iteration number true residual LABiOSstab1e-101e+10 iteration number true residual LABiOxMR21e-101e+10 iteration number true residual LABiOS Fig. 7.1. The true residual norm history (i.e. log(kb \Gamma Ax l ae l n vs. n) for the linear system defined in (7.1) solved by different LTPMs with look-ahead. Example 2. Our second example is taken from [6], Example 5.2: the matrix of order 400 and the right-hand side imply that the solution is the Lanczos process, and thus the LTPMs break down in the first iteration. Our look-ahead versions perform two inner steps at the first and second iteration. All following iterations are regular steps. This is a typical behavior: there are only few and short look-ahead steps, and therefore the resulting mean overhead per iteration is nearly negligible.1e-101e+10 iteration number true residual Fig. 7.2. The true residual norm history (i.e. log(kb \Gamma Ax l ae l n vs. n) for the linear system defined in (7.2) solved by different LTPMs with look-ahead. Table Indices of regular steps in LTPMs for three problems with a p-cyclic system matrix. Example LABiOStab LABiOxMR2 LABiOS In the next set of examples we consider p-cyclic matrices of the form Hochbruck [27] showed that the computational work for solving a linear system with a p-cyclic system matrix by QMR with look-ahead can be reduced by approximately a factor 1=p (compared to a straight-forward implementation using sparse matrix-vector multiplications with A), if the initial Lanczos vectors have only one nonzero block conforming to the block structure of A, if the inner vectors are chosen so that the nonzero structure of is not destroyed, and if the blocks B k are used for generating only possibly nonzero components of the Krylov space basis. Then it can be proven that in each cycle of p steps there are at least consecutive exact breakdowns for p ? 2. But when using directly the system matrix A to generate the Krylov subspace, we have only in the first cycle of p steps consecutive exact breakdowns, while in the following cycles these will, in general, no longer persist, but must be expected to become near-breakdowns. Therefore, such problems provide good test examples for look-ahead algorithms. Example 3. In this example we consider a 5-cyclic matrix with right-hand side and as initial left Lanczos vector we choose entries. The convergence history for the different LTPMs applied to this problem are shown in Figure 7.3, and the indices of the regular steps are listed in Table 7.1. Those are found exactly where predicted. Example 4. We move now to a bigger 4-cyclic system matrix with is a 100 \Theta 100 matrix with random entries. The results for this problem are shown in Figure 7.4, and the indices of the regular steps are depicted also in Table 7.1. Again, they occur where predicted. Example 5. Finally, we consider an 8-cyclic system matrix with B defined as in Example 4. The convergence history plotted in Figure 7.5 shows oscillations in the residual norm history of LABiOS, but overall LABiOS needs one iteration step less than LABiOStab and LABiOxMR2 to fulfill the convergence condition. For all methods the same look-ahead criterion ((5.3) with tol 2 defined as in (5.6) and used. Especially for LABiOS the correct choice of the look-ahead criterion seems to be crucial. While with the above values of C 1 and C 2 the breakdowns occurred only where expected, we discovered for this larger problem Fig. 7.3. The true residual norm history (i.e. log(kb \Gamma Ax l ae l n vs. n) for the linear system with the 5-cyclic system matrix defined in Example 3 solved by different LTPMs with look-ahead. Fig. 7.4. The true residual norm history (i.e. log(kb \Gamma Ax l ae l n vs. n) for the linear system with the 4-cyclic system matrix defined in Example 4 solved by different LTPMs with look-ahead. that in BiOS the maximal user specified block length of 10 was reached very often for which indicates, that the constructed inner vectors become more and more linear dependent. Therefore, further investigations are needed to figure out a better choice for the coefficient vector ff n in the inner case. For example, following the proposal in [21], Hochbruck used in [27] a Chebyshev iteration for the generation of the inner vectors instead of ff n which we adapted from [15]. Example 6. In our last example we take a real world problem from the Harwell- Fig. 7.5. The true residual norm history (i.e. log(kb \Gamma Ax l ae l n vs. n) for the linear system with the 8-cyclic system matrix defined in Example 5 solved by different LTPMs with look-ahead. Boeing Sparse Matrix Collection, namely SHERMAN1, a matrix of order 1000 with 3750 nonzero entries. The right-hand side b and the initial left Lanczos vector e z 0 were generated as different unit vectors with random entries. Without look-ahead, all LTPMs introduced here break down (BiOStab at step 186, BiOStab2 at step 176, BiOxMR2 at step 145 and BiOS at step 352). On the contrary, the look-ahead versions in combination with the look-ahead criterion (5.3) with tol 2 defined as in converge as shown in Figure 7.6. Due to the real spectrum of the system matrix A, there is only a slight difference in the convergence of LABiOStab and LABiOxMR2. It was reported to us that BiCGStab(') with and no look-ahead can handle this problem in about the same number of MVs as our BiCGStab with look-ahead. 8. Conclusions. We have proposed look-ahead versions for various Lanczos- type product methods that make use of the Lanczos three-term recurrences. Since they are based on the Lanczos look-ahead version of Gutknecht [23] and Freund et al. [15], they can handle look-ahead steps of any length and avoid steps that are longer than needed. The algorithms proposed in this work should be easy to understand due to the introduction of an array of product vectors, symbolically displayed in the w- table, and the visualization of the progress in this w-table. Furthermore, the w-table proved to be a useful tool to derive optimal variants, for which the computational work in terms of MVs is minimized. A variety of numerical examples demonstrate the practical performance of the proposed algorithms. However, larger problems indicate that further work should be directed to finding an improved look-ahead criterion that more reliably avoids critical perturbations of the Lanczos coefficients by roundoff errors. Moreover, one should investigate if there is a better way of constructing inner vectors than the choice adapted from [15]. Alternatives would be to orthogonalize them within each block [2] or to construct them by Chebyshev iteration [21]; but it is not clear if the additional cost involved pays off. Fig. 7.6. The true residual norm history (i.e. log(kb \Gamma Ax l ae l n vs. n) for a real-world problem with the SHERMAN1 matrix from the Harwell-Boeing collection solved by different LTPMs with look-ahead. The look-ahead process in an LTPM stabilizes primarily the vertical movement in the w-table, except in the BiOS algorithm where the w-table is symmetric. For the horizontal movement it is also important to generate the Krylov space stably, and both BiOStab2 and BiOxMR2 (in particular when suitably modified) do that more reliably than BiCGStab, since the two-dimensional steps offer more flexibility. This has also a lasting positive effect on the roundoff in the vertical movement. To stabilize the horizontal movement further, a local minimal residual polynomial of degree ' 1 with an adaptive choice of ', as in BiCGStab(') could be used. A further possibility is to adapt ' to the size h j of the current Lanczos block, which would mean to perform in each regular step an h j -dimensional local minimization of the residual. An alternative is to trade in the local residual minimization for a more stable Krylov space generation whenever the former causes a problem. Yet another possibility, indicated in Section 4.2, is to combine the Lanczos process with a hybrid Chebyshev iteration. It is known that in finite-precision arithmetic BiORes is usually more affected by roundoff than the standard BiOMin version of BiCG, at least with regard to the gap between recursively and explicitly computed residuals. Therefore, we are in the process to extend this work to look-ahead procedures for LTPMs that are based on coupled two-term recurrences. --R Avoiding the look-ahead in the Lanczos method Nonsymmetric Lanczos and finding orthogonal polynomials associated with indefinite weights CGM: a whole class of Lanczos-type solvers for linear systems Breakdowns in the computation of orthogonal poly- nomials New look-ahead Lanczos-type algorithms for linear systems Avoiding breakdown in the CGS algorithm Avoiding breakdown in variants of the BI-CGSTAB algorithm A quasi-minimal residual variant of the Bi-CGSTAB algorithm for nonsymmetric systems Working Note 78: Computational variants of the CGS and BiCGstab methods Numerical determination of fundamental modes A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices QMR: a quasi-minimal residual method for non-Hermitian linear systems Matrix Computations "look-around Lanczos" The unsymmetric Lanczos algorithms and their relations to Pad'e approx- imation Generalized conjugate gradient and Lanczos methods for the solution of non-symmetric systems of linear equations An iteration method for the solution of the eigenvalue problem of linear differential and integral operators The Tchebyshev iteration for nonsymmetric linear systems Reduction to tridiagonal form and minimal realizations Scientific Computing on Vector Computers BiCGstab(l) for linear equations involving unsymmetric matrices with complex spectrum Maintaining convergence properties of BiCGstab methods in finite precision arithmetic BiCGstab(l) and other hybrid Bi-CG methods Analysis of the Look Ahead Lanczos Algorithm Accelerating the Jacobi method for solving simultaneous equations by Chebyshev extrapolation when the eigenvalues of the iteration matrix are complex Residual smoothing techniques for iterative methods Generalized biorthogonal bases and tridiagonalisation of matrices --TR

lanczos-type product methods;look-ahead;sparse linear systems;non-Hermitian matrices;iterative methods

354387

A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra.

The matrix 1-norm estimation algorithm used in LAPACK and various other software libraries and packages has proved to be a valuable tool. However, it has the limitations that it offers the user no control over the accuracy and reliability of the estimate and that it is based on level 2 BLAS operations. A block generalization of the 1-norm power method underlying the estimator is derived here and developed into a practical algorithm applicable to both real and complex matrices. The algorithm works with n t matrices, where t is a parameter. For t=1 the original algorithm is recovered, but with two improvements (one for real matrices and one for complex matrices). The accuracy and reliability of the estimates generally increase with t and the computational kernels are level 3 BLAS operations for t > 1. The last t-1 columns of the starting matrix are randomly chosen, giving the algorithm a statistical flavor. As a by-product of our investigations we identify a matrix for which the 1-norm power method takes the maximum number of iterations. As an application of the new estimator we show how it can be used to efficiently approximate 1-norm pseudospectra.

Introduction . Research in matrix condition number estimation began in the 1970s with the problem of cheaply estimating the condition number and an approximate null vector of a square matrix A given some factorization of it. The earliest algorithm is one of Gragg and Stewart [10]. It was improved by Cline, Moler, Stewart and Wilkinson [4], leading to the 1-norm condition estimation algorithm used in LINPACK [8] and later included in Matlab (function rcond). During the 1980s, attention was drawn to various componentwise condition numbers and it was recognized that most condition estimation problems can be reduced to the estimation of kAk when matrix-vector products Ax and A T x can be cheaply computed [2], [16, Sec. 14.1]. Hager [12] derived an algorithm for the 1-norm that is a special case of the more general p-norm power method proposed by Boyd [3] and later investigated by Tao [18]. Hager's algorithm was modified by Higham [14] and incorporated in LAPACK (routine xLACON) [1] and Matlab (function condest). The LINPACK and LAPACK estimators both produce estimates that in practice are almost always within a factor 10 and 3, respectively, of the quantities they are estimating [13], [14], [15]. This has been entirely adequate for applications where only an order of magnitude estimate is required, such as the evaluation of error bounds. However, in some applications an estimate with one or more correct digits is required (see, for example, the pseudospectra application described in section 4), and for these the LINPACK and LAPACK estimators have the drawback that they offer the user no way to control or improve the accuracy of the estimate. Here, "accuracy" refers to average case behaviour. Also of interest for a norm estimator is its worst case behaviour, that is, its "reliability". This work was supported by Engineering and Physical Sciences Research Council grants GR/L76532 and GR/L94314. y Department of Mathematics, University of Manchester, Manchester, M13 9PL, England (higham@ma.man.ac.uk, http://www.ma.man.ac.uk/~higham/). z Department of Mathematics, University of Manchester, Manchester, M13 9PL, England (ftisseur@ma.man.ac.uk, http://www.ma.man.ac.uk/~ftisseur/). N. J. HIGHAM AND F. TISSEUR Table Empirical probabilities that minf e OE s one A of the form inv(randn(100)) and N(0; 1) vectors x j . Since estimating kAk cheaply appears inevitably to admit the possibility of arbitrarily poor estimates (although proving so is an open problem [6]), one might look for an approach for which probabilistic statements can be made about the accuracy of the estimate. The definition of a subordinate matrix norm suggests the estimate ae oe where s is a parameter and the x j are independently chosen random vectors. In the case of the 2-norm (kxk and an appropriate distribution of the x j , explicit bounds are available on the probability of such estimates being within a given factor of kAk [7]. As is done in [11] for certain estimates of the Frobenius norm we can scale our estimates e OE s / ' s OE s , where the constant ' s is chosen so that the expected value of e OE s is kAk (note that e OE s can therefore be greater or less than kAk). For the 1-norm (kxk which is our interest here, we investigate this approach empirically. For a fixed matrix A of the form, in Matlab notation, inv(randn(100)), Table 1.1 shows the observed probabilities that minf e various ff and s, based on 1000 separate evaluations of the OE s with vectors x j from the normal N(0; 1) distribution, and where ' s is determined empirically so that the mean of the e OE s is The table shows that even with only 35% of the estimates were within a factor 0.9 of the true norm. The statistical sampling technique is clearly too crude to be useful for obtaining estimates with correct digits. One way to exploit the information contained in the vectors Ax j is to regard them as first iterates from the 1-norm power method with starting vectors x j and to continue to iterate. These considerations motivate the block generalization of the 1-norm power method that we present in this paper. Our block power method works with a matrix with t columns instead of a vector. To give a feel for how our new estimator compares with the sampling technique, we applied the estimator (Algorithm 2.4) to 1000 random matrices of the form inv(randn(100)). The results are shown in Table 1.2; the estimates for are obtained at approximately the same cost as the estimates e OE s for respectively. The superiority of the new estimator is clear. In section 2 we derive the block 1-norm power method and develop it into a practical algorithm for both real and complex matrices. In section 3 we present numerical experiments that give insight into the behaviour of the algorithm. An application involving complex matrices is given in section 4, where we describe how the algorithm can be used to approximate 1-norm pseudospectra. Conclusions are presented in section 5. Table Empirical probabilities that est - for est from Algorithm 2.4 and A of the form inv(randn(100)). Finally, we note that although our work is specific to the 1-norm, the 1-norm can be estimated by applying our algorithm to A , since 2. Block 1-Norm Power Method. The 1-norm power method is a special case of Boyd's p-norm power method [3] and was derived independently by Hager [12]. For a real matrix A we denote by sign(A) the matrix with (i; according as a ij - 0 or a ij ! 0. The jth column of the identity matrix is denoted by e j . Algorithm 2.1 (1-norm power method). Given A 2 R n\Thetan this algorithm computes and x such that repeat quit (smallest such Algorithm 2.1 was modified by Higham [14, Alg. 4.1] (see also [15], [16, Alg. 14.4]) to improve its reliability and efficiency. The modifications that improve the reliability are, first, to force at least two iterations and, second, to take as the final estimate the maximum of that produced by the algorithm and The vector b is a heuristic choice intended to "pick out" any large elements of A in those cases where such elements fail to be revealed during the course of the algorithm. Efficiency is improved by terminating the algorithm after computing - if it is the same as the previous -, since it can be shown that convergence would otherwise be declared after the subsequent computation of z. To obtain a more accurate and reliable estimate than that provided by Algorithm 2.1 we could run the algorithm t times in succession on t different starting vectors. This idea was suggested in [12], with each starting vector being the mean of the unit vectors e j not already visited and with the algorithm being prohibited from visiting unit vectors previously visited. Note that the estimates obtained this way are nondecreasing in t. This approach has two weaknesses: it allows limited communication of information between the t different iterations and the highest level computational kernel remains matrix-vector multiplication. We therefore develop a block algorithm that works with an n \Theta t matrix as a whole instead of t separate 4 N. J. HIGHAM AND F. TISSEUR n-vectors. The block approach offers the potential of better estimates, through providing more information on which to base decisions, and it allows the use of level 3 BLAS operations, thus promising greater efficiency. The following algorithm estimates not just the 1-norm of A, but, as a by-product, the 1-norms of the t columns of A having largest 1-norms. Algorithm 2.2 (block 1-norm power method). Given A 2 R n\Thetan and a positive integer t, this algorithm computes vectors g and ind with t such that g j is a lower bound for the 1-norm of the column of A of jth largest 1-norm. Choose starting matrix X 2 R n\Thetat with columns of unit 1-norm. repeat Sort g so that ind best re-order ind correspondingly. Like the basic 1-norm power method Algorithm 2.2 has the attractive property that it generates increasing sequences of estimates. Denote with a superscript "(k)" quantities from the kth iteration of the loop in Algorithm 2.2 and let a j denote the jth column of A. Lemma 2.3. The sorted vectors g (k) and h (k) satisfy and 2: Proof. First, we have 1-j-t r But r y r r Thus r r k1 kx (k) r k1 - max Furthermore, if h (k) r s (k) To prove (2.3), assume without loss of generality that x (k\Gamma1) t. Then has the form Z ff t+1 ff t+1 where each ff i in row i is, in general, different, and ff i - ka n. The algorithm chooses the ind values corresponding to the t rows of Z (k) with largest 1-norm, and the g (k+1) on the next stage are at least as large as these 1-norms. Since row j of Z (k) has 1-norm ka follows. Algorithm 2.2 has three possible sources of inefficiency. First, the columns of S are vectors of \Sigma1s and so a pair of columns s i and s j may be parallel case z and in computing Z a matrix-vector multiplication is redundant. There can be up to t\Gamma1 redundant matrix-vector products per iteration, this maximum being achieved on the second iteration with S the matrix of ones when A has nonnegative elements. The second possible inefficiency is that a column of S may be parallel to one from the previous iteration, in which case the formation of redundant computation. We choose to detect parallel columns and replace them by random vectors randf\Gamma1; 1g not already in S or the previous denotes a vector with elements from the uniform distribution on the set f\Gamma1; 1g. The detection is done by forming inner products between columns and looking for elements of magnitude n. The total cost of these computations is O(nt 2 ) flops, which is negligible compared with the 2n 2 t flops required for each matrix product in the algorithm, since t - n in practice. Our strategy could be extended to check for parallel columns between the current S and all previous S; we return to this possibility in section 3. If all the columns of S are parallel to columns from the previous iteration then it is easy to see that Algorithm 2.2 is about to converge; we therefore immediately terminate the iteration without computing Z. Finally, on step k we can have x (k) so that the computation of Ax (k) then repeats an earlier computation. Repeated vectors e j are easily avoided by keeping track of the indices of all the previously used e j and selecting ind ind t from among the indices not previously used. If all the indices are repeats, then again we prematurely terminate the iteration, saving a matrix product. Note that the first and third inefficiencies are possibly only for t ? 1, while the second can happen for the original 1-norm power method. Our strategy of detecting redundant computations and replacing them by ones that provide new information has three benefits. First, it can reduce the amount of computation, through premature detection of convergence. Second, it can lead to better estimates. For the columns of S this depends on the random replacement vectors generated, but for the e j the improvement is deterministic up to future replacements 6 N. J. HIGHAM AND F. TISSEUR of columns of S. The third benefit is that the dimensions of the matrix multiplications remain constant at each iteration, as opposed to varying if we simply skip redundant computations; this helps us to make efficient use of the computing resources. The next algorithm incorporates these modifications. The algorithm is forced to take at least 2 (and at most itmax) iterations so that it computes at least t columns of A; it also explicitly identifies the approximate maximizing vector that achieves the norm estimate. Algorithm 2.4 (practical block 1-norm estimator). Given A 2 R n\Thetan and positive integers t and itmax - 2, this algorithm computes a scalar est and vectors v and w such that est - Choose starting matrix X 2 R n\Thetat with columns of unit 1-norm. recording indices of used unit vectors e j . est est est ? est old or ind best est - est old , est = est old , goto (6), end est (2) If every column of S is parallel to a column of S old , goto (6), end Ensure that no column of S is parallel to another column of S or to a column of S old by replacing columns of S by randf\Gamma1; 1g. best , goto (6), end re-order ind correspondingly. (5) If ind(1: t) is contained in ind \Gamma hist, goto (6), end Replace ind(1: t) by the first t indices in ind(1: n) that are not in ind \Gamma hist. hist ind(1: t)] best Note that this algorithm does not explicitly compute lower bounds for the 1-norms of all t largest columns of A. If this information is required (and we are not aware of any applications in which it is needed) then it can be obtained by keeping track of the largest Inequality (2.2), now expressed as est (k) - h (k) est (k+1) , is still valid, except that h (k) est (k+1) is possible on the last iteration if the original ind (k) 1 is a repeat (this event is handled by the test (1)). However, (2.3) is no longer true, because of the avoidance of repeated indices. How do Algorithms 2.4 and 2.2 compare? Ignoring the itmax test, Algorithm 2.4 terminates in at most n=t+1 iterations, since t vertices e j are visited on each iteration after the first and no vertex can be visited more than once. The same cannot be said of Algorithm 2.2 because of the possibility of repeated vertices. Algorithm 2.4 can produce a smaller estimate than Algorithm 2.2: when a redundant computation is avoided, the new information computed can lead to an apparently more promising vertex (based on the relative sizes of the h i ) replacing one that actually corresponds to a larger column of A. However it is more likely, when Algorithms 2.4 and Algorithm 2.2 produce different results, that Algorithm 2.4 produces a better estimate, as in the following example. Example 2.5. For a certain starting obtained the following results. For Algorithm 2.2: 1: (0, 8.10e-001) (0, 6.13e-001) 2: (10, 1.77e+000) (4, 1.76e+000) Underestimation ratio: 1.99e-001 The first column denotes the iteration number. The kth row gives the sorted g (k) t, each one preceded by the corresponding index ind i . (Since X 1 does not have columns e j , the ind i for the first iteration are shown as zero.) For Algorithm 2.4: 1: (0, 8.10e-001) (0, 6.13e-001) 2: (10, 1.77e+000) (4, 1.76e+000) parallel column between S and S-old 3: (2, 8.87e+000) (5, 4.59e+000) Exact estimate! Algorithm 2.2 converges after 2 iterations and produces an estimate too small by a factor 5. However, on the second iteration Algorithm 2.4 detects a column of S parallel to one of S old and replaces it. The new column produces a different Z matrix and causes the convergence test (4) to be failed. The extra iteration visits the (unique) column of maximum 1-norm, so an exact estimate is obtained. When 2.4 differs from the modified version of Algorithm 2.1 used in LAPACK 3.0 [14, Alg. 4.1], [16, Alg. 14.4] in two ways. First, Algorithm 2.4 does not use the "extra estimate" (2.1). Second, the LAPACK algorithm checks for but not for - does Algorithm 2.4. This is an oversight. We recommend that LAPACK's xLACON be modified to include the extra test. This change will not affect the estimates produced but will sometimes reduce the number of iterations. We now explain our choice of starting matrix. We take the first column of X to be the vector of 1s, which is the starting vector used in Algorithm 2.1. This has the advantage that for a matrix with nonnegative elements the algorithm converges with an exact estimate on the second iteration, and such matrices arise in applications, for example as a stochastic matrix or as the inverse of an M-matrix. The remaining columns are chosen as randf\Gamma1; 1g, with a check for and correction of parallel columns, exactly as for S in the body of the algorithm. We choose random vectors because it is difficult to argue for any particular fixed vectors and because randomness lessens the importance of counterexamples (see the comments in the next section). Next, we consider complex matrices, which arise in the pseudospectrum application of section 4. Everything in this section remains valid for complex matrices provided that sign(A) is redefined as the matrix (a transposes are replaced by conjugate transposes. The matrix S is now complex with elements of unit modulus and we are much less likely to find parallel columns of S 8 N. J. HIGHAM AND F. TISSEUR from one iteration to the next or within the current S. Therefore for complex matrices we omit the tests for parallel columns. However, we take the same, real, starting matrix. There is one further question in the complex case. In the analogue of Algorithm 2.1 for complex matrices in [14, Alg. 5.1] z is defined as z = Re(A -), based on subgradient considerations. In our block algorithm should we take The former can be justified from heuristic considerations and preserves more information about A. We return to this question in the next section. The motivation for Algorithm 2.4 is to enable more accurate and reliable estimates to be obtained than are provided by the 1-norm power method. The question arises of how the accuracy and reliability of the estimates varies with t. Little can be said theoretically because, unlike the approach of [12] mentioned at the start of this section, the estimates are not monotonic in t. If we run Algorithm 2.4 for t 1 and for using a common set of t 1 starting vectors, we can obtain a smaller estimate for t 2 than for t 1 , because a less promising choice of unit vector e j can turn out to be better than a more promising choice made with more available information. Non-monotonicity is unlikely, however, and we argue that it is a price worth paying for the other advantages that accrue. In the next section we investigate the behaviour of Algorithm 2.4 empirically. 3. Numerical Experiments. Our aim in this section is to answer the following questions about Algorithm 2.4, bearing in mind that for the algorithm is an implementation of the well understood 1-norm power method. 1. How does the accuracy and reliability of the norm estimates vary with t? 2. How good are the norm estimates in general? 3. How does the number of iterations behave for t ? 1? Note that we are not searching for counterexamples, as was done for previous condition estimators [4], [5], [14]. We know that for a fixed starting matrix and any t - n there must be families of matrices whose norm is underestimated by an arbitrarily large factor, since the algorithm samples the behaviour of the n \Theta n matrix A on fewer than vectors. But since the algorithm uses a random starting matrix for t ? 1, each counterexample will be valid only for particular starting matrices. All our tests have been performed with Matlab. Our first group of tests deals with random real matrices. Amongst the matrices A we used were: 1. A from the normal N(0; 1) distribution (denoted randn) and its inverse, orthogonal QR factor, upper triangular part, and inverse of the upper triangular part. 2. A from the uniform distribution on the set f\Gamma1; 0; 1g (denoted rand(-1,0,1-)), A \Gamma1 from the uniform distribution on the interval [0; 1]. 3. A and A \Gamma1 of the form U \Sigma V T where U and V are random orthogonal matrices and with the singular values oe i distributed exponentially, arithmetically or with all except the smallest equal to unity, and with 2-norm condition number ranging from 1 to 10 16 . Note that we omit, for example, matrices from the uniform [0; 1] and uniform f\Gamma1; 1g distributions, because for such matrices Algorithm 2.4 is easily seen to produce the exact norm for all t. We chose n and t in the range For each test matrix we recorded a variety of statistics including the underestimation ratio averaged and minimized over each type of A for fixed n and t, the relative error jest and the number of iterations. We declared an estimate exact if the relative error was no larger than 10 \Gamma14 (the unit roundoff is of For a given matrix A we first generated a starting columns, where t max is the largest value of t to be used, and then ran Algorithm 2.4 using starting t). In this way we could see the effect of increasing t. In particular, we checked what percentage of the estimates for a given t were at least as large as the estimates for all smaller t; we denote this "improve%". We set Algorithm 2.4. First, we give some general comments on the results. 1. Increasing t usually gave larger average and minimum underestimation ratios, though there were exceptions. The quantity improve% was 100 about half the time and never less than 78. 2. The number of iterations averaged between 2 and 3 throughout, with maxima ranging from 2 to 5 depending on the type of matrix. Thus increasing t from 1 has little effect on the number of iterations-an important fact that could not be predicted from the theory. Although there are specially constructed examples for which the 1-norm power method requires many iterations (one is described below), it is rare for the limit of 5 iterations to come into effect. 3. Throughout the tests we also computed the extra estimate (2.1) used by the LAPACK norm estimator [14]. As expected, in none of our tests (with random matrices) was the extra estimate larger than the estimate provided by Algorithm 2.4 with In Tables 3.1 and 3.2 we show detailed results for two particular types of random matrix from among those described above, with rand(-1,0,1-). The columns headed "Products" show the average and maximum total number of matrix products AX and A T S. In each case 5000 matrices were used. For the matrices inv(randn) taking significantly improves the worst-case and average estimates and the proportion of exact estimates over estimate is exact almost 98 percent of the time. For the matrices rand(-1,0,1-) the improvements as t increases are less dramatic but still useful; notice that exactly four matrix products were required in every case. As well as recording the number of products, we checked how convergence was achieved. For the matrices rand(-1,0,1-) convergence was always achieved at the test (4) in Algorithm 2.4, while for inv(randn) convergence was declared at tests (4) and (2) in approximately 96 and 4 percent of the cases, respectively (with just a few instances of convergence at (5) for t - 2). The last two columns of Table 3.1 show the average and maximum number of parallel columns of S that were detected. A small number of repeated e j vectors were detected and replaced (the largest average was 0.03, occurring for and the maximum number of 7 occurred for columns or repeated were detected for the matrices in Table 3.2. The strategy of replacing parallel columns of S and repeated e j vectors has little effect on the overall performance of Algorithm 2.4 in our tests with random matrices. Since particular examples can be found where it is beneficial (see Example 2.5) and the cost is negligible, we feel its use is worthwhile. However, we see no advantage to extending the strategy to compare the columns of S with those of all previous S matrices. Higham [15] gives a tridiagonal matrix for which the 1-norm power method (Algo- rithm 2.1) requires n iterations to converge. We have constructed a matrix for which N. J. HIGHAM AND F. TISSEUR Table Results for 5000 matrices inv(randn) of dimension 100. Underest. ratio Products Parallel cols. t min average % exact average max improve% average max 9 0.893 1.000 99.88 4.0 4 100.00 1.22 15 Table Results for 5000 matrices rand(f-1,0,1g) of dimension 100. Underest. ratio Products t min average % exact average max improve% 9 0.775 0.951 28.56 4 4 90.74 the maximum iterations is required. It is the inverse of a bidiagonal matrix: An 1 . . ff3 1 . . \Gammaff3 (The minus sign in front of the matrix is necessary!) It is straightforward to show that when Algorithm 2.1 is applied to An (ff) it produces, for Table Results for matrix A 100 repetitions. Underest. ratio Products t min average % exact average max improve% 6 1.000 1.000 100.00 4.6 11 100.00 7 1.000 1.000 100.00 4.3 11 100.00 8 1.000 1.000 100.00 4.2 8 100.00 9 1.000 1.000 100.00 4.1 8 100.00 Thus every column of An (ff) is computed, in order from first to last, and the exact norm is obtained. If the algorithm is terminated after p iterations then it produces the estimate (1 \Gamma ff behaves in exactly the same way. We applied Algorithm 2.4 1000 times to A 100 as in all our tests). The results are shown in Table 3.3; in the 6th column "11 " denotes that convergence was declared because the iteration limit was reached (the percentage of such occurrences varied from 100% for to 0.1% for 7). The underestimation ratio for agrees with the theory and is unacceptably small (note that there is no randomness, and hence only one estimate, for 1). But for all t - 2 the average norm estimates are satisfactory. The extra estimate (2.1) has the value 0.561; thus it significantly improves the estimate for but is worse than the average estimates for all greater t. For complex matrices we have tried both at (3) in Algorithm 2.4. Tables 3.4 and 3.5 compare the two choices for 5000 100 \Theta 100 random complex matrices of the form inv(rand+i*rand), where rand is a matrix from the uniform distribution on the interval [0; 1]. In this test, larger underestimation ratios are obtained for the percentage of exact estimates is higher, and the statistics on the number of matrix products are slightly better. In other tests we have found the complex choice of Z always to perform at least as well, overall, as the real choice (see, for example, the riffle shuffle example in section 4). We therefore keep Z complex in Algorithm 2.4. In version 3.0 of LAPACK the norm estimator has been modified from that in version 2.0 to keep the vector z complex. Finally, how does Algorithm 2.4 compare with the suggestion of Hager mentioned at the beginning of section 2 of running the 1-norm power method t times in suc- cession? In tests with random matrices we have found Hager's approach to produce surprisingly good norm estimates, but they are generally inferior to those from Algorithm 2.4. Since Hager's approach is based entirely on level 2 BLAS operations Algorithm 2.4 is clearly to be preferred. 4. Computing 1-Norm Pseudospectra. In this section we apply the complex version of Algorithm 2.4 to the computation of 1-norm pseudospectra. For ffl - 0 and N. J. HIGHAM AND F. TISSEUR Table Results for 5000 matrices inv(rand+i*rand) of dimension 100, with Underest. ratio Products t min average % exact average max improve% 9 0.859 1.000 99.86 4.0 4 100.00 Table Results for 5000 matrices inv(rand+i*rand) of dimension 100, with Underest. ratio Products t min average % exact average max improve% 9 0.819 0.999 98.12 4.0 6 99.74 any subordinate matrix norm the ffl-pseudospectrum of A 2 C n\Thetan is defined by [22] z is an eigenvalue of A +E for some E withkEk - ffl g or, equivalently, in terms of the resolvent (zI Most published work on pseudospectra has dealt with the 2-norm and the utility of 2- norm pseudospectra in revealing the effects of non-normality is well appreciated [19], The 2-norm and any other p-norm of an n \Theta n matrix differ by a factor at most n. For small n, pseudospectra therefore do not vary much between different p-norms. However, J'onsson and Trefethen have shown [17] that in Markov chain applications the choice of norm for pseudospectra can be crucial. In Markov chains representing a random walk on an m-dimensional hypercube and a riffle shuffle of m cards, the transition matrices are of dimension exponential in m and factorial in m, respectively. It is known that when measured in an appropriate way, these random processes converge to a steady state not gradually but suddenly after a certain number of steps. As these processes involve powers of matrices of possibly huge dimension, and as the 1-norm is the natural norm for probability 1 , 1-norm pseudospectra are an important tool for explaining the transient behaviour [17]. One of the most useful graphical representations of pseudospectra is a plot of level curves of the resolvent. We therefore consider the standard approach of evaluating on an equally spaced grid of points z in some region of interest in the complex plane and sending the results to a contour plotter. A variety of methods for carrying out these computations for the 2-norm are surveyed by Trefethen [21], but most of the ideas employed are not directly applicable to the 1-norm. Explicitly forming (zI at each grid point is computationally expensive. A more efficient approach is to factorize P LU at each point z by LU factorization with partial pivoting and to use Algorithm 2.4 to estimate k(zI \Gamma the matrix multiplications in the algorithm become triangular solves with multiple right-hand sides. This approach can take advantage of sparsity in A. However, the method still requires O(n 3 ) operations per grid point when A is full. We consider instead a more efficient approach that is applicable when we can compute a Schur factorization [9, Chap. 7] where Q is unitary and T is upper triangular. Given this factorization we have so that forming a matrix product with (zI its conjugate transpose reduces to solving a multiple right-hand side triangular system and multiplying by Q and . Given this initial decomposition the cost per grid point of estimating the resolvent norm using Algorithm 2.4 is just O(n 2 t) flops-a substantial saving over the first approach, and of the same order of magnitude as the cost of standard methods for computing 2-norm pseudospectra (provided t is small). In place of the Schur decomposition we could use a Hessenberg decomposition, computed either by Gauss transformations or Householder transformations [9, Sec. 7.4]; these decompositions are less expensive but the cost per grid point is a larger multiple of n 2 flops because of the need to factorize a Hessenberg matrix. In more detail our approach is as follows. Algorithm 4.1 (1-norm pseudospectra estimation). Given A 2 C n\Thetan and a positive integer t, this algorithm estimates k(zI on a specified grid of points in the complex plane, using Algorithm 2.4 with parameter t. Compute the Schur factorization for each grid point z Apply the complex version of Algorithm 2.4 to (zI with parameter t, using the representation (4.2). Our experience is that Algorithm 4.1 frequently leads to visually acceptable contour plots even for 1. As the following example shows, however, a larger value of t may be needed. We take an example from Markov chains: the Gilbert-Shannon-Reeds model of a riffle shuffle on a deck of n cards. The transition matrix P is of dimension n!. Remarkably, as J'onsson and Trefethen explain [17], the dimension of the matrix 1 Because of the use of row vectors in the Markov chain literature, it is actually the 1-norm that is relevant. By using k1 we can continue to work with the 1-norm. 14 N. J. HIGHAM AND F. TISSEUR can be reduced to n by certain transformations that preserve the 1-norms of powers and of the resolvent. For this experiment we took and, as in [17], we computed pseudospectra of the "decay matrix" working with the reduced form of A. Figure 4.1 shows approximations to the 1-norm pseudospectra computed on a 100 \Theta 100 grid, with 2.4. Contours are plotted for the dashed line marks the unit circle, and the eigenvalues are plotted as dots. We did not exploit the fact that, since A is real, the pseudospectra are symmetric about the real axis. The contour plot for clearly incorrect in the outer contour, while yields an improvement and the plot for to visual accuracy. Table 4.1 summarizes the key statistics from these computations, showing that on average the norm estimates had about t correct significant digits for 3. The figure and the table confirm that it is better to keep Z complex in Algorithm 2.4. We give a further example, in which A is a spectral discretization of an integral operator of Landau [21, Sec. 21]; like the operator, A is complex and symmetric. We took dimension Fresnel number 8. As noted in [21] for the 2-norm, a fine grid is needed to resolve the details for this example; we used a 200 \Theta 200 grid. Figure 4.2 shows the computed pseudospectra for summarizes the statistics for these values and for 2. Contours are plotted for the dashed line marks the unit circle, and the eigenvalues are plotted as dots. In Table 4.2, "11 " denotes that convergence was declared because the iteration limit was reached (the percentage of such occurrences was 0.12% for and 0.025% for one of the contour lines misses an eigenvalue in the north-west corner of the plot. For the plot differs from the exact one only by some tiny oscillations in two outer contours. While Algorithm 2.4 performs well even for small t, as measured by the underestimation ratio, quite accurate values of the 1-norm of the resolvent are needed in this example in order to produce smooth contours. 5. Conclusions. We have derived a new matrix 1-norm estimation algorithm, Algorithm 2.4, with a number of key features. Most importantly, the algorithm has a parameter t that can be used to control the accuracy and reliability of the estimate (which is actually a lower bound). While there is no guarantee that increasing t increases the estimate (leaving aside the fact that the starting matrix is partly random), the estimate typically does increase with t, leading quickly to one or more correct significant digits in the estimate. A crucial property of the algorithm is that the number of iterations and matrix products required for convergence is essentially independent of t (for random matrices about 2 iterations are required on average, corresponding to 4 products of n \Theta n and n \Theta t matrices). The algorithm avoids redundant computations and keeps constant the size of the matrix multiplications. In future work we intend to investigate how the choice of t affects the efficiency of the algorithm in a high performance computing environment. Unlike the statistically-based norm estimation techniques in [7], [11] which currently apply only to real matrices, our algorithm handles both real and complex matrices. Since our algorithm uses a partly random starting matrix for t - 2, it is natural to ask whether bounds, valid for all A, can be obtained on the probability of the estimate being within a certain factor of kAk 1 . We feel that the very features of the algorithm that make it so effective make it difficult or impossible to derive useful bounds of this type. For our algorithm is very similar to the estimator in LAPACK, the differences being that our algorithm omits the extra estimate (2.1) and for real matrices we test for parallel rather than simply repeated sign vectors (which improves the ef- ficiency). Unlike in the estimator in LAPACK 2.0, for complex matrices we do not take the real part of the z vector (which improves both the quality of the estimates and the efficiency), and this change has been incorporated into LAPACK 3.0. The new algorithm makes an attractive replacement for the existing LAPACK estimator. The value would be the natural default choice (with the extra estimate (2.1) included for extra reliability and backward compatibility) and a user willing to pay more for more accurate and reliable 1-norm estimates would have the option of choosing a larger t. Acknowledgements . We thank Nick Trefethen for providing M-files that compute the riffle shuffle and Landau matrices used in section 4. N. J. HIGHAM AND F. TISSEUR Table Results for riffle shuffle example. Underest. ratio Products t min average % exact average max Fig. 4.1. 1-norm pseudospectra for the riffle shuffle. Clockwise from top left: Table Results for Landau matrix example. Underest. ratio Products t min average % exact average max Fig. 4.2. 1-norm pseudospectra for the Landau matrix. Clockwise from top left: N. J. HIGHAM AND F. TISSEUR --R Solving sparse linear systems with sparse backward error. The power method for An estimate for the condition number of a matrix. A set of counter-examples to three condition number estimators Open problems in numerical linear algebra. Estimating extremal eigenvalues and condition numbers of matrices. Matrix Computations. A stable variant of the secant method for solving nonlinear equations. Condition estimates. A survey of condition number estimation for triangular matrices. FORTRAN codes for estimating the one-norm of a real or complex matrix Experience with a matrix norm estimator. Accuracy and Stability of Numerical Algorithms. A numerical analyst looks at the Convergence of a subgradient method for computing the bound norm of matrices. Pseudospectra of matrices. Pseudospectra of linear operators. Computation of pseudospectra. Spectra and Pseudospectra: The Behavior of Non-Normal Matrices and Operators --TR --CTR J. R. Cash , F. Mazzia , N. Sumarti , D. Trigiante, The role of conditioning in mesh selection algorithms for first order systems of linear two point boundary value problems, Journal of Computational and Applied Mathematics, v.185 n.2, p.212-224, 15 January 2006 J. R. Cash , F. Mazzia, A new mesh selection algorithm, based on conditioning, for two-point boundary value codes, Journal of Computational and Applied Mathematics, v.184 n.2, p.362-381, 15 December 2005

LAPACK;matrix condition number;p-norm power method;level 3 BLAS;1-norm pseudospectrum;condition number estimation;matrix 1-norm;matrix norm estimation

354402

Constraint Preconditioning for Indefinite Linear Systems.

The problem of finding good preconditioners for the numerical solution of indefinite linear systems is considered. Special emphasis is put on preconditioners that have a 2 2 block structure and that incorporate the (1,2) and (2,1) blocks of the original matrix. Results concerning the spectrum and form of the eigenvectors of the preconditioned matrix and its minimum polynomial are given. The consequences of these results are considered for a variety of Krylov subspace methods. Numerical experiments validate these conclusions.

Introduction In this paper we are concerned with investigating a new class of preconditioners for indefinite systems of linear equations of a sort which arise in constrained optimization as well as in least-squares, saddle-point and Stokes problems. We attempt to solve the indefinite linear system A where A 2 IR n\Thetan is symmetric and B 2 IR m\Thetan . Throughout the paper we shall assume that m - n and that A is non-singular, in which case B must be of full rank. Example 1. Consider the problem of minimizing a function of n variables subject to m linear equality constraints on the variables, i.e. minimize subject to Any finite solution to (1.2) is a stationary point of the Lagrangian function where the - i are referred to as Lagrangian multipliers. By differentiating L with respect to x and - the solution to (1.2) is readily seen to satisfy linear equations of the form (1.1) with x d. For this application these are known as the Karush-Kuhn-Tucker (KKT) conditions. 2 Example 2 (The Stokes Problem). The Stokes equations in compact form are defined by div Discretising equations (1.3) together with the boundary conditions defines a linear system of equations of the form (1.1), where b Carsten Keller, Nick Gould and Andy Wathen Among the most important iterative methods currently available, Krylov sub-space methods apply techniques that involve orthogonal projections onto sub-spaces of the form The most common schemes that use this idea are the method of conjugate gradients (CG) for symmetric positive definite matrices, the method of minimum residuals (MINRES) for symmetric and possibly indefinite matrices and the generalised minimum residual method (GMRES) for unsymmetric matrices, although many other methods are available-see for example Greenbaum [12]. One common feature of the above methods is that the solution of the linear system (1.1) is found within n +m iterations in exact arithmetic-see Joubert and Manteuffel [14, p. 152]. For very large (and possibly sparse) linear systems this upper limit on the number of iterations is often not practical. The idea of preconditioning attempts to improve on the spectral properties, i.e. the clustering of the eigenvalues, such that the total number of iterations required to solve the system to within some tolerance is decreased substantially. In this paper we are specifically concerned with non-singular preconditioners of the form n\Thetan approximates, but is not the same as A. The inclusion of the exact representation of the (1; 2) and (2; 1) matrix blocks in the preconditioner, which are often associated with constraints (see Example 1), leads one to hope for a more favourable distribution of the eigenvalues of the (left-)preconditioned linear system Since these blocks are unchanged from the original system, we shall call G a constraint preconditioner. A preconditioner of the form G has recently been used by Luk-san and Vl-cek [16] in the context of constrained non-linear programming problems-see also Coleman [4], Polyak [18] and Gould et al. [11]. Here we derive arguments that confirm and extend some of the results in [16] and highlight the favourable features of a preconditioner of the form G. Note that Golub and Wathen [10] recently considered a symmetric preconditioner of the form (1.4) for problems of the form (1.1) where A is non-symmetric. In Section 2 we determine the eigensolution distribution of the preconditioned system and give lower and upper bounds for the eigenvalues of G \Gamma1 A Constraint Preconditioning 3 in the case when the submatrix G is positive definite. Section 3 describes the convergence behaviour of a Krylov subspace method such as GMRES, Section 4 investigates possible implementation strategies, while in Section 5 we give numerical results to support the theory developed in this paper. Preconditioning A For symmetric (and in general normal) matrix systems, the convergence of an applicable iterative method is determined by the distribution of the eigenvalues of the coefficient matrix. In particular it is desirable that the number of distinct eigenvalues, or at least the number of clusters, is small, as in this case convergence will be rapid. To be more precise, if there are only a few distinct eigenvalues then optimal methods like CG, MINRES or GMRES will terminate (in exact arithmetic) after a small and precisely defined number of steps. We prove a result of this type below. For non-normal systems convergence as opposed to termination is not so readily described-see Greenbaum [12, p. 5]. 2.1 Eigenvalue Distribution The eigenvalues of the preconditioned coefficient matrix G \Gamma1 A may be derived by considering the general eigenvalue problem x y x y Y Z be an orthogonal factorisation of B T , where n\Thetam and Z 2 IR n\Theta(n\Gammam) is a basis for the nullspace of B. Premultiplying (2.6) by the non-singular and square matrix6 6 4 and postmultiplying by its transpose gives6 6 4 x z x y x z x y with and where we made use of the equalities Performing a simultaneous sequence of row and column inter- 4 Carsten Keller, Nick Gould and Andy Wathen changes on both matrices in (2.7) reveals two lower block-triangular matrices ~ and thus the preconditioned coefficient matrix G \Gamma1 A is similar to I \Theta (Z T GZ) Here the precise forms of \Theta, \Upsilon and \Gamma are irrelevant for the argument that they are in general non-zero. We just proved the following theorem. Theorem 2.1. Let A 2 IR (n+m)\Theta(n+m) be a symmetric and indefinite matrix of the form where A 2 IR n\Thetan is symmetric and B 2 IR m\Thetan is of full rank. Assume Z is an n \Theta (n \Gamma m) basis for the nullspace of B. Preconditioning A by a matrix of the form n\Thetan is symmetric, G 6= A and B 2 IR m\Thetan is as above, implies that the matrix G \Gamma1 A has (1) an eigenvalue at 1 with multiplicity 2m; and eigenvalues which are defined by the generalised eigenvalue problem Z T AZx Note that the indefinite constrained preconditioner applied to the indefinite linear system (1.1) yields the preconditioned matrix P which has real eigenvalues. Remark 1. In the above argument we assumed that B has full row rank and consequently applied an orthogonal factorisation of B T which resulted in a Constraint Preconditioning 5 upper triangular matrix R 2 IR m\Thetam . If B does not have full row rank, i.e. rows and columns can be deleted from both matrices in (2.7), thus giving a reduced system of dimension This removal of the redundant information does not impose any restriction on the proposed preconditioner, since all mathematical arguments equivalently apply to the reduced system of equations. 2.2 Eigenvector Distribution We mentioned above that the termination for a Krylov subspace method is related to the location of the eigenvalues and the number of corresponding linearly independent eigenvectors. In order to establish the association between eigenvectors and eigenvalues we expand the general eigenvalue problem (2.7), yielding From (2.11) it may be deduced that either In the former case equations (2.9) and (2.10) simplify to which can consequently be written as Y Z and z . Since Q is orthogonal, the general eigenvalue problem (2.12) is equivalent to considering with w 6= 0 if and only if oe = 1. There are m linearly independent eigenvectors corresponding to linearly independent eigenvectors (corresponding to eigenvalues Now suppose - 6= 1, in which case x Equations (2.9) and (2.10) yield 6 Carsten Keller, Nick Gould and Andy Wathen The general eigenvalue problem (2.14) defines m) of these are not equal to 1 and for which two cases have to be distinguished. If x z 6= 0, y must satisfy from which follows that the corresponding eigenvectors are defined by If x we deduce from (2.15) that and hence that As z x T 0 in this case, no extra eigenvectors arise. Summarising the above, it is evident that P has We now show that, under realistic assumptions, these eigenvectors are in fact linearly independent. Theorem 2.2. Let A 2 IR (n+m)\Theta(n+m) be a symmetric and indefinite matrix of the form where A 2 IR n\Thetan is symmetric and B 2 IR m\Thetan is of full rank. Assume the preconditioner G is defined by a matrix of the form n\Thetan is symmetric, G 6= A and B 2 IR m\Thetan is as above. Let Z denote an n \Theta (n \Gamma m) basis for the nullspace of B and suppose that Z T GZ is positive definite. The preconditioned matrix G \Gamma1 A has n+m eigenvalues as defined by Theorem 2.1 and linearly independent eigenvectors. There are eigenvectors of the form that correspond to the case Constraint Preconditioning 7 eigenvectors of the form z x T arising from z linearly independent, m) eigenvectors of the form that correspond to the case - 6= 1. Proof. To prove that the m eigenvectors of P are linearly independent we need to show y (1) a (1)a (1) x z (2) (2) x y (2) (2) y (2) a (2)a (2) x z y (3) a (3)a (3) implies that the vectors a are zero vectors. Multiplying by A and G \Gamma1 , and recalling that in the previous equation the first matrix arises from the case - m), the second matrix from the case - the last matrix arises from - k y (1) a (1)a (1) x z (2) (2) x y (2) (2) y (2) a (2)a (2) x z y (3) a (3)a (3) Subtracting Equation (2.16) from (2.17) we obtain6 6 4 x z y (3) a (3)a (3) 8 Carsten Keller, Nick Gould and Andy Wathen which simplifies to6 6 4 x z y (3) a (3)a (3) since - k The assumption that Z T GZ is positive definite implies that x z in (2.18) are linearly independent and thus a (3) Similarly, a (2) follows from the linear independence of x z (2) x y (2) thus (2.16) simplifies to6 6 4 y (1) a (1)a (1) But y (1) are linearly independent and thus a (1) Remark 2. Note that the result of Theorem 2.2 remains true if Z T (flA is positive definite for some scalars fl and oe-see Parlett [17, p. 343] for details. To show that the eigenvector bounds of Theorem 2.2 can in fact be attained, consider the following two examples. Example 3 (Minimum bound). Consider the matrices so that 2. The preconditioned matrix P has an eigenvalue at 1 with multiplicity 3, but only one eigenvector arising from case (1) in Theorem 2.2. This eigenvector may be taken to be . 2 Example 4 (Maximum bound). Let A 2 IR 3\Theta3 be defined as in Example 3, but assume A. The preconditioned matrix P has an eigenvalue at 1 with multiplicity 3 and clearly a complete set of eigenvectors. These may be taken to be , and Constraint Preconditioning 9 2.3 Eigenvalue Bounds It is apparent from the calculations in the previous section that the eigenvalue at 1 with multiplicity 2m is independent of the choice of G in the preconditioner. On the contrary, the eigenvalues that are defined by (2.14) are highly sensitive to the choice of G. If G is a close approximation of A, we can expect a more favourable distribution of eigenvalues and consequently may expect faster convergence of an appropriate iterative method. In order to determine a good factorisation of A it will be helpful to find intervals in which the are located. If G is a positive definite matrix one possible approach is provided by Cauchy's interlace theorem. Theorem 2.3 (Cauchy's Interlace Theorem, see [17, Theorem 10.1.1]). Suppose n\Thetan is symmetric, and that Label the eigenpairs of T and H as Then Proof. See Parlett [17, p. 203]. 2 The applicability of Theorem 2.3 is verified by recalling the definitions of Q and Z given in the previous section, and by considering the generalised eigenvalue problems and Carsten Keller, Nick Gould and Andy Wathen Since G is positive definite so is Q T GQ, and we may therefore write RR T . Rewriting and (2.20) gives and where Now, since the matrix M \Gamma1 Q T AQM \GammaT is similar to G defines the same eigenvalues ff i A. We may therefore apply Theorem 2.3 directly. The result is that the In particular, the - i are bounded by the extreme eigenvalues of G \Gamma1 A so that the - i will necessarily be clustered if G is a good approximation of A. Furthermore, a good preconditioner G for A implies that Z T GZ is at least as good a preconditioner for Z T AZ. To show that the preconditioner Z T GZ can in fact be much better, consider the following example, taken from the CUTE collection [3]. Example 5. Consider the convex quadratic programming problem BLOWEYC which may be formulated as subject to Z Selecting a size parameter of 500 discretisation intervals defines a set of linear equations of the form (1.1), where Letting G be the diagonal of A, we may deduce by the above theory that the extreme eigenvalues of G \Gamma1 A give a lower and upper bound for the defined by the general eigenvalue problem (2.14). In Figure 2.1 (a) the 1002 eigenvalues of G \Gamma1 A are drawn as vertical lines, whereas Figure 2.1 (b) displays the 500 eigenvalues of (Z T GZ) The spectrum of Figure 2.1 (a) is equivalent to a graph of the entire spectrum of P, but with an eigenvalue at 1 and multiplicity 502 removed. Rounded to two decimal places the numerical values of the two extreme eigenvalues of G \Gamma1 A are Constraint Preconditioning 11 -0.4 size of eigenvalue (a) Eigenvalues of G \Gamma1 A -0.4 size of eigenvalue (b) Eigenvalues of (Z T GZ) Figure 2.1: Continuous vertical lines represent the eigenvalues of (a) G \Gamma1 A and 0:02 and 1:98, whereas the extreme eigenvalues of (Z T GZ) are given by 0:71 and 1. Note that for this example a large number of eigenvalues of are clustered in the approximate intervals [0:02; 0:38] and [1:65; 1:97]. The eigenvalue distribution in Figure (2.1) (b) reveals that there is one eigenvalue near 0:71 and a group of eigenvalues near 1. It follows that any appropriate iterative method that solves (1.5) can be expected to converge in a very small number of steps; this is verified by the numerical results presented in Section 5.It is readily seen from Example 5 that in this case the bounds provided by Theorem 2.3 are not descriptive in that there is significantly more clustering of the eigenvalues than implied by the theorem. Convergence In the context of this paper, the convergence of an iterative method under pre-conditioning is not only influenced by the spectral properties of the coefficient matrix, but also by the relationship between the dimensions n and m. In par- ticular, it follows from Theorem 2.1 that in the special case when the preconditioned linear system (1.5) has only one eigenvalue at 1 with multiplicity 2n. For gives an eigenvalue at 1 with multiplicity 2m and (generally distinct) eigenvalues whose value may or may not be equal to 1. Before we examine how these results determine upper bounds on the number 12 Carsten Keller, Nick Gould and Andy Wathen of iterations of an appropriate Krylov subspace method, we recall the definition of the minimum polynomial of a matrix. Definition 1. Let A 2 IR (n+m)\Theta(n+m) . The monic polynomial f of minimum degree such that is called the minimum polynomial of A. The importance of this definition becomes apparent when considering subsequent results and by recalling that similar matrices have the same minimum polynomial. The Krylov subspace theory states that the iteration with any method with an optimality property such as GMRES will terminate when the degree of the minimum polynomial is attained-see Axelsson [1, p. 463] (To be precise, the number may be less in special cases where b is a combination of a few eigenvectors that affect the 'grade' of A with respect to b). In particular, the degree of the minimum polynomial is equal to the dimension of the corresponding Krylov subspace (for general b) and so the following theorems are relevant. Theorem 3.1. Let A 2 IR (n+m)\Theta(n+m) be a symmetric and indefinite matrix of the form where A 2 IR n\Thetan is symmetric and B 2 IR m\Thetan is of full rank. Let If A is preconditioned by a matrix of the form where G 2 IR n\Thetan , G 6= A and B 2 IR m\Thetan is as above, then the Krylov subspace K(P; b) is of dimension at most 2 for any b. Proof. Writing the preconditioned system (2.8) in its explicit form we observe that P is in fact given by I 0 \Upsilon I where \Upsilon is non-zero if and only if A 6= G. To show that the dimension of the corresponding Krylov subspace is at most 2 we need to determine the minimum polynomial of the system. It is evident from (3.23) that the Constraint Preconditioning 13 eigenvalues of P are all 1 and so the minimum polynomial is of order 2. 2 Remark 3. It is of course possible in the case to solve the (square) constrained equation Bx and then to obtain x gives motivation for why the result of Theorem 3.1 is independent of G. Remark 4. The important consequence of Theorem 3.1 is that termination of an iteration method such as GMRES will occur in at most 2 steps for any choice of b, even though the preconditioned matrix is not diagonalisable (unless Theorem 3.2. Let A 2 IR (n+m)\Theta(n+m) be a symmetric and indefinite matrix of the form where A 2 IR n\Thetan is symmetric and B 2 IR m\Thetan is of full rank. Assume n and that A is non-singular. Furthermore, assume A is preconditioned by a matrix of the form n\Thetan is symmetric, G 6= A and B 2 IR m\Thetan is as above. If Z T GZ is positive definite, where Z is an n \Theta (n \Gamma m) basis for the nullspace of B, then the dimension of the Krylov subspace K(P; b) is at most n \Gamma m+ 2. Proof. From the eigenvalue derivation in Section 2.1 it is evident that the characteristic polynomial of the preconditioned linear system (1.5) is To prove the upper bound on the dimension of the Krylov subspace we need to show that the order of the minimum polynomial is less than or equal to 2. 14 Carsten Keller, Nick Gould and Andy Wathen Expanding the polynomial obtain a matrix of the form6 6 4 Here \Psi n\Gammam is defined by the recursive formula \Theta with base cases \Psi Note that the (2; 1), (2; 2) and (3; 2) entries of matrix (3.24) are in fact zero, since the - i m) are the eigenvalues of S, which is similar to a symmetric matrix and is thus diagonalisable. Thus (3.24) may be written as 2 and what remains is to distinguish two different cases for the value of \Phi n\Gammam , that is \Phi In the former case the order the minimum polynomial of P is less than or equal to thus the dimension of the Krylov subspace K(P; b) is of the same order. In the latter case the dimension of K(P; b) is less than or equal to n \Gamma m+ 2 since multiplication of (3.25) by another factor (P \Gamma I) gives the zero matrix. The upper bound on the dimension of the Krylov subspace, as stated in Theorem 3.2, can be reduced in the special case when (Z repeated eigenvalues. This result is stated in Theorem 3.3. The following (ran- domly generated) example shows that the bound in Theorem 3.2 is attainable. Example 6. Let A 2 IR 6\Theta6 and B T 2 IR 6\Theta2 be given by 2:69 1:62 1:16 1:60 0:81 \Gamma1:97 1:62 6:23 \Gamma1:90 1:89 0:90 0:05 0:81 0:90 \Gamma0:16 0:01 1:94 0:38 Constraint Preconditioning 15 and assume that diag(A). For the above matrices the (3; 1) entry of (3.25) is \Gamma0:22 \Gamma0:02 It follows that the minimum polynomial is of order 6 and thus the bound given in Theorem 3.2 is sharp. 2 Theorem 3.3. Let A 2 IR (n+m)\Theta(n+m) be a symmetric and indefinite matrix of the form where A 2 IR n\Thetan is symmetric and B 2 IR m\Thetan is of full rank. Assume non-singular and that A is preconditioned by a matrix of the n\Thetan is symmetric, G 6= A and B 2 IR m\Thetan is as above. Furthermore, let Z be an n \Theta (n \Gamma m) basis for the nullspace of B and assume (Z T GZ) m) distinct eigenvalues of respective multiplicity - i , where the dimension of the Krylov subspace K(P; b) is at most k 2. Proof. The proof is similar to the one for Theorem 3.2. In the case when (Z distinct eigenvalues of multiplicity - i we may, without loss of generality, write the characteristic polynomial of P as Y Y Expanding (y) we obtain the matrix6 6 4 Carsten Keller, Nick Gould and Andy Wathen Y Here \Psi k is given by the recursive formula \Theta with base cases \Psi Note that the (2; 1), (2; 2) and (3; 2) blocks of matrix (3.26) are in fact zero. It follows that, for \Phi k 6= 0, a further multiplication of (3.26) by the zero matrix and thus the dimension of Krylov subspace K(P; b) is less then or equal to k 2. 2 To verify that the bound in Theorem 3.3 is attainable consider the following example. Example 7. Let A 2 IR 4\Theta4 , G 2 IR 4\Theta4 and B T 2 IR 4\Theta1 be given by . Then two of the eigenvalues that are defined by the generalised eigenvalue problem (2.14) are distinct and given by [2; 4]. It follows that the (3; 1) entry of (3.26) is non-zero with and so the minimum polynomial is of order 4. 2 Implementation There are various strategies that can be used to implement the proposed pre- conditioner, two of which are used in the numerical results in Section 5. The first strategy applies the standard (preconditioned) GMRES algorithm [20], where the preconditioner step is implemented by means of a symmetric indefinite factorisation of (1.4). Such a factorisation of the preconditioner may be much less demanding than the factorisation of the initial coefficient matrix if G is a considerably simpler matrix than A. The second approach, discussed in the next section, is based on an algorithm that solves a reduced linear system Constraint Preconditioning 17 4.1 Conjugate Gradients on a Reduced Linear System In [11] Gould et al. propose a Conjugate Gradient like algorithm to solve equality constrained quadratic programming problems such as the one described in Example 1. The algorithm is based on the idea of computing an implicit basis Z which spans the nullspace of B. The nullspace basis is then used to remove the constraints from the system of equations, thus allowing the application of the Conjugate Gradients method to the (positive definite) reduced system. Assume that W GZ is a symmetric and positive definite preconditioner matrix of dimension (n \Gamma m) \Theta (n \Gamma m) and Z is an n \Theta (n \Gamma m) matrix. The algorithm can then be stated as follows. Algorithm 4.1: Preconditioned CG for a Reduced System. (1) Choose an initial point x satisfying (2) Compute \Gammag (3) Repeat the following steps until x The computation of the preconditioned residual in (4.29) is often the most expensive computational factor in the algorithm. Gould et al. suggest avoiding the explicit use of the nullspace Z, but instead to compute g + by applying a Carsten Keller, Nick Gould and Andy Wathen symmetric indefinite factorisation of r +# In practice (4.30) can often be factored efficiently by using the MA27 package of the Harwell Subroutine Library when G is a simple matrix block, whereas the direct application of MA27 to the original system (1.1) is limited by space requirements as well as time for large enough systems [6]. In this context the factorisation consists of three separate routines, the first two of which analyse and factorise the matrix in (4.30). They need to be executed only once in Step (1) of Algorithm 4.1. Repeated calls to the third routine within MA27 apply forward- and backward-substitutions to find the initial point x in Step (1), solve for g in (4.27) and also to find g + in (4.29). Remark 5. The computation of the projected residual g + is often accompanied by significant roundoff errors if this vector is much smaller than the residual Iterative refinement is used in (4.28) to redefine r + so that its norm is closer to that of g + . The result is a dramatic reduction of the roundoff errors in the projection operation-see Gould et al. [11]. 5 Numerical Results We now present the results of numerical experiments that reinforce the analysis given in previous sections. The test problems we use are partly randomised sparse matrices (Table 5.1) and partly matrices that arise in linear and non-linear optimization (Table 5.2)-see Bongartz et al. [3]. As indicated through- out, all matrices are of the form where A 2 IR n\Thetan is symmetric, B 2 IR m\Thetan has full rank and m - n. Four different approaches to finding solutions to (1.1) are compared-three iterative algorithms based on Krylov subspaces, and the direct solver MA27 which applies a sparse variant of Gaussian elimination-see Duff and Reid [6]. To investigate possible favourable aspects of preconditioning it makes sense to compare unpreconditioned with preconditioned solution strategies. The indefinite nature of matrix (5.31) suggests the use of MINRES in the unpreconditioned case. As outlined in Section 4 we employ two slightly different strategies in order to implement the preconditioner G. The first method applies a standard (full) GMRES(A) code (PGMRES in Tables 5.1 and 5.2 below), which is Constraint Preconditioning 19 mathematically equivalent to MINRES(A) for symmetric matrices A, whereas the second approach implements Algorithm 4.1 (RCG in Tables 5.1 and 5.2 be- low). The choice in the preconditioner is made for both PGMRES and RCG. Random I Random II Random III Random IV non-zero entries in A 2316 9740 39948 39948 non-zero entries in B 427 1871 3600 686 MINRES # of iterations 174 387 639 515 time in seconds 0:4 3:1 17:5 13:1 PGMRES # of iterations 46 87 228 242 time in seconds 0:2 3:9 96:0 108:9 RCG # of iterations 36 67 197 216 time in seconds 0:1 1:0 5:9 5:9 time in seconds 0:1 0:9 5:4 2:9 Table 5.1: Random test problems All tests were performed on a SUN Ultra SPARCII-300MhZ (ULTRA-30) workstation with 245 MB physical RAM and running SunOS Release 5:5:1. Programs were written in standard Fortran 77 using the SUN WorkShop f77 compiler (version 4:2) with the -0 optimization flag set. In order to deal with large sparse matrices we implemented an index storage format that only stores non-zero matrix elements-see Press et al. [19]. The termination criterion for all iterative methods was taken to be a residual vector of order less than 10 \Gamma6 in the 2-norm. As part of its analysis procedure, MA27 accepts the pattern of some coefficient matrix and chooses pivots for the factorisation and solution phases of subsequent routines. The amount of pivoting is controlled by the special parameter Modifying u within its positive range influences the accuracy of the resulting solution, whereas a negative value prevents any pivoting-see Duff and Reid [6]. In this context, the early construction of some of the test examples with the default value accompanied by difficulties in the form of memory limitations. We met the trade-off between less use of memory and solutions of high enough accuracy by choosing the parameter value Tables 5.1 and 5.2 . The time measurements for the eight test examples indicate that the itera- Carsten Keller, Nick Gould and Andy Wathen non-zero entries in A 3004 672 1020 13525 non-zero entries in B 2503 295 148 50284 MINRES # of iterations 363 no convergence 51 180 time in seconds 1:8 no convergence 0:1 14:2 PMGREM # of iterations time in seconds 0:6 0:1 0:2 13:2 RCG # of iterations time in seconds 0:6 0:1 0:2 16:2 time in seconds 0:5 0:1 0:1 15:6 Table 5.2: CUTE test problems tion counts for each of the three proposed iterative methods are comparable as far as operation counts, i.e. work, is concerned. The numerical results suggest that the inclusion of the (1; 2) and (2; 1) block of A into the preconditioner, together with results in a considerable reduction of iterations, where the appropriate bounds of Theorems 3.1, 3.2 and 3.3 are attained in all cases. Specifically, Theorem 3.1 applies in context of problem CVXQP1. Test problems RANDOM III and RANDOM IV in Table 5.1 emphasise the storage problems that are associated with the use of long recurrences in the PGMRES algorithm. The time required to find a solution to both RANDOM III and RANDOM IV via the PGMRES algorithm is not comparable to any of the other methods, which is due to the increased storage requirements and the data trafficking involved. A solution to the memory problems is to restart PGMRES after a prescribed number of iterations, but the iteration counts for such restarts would not be comparable with those of full PGMRES. The relevance of the time measurements for MA27 are commented on in the next section. 6 Conclusion In this paper we investigated a new class of preconditioner for indefinite linear systems that incorporate the (2; 1) and (2; 2) blocks of the original matrix. These blocks are often associated with constraints. In our numerical results we used a simple diagonal matrix G to approximate the (1; 1) block of A, Constraint Preconditioning 21 even though other approximations, such as an incomplete factorisation of A, are possible. We first showed that the inclusion of the constraints into the preconditioner clusters at least 2m eigenvalues at 1, regardless of the structure of G. However, unless G represents A exactly, P does not have a complete set of linearly independent eigenvectors and thus the standard convergence theory for Krylov subspace methods is not readily applicable. To find an upper bound on the number of iterations, required to solve linear system of the form (1.1) by means of appropriate subspace methods, we used a minimum polynomial argument. Theorem 3.1 considers the special condition m, in which case termination is guaranteed in two iterations. For Theorem 3.2 gives a general (sharp) upper bound on the dimension of the Krylov subspace, whereas Theorem 3.3 defines a considerably stronger result if some of the defined by (Z T GZ) are repeated. In the special case when G is a positive definite matrix block we were able to apply Cauchy's interlacing theorem in order to give an upper and lower bound for the eigenvalues that are defined by the (2; 2) block of matrix (2.8). To confirm the analytical results in this paper we used three different sub-space methods, MINRES of Paige and Saunders for the unpreconditioned matrix system and RCG of Gould et al. and also PGMRES of Saad and Schultz for the preconditioned case. Overall, the results show that the number of iterations is decreased substantially if preconditioning is applied. The Krylov subspaces that are built during the execution of the two preconditoned implementations are in theory of equal dimension for any of the eight test examples, and thus PGMRES and RCG can be expected to terminate in the same number of steps. However, convergence to any prescribed tolerance may occur for a different number of steps since PGMRES and RCG minimize different quantities. This can be seen in some of the examples. Nevertheless, we note that convergence for both methods is attained much earlier than suggested by the bounds in Theorems 3.1, 3.2 and 3.3. The time measurements for MA27 in the last section suggest that the preconditioned conjugate gradients algorithm, discussed in Section 4.1, is a suitable alternative to the direct solver. Whereas both MINRES and especially PGM- RES are considerably slower than MA27, the timings for RCG are in virtually all cases comparable. For problems of large enough dimension or bandwidth the resources required by MA27 must become prohibitive in which case RCG becomes even more competitive. 22 Carsten Keller, Nick Gould and Andy Wathen Acknowledgments The authors would like to to thank Gene. H. Golub for his insightful comments during the process of the work. --R Cambridge University Press CUTE: Constrained and Unconstrained Testing Environment Linearly constrained optimization and projected preconditioned conjugate gradients Direct Methods for Sparse Matrices The multifrontal solution of indefinite sparse symmetric linear equations Perturbation of eigenvalues of preconditioned Navier-Stokes operators Polynomial Based Iteration Methods for Symmetric Linear Systems Matrix Computations An iteration for indefinite system and its application to the Navier-Stokes equations On the solution of equality constrained quadratic programming problems arising in optimiza- tion Iterative Methods for Solving Linear Systems Iterative methods for non-symmetric linear systems Iterative Methods for Linear and Nonlinear Equations The Symmetric Eigenvalue Problem Numerical Recipes in Fortran: The Art of Scientific Computing GMRES: a generalised minimal residual algorithm for solving nonsymmetric linear systems --TR --CTR Joo-Siong Chai , Kim-Chuan Toh, Preconditioning and iterative solution of symmetric indefinite linear systems arising from interior point methods for linear programming, Computational Optimization and Applications, v.36 n.2-3, p.221-247, April 2007 Luca Bergamaschi , Jacek Gondzio , Manolo Venturin , Giovanni Zilli, Inexact constraint preconditioners for linear systems arising in interior point methods, Computational Optimization and Applications, v.36 n.2-3, p.137-147, April 2007 Z. Dostl, An optimal algorithm for a class of equality constrained quadratic programming problems with bounded spectrum, Computational Optimization and Applications, v.38 n.1, p.47-59, September 2007 H. S. Dollar , N. I. Gould , W. H. Schilders , A. J. Wathen, Using constraint preconditioners with regularized saddle-point problems, Computational Optimization and Applications, v.36 n.2-3, p.249-270, April 2007 Luca Bergamaschi , Jacek Gondzio , Giovanni Zilli, Preconditioning Indefinite Systems in Interior Point Methods for Optimization, Computational Optimization and Applications, v.28 n.2, p.149-171, July 2004 S. Cafieri , M. D'Apuzzo , V. Simone , D. Serafino, Stopping criteria for inner iterations in inexact potential reduction methods: a computational study, Computational Optimization and Applications, v.36 n.2-3, p.165-193, April 2007 S. Bocanegra , F. F. Campos , A. R. Oliveira, Using a hybrid preconditioner for solving large-scale linear systems arising from interior point methods, Computational Optimization and Applications, v.36 n.2-3, p.149-164, April 2007 S. Cafieri , M. D'Apuzzo , V. Simone , D. Serafino, On the iterative solution of KKT systems in potential reduction software for large-scale quadratic problems, Computational Optimization and Applications, v.38 n.1, p.27-45, September 2007 Silvia Bonettini , Emanuele Galligani , Valeria Ruggiero, Inner solvers for interior point methods for large scale nonlinear programming, Computational Optimization and Applications, v.37 n.1, p.1-34, May 2007

indefinite matrices;preconditioning;krylov subspace methods

354647

A bandwidth analysis of reliable multicast transport protocols.

Multicast is an efficient communication technique to save bandwidth for group communication purposes. A number of protocols have been proposed in the past to provide a reliable multicast service. Briefly classified, they can be distinguished into sender-initiated, receiver-initiated and tree-based approaches. In this paper, an analytical bandwidth evaluation of generic reliable multicast protocols is presented. Of particular importance are new classes with aggregated acknowledgments. In contrast to other approaches, these classes provide reliability not only in case of message loss but also in case of node failures. Our analysis is based on a realistic system model, including data packet and control packet loss, asynchronous local clocks and imperfect scope-limited local groups. Our results show that hierarchical approaches are superior. They provide higher throughput as well as lower bandwidth consumption. Relating to protocols with aggregated acknowledgments, the analysis shows only little additional bandwidth overhead and therefore high throughput rates.

INTRODUCTION A number of reliable multicast transport protocols have been proposed in the literature, which are based on the acknowledgment scheme. Reliability is ensured by replying acknowledgment messages from the receivers to the sender, either Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Second International Workshop on Networked Group Communication '00, Palo Alto, CA USA Copyright 2000 ACM 0-89791-88-6/97/05 .$5.00 to conrm correct data packet delivery or to ask for a re- transmission. Reliable multicast protocols are usually clas- sied into sender-initiated, receiver-initiated and tree-based ones. Brie y characterized, in sender-initiated approaches receivers reply positive acknowledgments (ACKs) to conrm correct message delivery in contrast to receiver-initiated protocols, which indicate transmission errors or losses by negative acknowledgments (NAKs). Both classes can result in an overwhelming of the sender and the network around the sender by a large number of ACK or NAK messages. This problem is the well-known acknowledgment implosion problem, which is a vital challenge for the design of reliable multicast protocols, since it limits the scalability for large receiver groups. Tree-based approaches promise to be scalable even for a large number of receivers, since they arrange receivers into a hierarchy, called ACK tree [10]. Leaf node receivers send their positive or negative acknowledgments to their parent node in the ACK tree. Each non-leaf receiver is responsible for collecting ACKs or NAKs only from their direct child nodes in the hierarchy. Since the maximum number of child nodes is limited, no node is overwhelmed with messages and scalability for a large receiver group is ensured. The maximum number of child nodes can be determined according to the processing performance of a node, its available network bandwidth, its memory equip- ment, and its reliability. In this paper we present a throughput analysis based on bandwidth requirements as well as the overall bandwidth consumption of all group members, which refer to the data transfer costs. One characteristic of multicast transmissions is that the component with the weakest performance may determine the transmission speed. This means, a group member with a low bandwidth connection, low processing power, high packet loss rate or high packet delay may prevent high transmission rates. Therefore, it is very useful to be able to quantify the necessary requirements for a given multicast protocol. The remainder of this paper is structured as follows. In Section 2 we discuss the background of our throughput analysis and take a look at related work. In Section 3 we brie y classify the analyzed protocols. Our bandwidth evaluation in Section 4 starts with a denition of the assumed system model before the various protocol classes are analyzed in detail. To illustrate the results, some numerical evaluations are presented in Section 5 before we conclude with a brief summary. 2. RELATED WORK Reliable multicast protocols were already analyzed in previous work. The rst processing requirements analysis of generic reliable multicast protocols was presented by Pingali et al. [8]. They compared the class of sender- and receiver-initiated protocols. Following analytical papers are often based on the model and analytical methods introduced by [8]. Levine et al. [3] have extended the analysis to the class of ring- and tree-based approaches. In Maihofer et al. [5] protocols with aggregated acknowledgments are considered. A bandwidth analysis of generic reliable multicast protocols was done by Kasera et al. [2], Nonnenmacher et al. [6] and Poo et al. [9]. In [2], local recovery techniques are analyzed and compared. The system model is based on a special topology structure consisting of a source link from the sender to the backbone, backbone links and nally tail links from the backbone to the receivers. In [6] a similar topology structure is used. They studied the performance gain of protocols using parity packets to recover from transmission errors. The protocols use receiver-based loss detection with multicasted NAKs and NAK avoidance. In [9], non-hierarchical protocols are compared. In contrast to previous work, not only stop-and-wait error recovery is considered in the analysis but also go-back-N and selective-repeat schemes. Our paper diers from previous work in the following ways. First, we consider the loss of data packets and control pack- ets. Second, we assume that local clocks are not synchronized which aects the NAK-avoidance scheme (see Section 3.2). NAK-avoidance works less e-ciently with this more realistic assumption. Third, our analysis considers that local groups may not be conned perfectly, so that local data or control packets may reach nodes in other local groups. Finally, our work extends previous analysis by two new tree-based protocol classes. They are based on aggregated ACKs to be able to cope with node failures. 3. CLASSIFICATIONOFRELIABLEMUL- In this section we brie y classify the reliable multicast protocols analyzed in this paper. A more detailed and more general description for some of these classes can be found in [8], [3] and [6]. 3.1 Sender-Initiated Protocols The class of sender-initiated protocols is characterized by positive acknowledgments (ACKs) returned by the receivers to the sender. A missing ACK detects either a lost data packet at the corresponding receiver, a lost ACK packet or a crashed receiver, which cannot be distinguished by the sender. Therefore, a missing ACK packet leads to a data packet retransmission from the sender. We assume that such a retransmission is always sent using multicast. This protocol class will be referred to as (A1). Note that the use of negative acknowledgments, for example to speed up retrans- missions, does not necessarily mean that a protocol is not of class (A1). Important is that positive acknowledgments are necessary, for example to release data from the sender's buer space. An example for a sender-initiated protocol is the Xpress Transport Protocol (XTP) [12]. 3.2 Receiver-Initiated Protocols In contrast to sender-initiated protocols, receiver-initiated protocols return only negative acknowledgments (NAKs) instead of ACKs. As in the sender-initiated protocol class, we assume that retransmissions are sent using multicast. When a receiver detects an error, e.g. by a wrong checksum, a skip in the sequence number or a timeout while waiting for a data packet, a NAK is returned to the sender. Pure receiver-initiated protocols have a non-deterministic characteristic, since the sender is unable to decide when all group members have correctly received a data packet. Receiver-initiated protocols can either send NAKs using unicast or multicast transmission. The protocol class sending unicast NAKs will be called (N1). An example for (N1) is PGM [11]. The approach using multicast NAKs (N2) is known as NAK-avoidance scheme. A receiver that has detected an error sends a multicast NAK provided that it has not already received a NAK for this data packet from another receiver. Thus, in optimum case, only one NAK is received by the sender for each lost data packet. An example for such a protocol is the scalable reliable multicasting protocol (SRM) [1]. 3.3 Tree-Based Protocols Tree-based approaches organize the receivers into a tree structure called ACK tree, which is responsible for collecting acknowledgments and sending retransmissions. We assume that the sender is the root of the tree. If a receiver needs a retransmission, the parent node in the ACK tree is informed rather than the sender. The parent nodes are called group leaders for their children which form a local group. Note that a group leader may also be a child of another local group. A child which is only a receiver rather than a group leader is called leaf node. The rst considered scheme of this class (H1) is similar to sender-initiated protocols since it uses ACKs sent by the receivers to their group leaders to indicate correctly received packets. Each group leader that is not the root node also sends an ACK to its parent group leader until the root node is reached. If a timeout for an ACK occurs at a group leader or the root, a multicast retransmission is invoked. An example of a protocol similar to our denition of (H1) is RMTP [7]. The second scheme (H2) is based on NAKs with NAK suppression similar to (N2) and selective ACKs (SAKs), which are sent periodically for deciding deterministically when packets can be removed from memory. A SAK is sent to the parent node after a certain number of packets are received or after a certain time period has expired, to propagate the state of a receiver to its group leader. TMTP [14] is an example for class (H2). Before the next scheme will be introduced, it is necessary to understand that (H1) and (H2) can guarantee reliable delivery only if no group member fails in the system. Assume for example that a group leader G1 fails after it has acknowledged correct reception of a packet to its group leader G0 which is the root node. If a receiver of G1 's local group needs a retransmission, neither G1 nor G0 can resend the data packet since G1 has failed and G0 has removed the packet from memory. This problem is solved by aggregated hierarchical ACKs (AAKs) of the third scheme (H3). A group leader sends an AAK to its parent group leader after all children have acknowledged correct reception. After a group leader or the root node has received an AAK, it can remove the corresponding data from memory because all members in this subhierarchy have already received it cor- rectly. Lorax [4] and RMTP-II [13] are examples for AAK protocols. Our denition of (H3)'s generic behavior is as follows: 1. Group leaders send a local ACK after the data packet is received correctly. 2. Leaf node receivers send an AAK after the data packet is received correctly. 3. The root node and group leaders wait a certain time to receive local ACKs from their children. If a timeout occurs, the packet is retransmitted to all children or selective to those whose ACK is missing. Since leaf node receivers send only AAKs rather than local ACKs, a received AAK from a receiver is also allowed to prevent the retransmission. 4. The root node and group leaders wait to receive AAKs from their children. Upon reception of all AAKs, the corresponding packet can be removed from memory and a group leader sends an AAK to its parent group leader. If a timeout occurs while waiting for AAKs, a unicast AAK query is sent to the aected nodes. 5. If a group leader or leaf node receiver receives further retransmissions after an AAK has been sent or the prerequisites for sending an AAK are met, these data packets are acknowledged by AAKs rather than ACKs. The same applies for receiving an AAK query that is replied with an AAK if the prerequisites are met. In summary, ACKs are used for fast error recovery in case of message loss and AAKs to clear buer space. Besides the AAK scheme, we consider in our analysis of (H3) a threshold scheme to decide whether a retransmission is performed using unicast or multicast. The sender or group leader compares the number of missing ACKs with a threshold pa- rameter. If the number of missing ACKs is smaller than this threshold, the data packets are retransmitted using uni- cast. Otherwise, if the number of missing ACKs exceeds the threshold, the overall network and node load is assumed to be lower using multicast retransmission. Our next protocol will be denoted as (H4) and is a combination of the negative acknowledgment with NAK suppression scheme (H2) and aggregated acknowledgments (H3). Similar to (H2), NAKs are used to start a retransmission. Instead of selective periodical ACKs, aggregated ACKs are used to announce the receivers' state and allow group leaders and the sender to remove data from memory. Like SAKs, we assume that AAKs are sent periodically. We dene the generic behavior of (H4) as follows: 1. Upon detection of a missing or corrupted data packet, receivers send a NAK per multicast scheduled at a random time in the future and provided that not already a NAK for this data packet is received before the scheduled time. If no retransmission arrives within a certain time period, the NAK sending scheme is repeated. 2. Group leaders and the sender retransmit a packet per multicast if a NAK has been received. 3. After a certain number of correctly received data pack- ets, leaf node receivers send an AAK to its group leader in the ACK tree. A group leader forwards this AAK to its parent group leader or sender, respectively, as soon as the same data packets are correctly received and the corresponding AAKs from all child nodes are received. 4. The sender and group leaders initiate a timer to wait for all AAKs to be received. If the timer expires, an AAK query is sent to those child nodes whose AAK is missing. 5. If a group leader or leaf node receiver gets an AAK query and the prerequisites for sending an AAK are met, the query is acknowledged with an AAK. 4. BANDWIDTH ANALYSIS 4.1 Model Our model is similar to the one used by Pingali et al. [8] and Levine et al. [3]. A single sender is assumed, multicasting to R identical receivers. In case of tree-based protocols, the sender is the root of the ACK tree. We assume that nodes do not fail and the network is not partitioned, i.e. that retransmissions are nally successful. In contrast to previous work, packet loss can occur on both, data packets and control packets. Multicast packet loss probability is given by q and unicast packet loss probability by p for any node. Table 1 summarizes the notations for protocol classes (A1), (N1), (N2), (H1) and (H2). In Table 2 the additional notations for the protocol classes (H3) and (H4) are given. We assume that losses at dierent nodes are independent events. In fact, since receivers share parts of the multicast routing tree, this assumption does not hold in real networks. However, if all classes have similar trees no protocol class is privileged relative to another one by this assumption. In the following subsections, the generic protocol classes are analyzed in detail. Although our work considers more protocols and a more general system model, the notations and basic analyzing methods follows [8] and [3]. 4.2 Sender-Initiated Protocol (A1) We determine the bandwidth requirements at the sender S and each receiver W A1 R , based on the necessary band-width for sending a single data packet correctly to all re- ceivers. We assume that the sender waits until all ACKs are received and then sends a retransmission if necessary. The bandwidth consumption at the sender is: (receiving ACKs) e Wa (i) (1) E(W E( e Wd (m) and Wa(i) are the bandwidths required for a data packet or ACK packet for the m-th or i-th transmission, Table 1: Notations for the analysis of (A1), (N1), (N2), (H1) and (H2) R Size of the receiver set. Branching factor of a tree or the local group size. Bandwidth for a data packet, ACK, NAK and SAK, respectively. Bandwidth requirements for protocols w at the sender, receiver, group leader and overall bandwidth consumption. Throughput respectively relative bandwidth efciency for protocols w at the sender, receiver, group leader and overall system throughput. S Number of periodical SAKs received by the sender in the presence of control message loss. Probability for unicast or multicast data loss at a receiver, respectively. Probability for unicast ACK, NAK or multicast loss. ~ Probability that a retransmission is necessary for protocol (A1) or (N2), respectively. ps Probability for simultaneous and therefore un-necessary NAK sending in (N2), (H2) and (H4). l Probability for receiving a data or control packet from another local group. L w Number of ACKs or NAKs per data packet sent by receiver r that reach the sender or total number of ACKs or NAKs per data packet received from all receivers. r , N w Total number of transmissions per data packet received by receiver r from the parent node or total number of received data packets from all local groups, respectively. Total number of transmissions per data packet received by a group leader from all local groups. w Number of necessary transmissions for receiver r, to receive a data packet correctly in the presence of data and ACK or NAK loss or total number of transmissions for all receivers. O Number of necessary rounds to correctly deliver a packet to all receivers or to receiver r. O Total number of empty rounds or empty rounds for receiver r, respectively. k Number of NAKs sent in round k. respectively. M A1 is the total number of transmissions necessary to transmit a packet correctly to all receivers in the presence of data packet and ACK loss and e L A1 is the total number of ACKs received for this data packet. E(W A1 S ) is the expectation of the bandwidth requirement at the sender. The only unknowns are E(M A1 ) and E( e E( e L This means, the sender gets one ACK per data packet transmission E(M A1 ) from every receiver R, provided that the data packet is not lost with probability (1 qD ) and the ACK is not lost with probability (1 pA ). Now, the number of transmissions have to be analyzed. The probability for a retransmission is: i. e. either a data packet is lost (qD ) or the data packet is received correctly and the ACK is lost ((1 qD )pA ). So, the probability that the number of necessary transmissions r for receiver r is smaller or equal to m (m=1, 2, .) is: As the packet losses at dierent receivers are independent from each other: R Y R R R R R ~ R E(M A1 ) is the expected total number of necessary transmissions to receive the data packet correctly at all receivers. Now E(W A1 S ) is entirely determined. The bandwidth efciency respectively maximum throughput for the sender S , to send data packets successfully to a receiver is: Accordingly, the processing requirement for a packet at the receiver is: (sending ACKs) (10) E(W where a packet is received with probability (1 qD ) and each received packet is acknowledged. The maximum throughput of a receiver is: R =E(W A1 system throughput A1 is determined by the minimum of the throughput rates at the sender and receivers: R g: Now we are able to determine the total bandwidth consump- tion. In contrast to previous work, our denition of total bandwidth consumption is the bandwidth that is necessary at the sender and receivers to send and receive messages. This means, we assume that the internal network structure is not known and therefore not considered in the analysis. In [2] and [6], total bandwidth is dened on a per link basis. Such a denition encompasses the total costs within the net-work but has the disadvantage, that a network topology has to be dened with routers and links between routers. Here, we want to determine the total costs at the communication endpoints, i.e. the costs for the sender and receivers. The total bandwidth consumption of protocol (A1) is then the sum of the sender's and receivers' bandwidth consumptions E(W R 4.3 Receiver-Initiated Protocol (N1) As in the sender-initiated protocol, data packets are always transmitted using multicast. In (N1), error control is realized by unicast NAKs. The sender collects all NAKs received within a certain timeout period and sends only one retransmission independent of the number of received or lost control packets during that round. The bandwidth requirement at the sender is: (receiving NAKs) e Wn (i) (15) E(W N1 E( e The only unknowns are E(M N1 ) and E( e L N1 ). The number of transmissions, M N1 , until all receivers correctly receive a packet is only determined by the probability for data packet loss analogous to Eq. 8 of (A1) with qD instead of ~ p. To determine E( e steps have to be done. First, the number of transmissions, M N1 r , for a single receiver is given by the probability qD . This means, M N1 r counts the number of trials until the rst success occurs. The probability for the rst success in a Bernoulli experiment at trial k with probability for success (1 qD ) is: The necessary number of transmissions for a single receiver r follows from the Bernoulli distribution and [8]: E(M N1 E(M N1 r jM N1 E(M N1 r jM N1 r > r > 1)[E(M N1 r jM N1 Besides the necessary number of transmissions, we have to introduce the number of rounds, necessary to correctly deliver a data packet. A round starts with the sending of a data packet and ends with the expiration of a timeout at the sender. Normally, there will be one data transmission in each round. However, if the sender receives no NAKs due to NAK loss, no retransmission is made and new NAKs must be sent by the receivers in the next round. O N1 r is the number of necessary rounds for receiver r. The number of rounds is the sum of the number of necessary rounds for sending transmissions M N1 r and the number of empty rounds O N1 e;r in which all NAKs are lost and therefore no retransmission is made: O N1 r +O N1 E(M N1 r ) is given in Eq. 18. E(O N1 e;r ) can be determined analogous to E(M N1 r ), with probability pk for the loss of all sent NAKs in round k (see Eq. 25). The expected number of empty rounds E(O N1 e;r ) is the expected number of empty rounds after the rst transmission plus the expected number of empty rounds after the second transmission and so on: E(O N1 is the expectation for the number of empty rounds plus the last successful NAK reception at the sender which is subtracted. Nk , the number of NAKs sent in round k is given by: where qD k is the probability for a single receiver that until round k all data packets are lost. The number of empty rounds after transmission k is determined by the failure probability: pk is the probability that all sent NAKs in round k are lost. The number of sent NAKs is equal to the number of receivers qD k R that need a retransmission in round k (see Eq. 24). e L N1 is the number of NAKs received by the sender and #1 is the total number of NAKs sent in all rounds: E( e Finally, at the receiver we have: E(W N1 r > 1)[E(O N1 r jO N1 Note that the last, successful transmission is not replied with a NAK. The throughput rates are analogous to (A1): N1 R =E(W N1 R g: The total bandwidth consumption is: E(W R 4.4 Receiver-Initiated Protocol (N2) In contrast to (N1), this protocol class sends NAKs to all group members using multicast. Ideally, NAK suppression ensures that only one NAK is received by the sender. As in the previous protocol, the sender collects all NAKs belonging to one round and then starts a retransmission: (receiving NAKs) e L N2 Wn (i) (31) E(W N2 E( e E(M N2 ) is determined analogous to (A1) and (N1) with loss probability qD (see Eq. 8). e contains the number of necessary and additional NAKs received at the sender: E( e Nk , the number of NAKs sent in round k, is the sum of: NAK of the rst receiver that did not receive the data packet plus NAK of another unsuccessful receiver that did not receive the rst NAK packet and sends a second NAK and so on: The rst receiver sends a NAK provided that the data packet was lost with probability qD k . The second receiver sends a NAK provided that the data packet was lost and the NAK of the rst receiver was lost (Nk;1qN ) or the rst receiver sends no NAK (1 Nk;1 ), and so on. In Eq. 38, a perfect system model is assumed in which additional NAKs are only sent due to NAK loss at receivers. This means, receivers must have synchronized local clocks and a dened sending order for NAKs. However, since receivers are usually not synchronized in real systems it can occur that NAKs are sent simultaneously. Therefore, we extend Equations 36-38 with the probability for simultaneous sending (ps) to: ps qN ps The number of rounds O N2 r for receiver r is obtained analogous to protocol (N1). It is the sum of the number of necessary rounds for sending transmissions M N2 r and the number of empty rounds O N2 e;r in which all NAKs are lost and therefore no retransmission is made. The total number of rounds O N2 for all receivers can be dened analogous to O N2 O N2 r +O N2 e;r (41) O The number of necessary transmissions, M N2 r , for a single receiver r is given by the probability qD . Analogous to Eq. of protocol (N1) the expectation is: E(M N2 The number of empty rounds after transmission k is determined by the failure probability: pk is the probability that all sent NAKs in round k are lost. The expected number of empty rounds E(O N2 e ) is equal to the expected number of empty rounds after the rst transmission plus the expected number of empty rounds after the second transmission and so on. Now, E(O N2 e ) and E(O N2 can be determined analogous to M N2 r (see Eq. E(O E(O N2 is the expectation for the number of empty rounds plus the last successful NAK reception at the sender, which is subtracted. At the receiver we have: E(W N2 r > 1)[E(O N2 r jO N2 r > 1)[E(O N2 r jO N2 #2 is the average number of NAKs sent in each round and #3 is the mean number of receivers that did not receive a data packet and therefore want to send a NAK: where (1=1 pk ) is the number of empty rounds plus the last successful NAK sending (see Eq. 18, 45 and 46). The second term in E(W N2 R ) is the processing requirement to send NAKs, where the considered receiver r is only with probability #2/#3 the one that sends a NAK. In the third term the number of sent NAKs is subtracted from the number of total NAKs to get the number of received NAKs. The throughput rates are: N2 R =E(W N2 R g: (50) The total bandwidth consumption is: E(W R 4.5 Tree-Based Protocol (H1) Our analysis distinguishes between the three dierent kinds of nodes in the ACK tree, the sender at the root of the tree, the receivers that form the leaves of the ACK tree and the receivers that are inner nodes. We will call these inner receivers group leaders. Group leaders are sender and receiver as well. Our analysis of all tree-based protocols is based on the assumption that each local group consists of exactly B members and one group leader. We assume further, that when a group leader has to sent a retransmission, the group leader has already received this packet correctly. The following subsections analyze the bandwidth requirements at the sender W H1 S , receivers W H1 R and group leaders W H1 H . 4.5.1 Sender (root node) e Wa (i) (52) E(W H1 E( e M H1 is the number of necessary transmissions until all members of a local group have received a packet correctly. E(M H1 ) is determined analogous (B instead of R) to Eq. 8 of protocol (A1), since every local group is like a sender-based system. Furthermore, the number of ACKs received by the sender and group leaders in the presence of possible ACK loss E( e similar to E( e of R: E( e L H1 4.5.2 Receiver (leaf node) E(N H1 r;t ) is the total number of received transmissions at receiver r and consists mainly of the sent messages from the parent E(N H1 r ), provided that each local group has its own multicast address. However, if the whole multicast group has only one multicast address, retransmissions may reach members outside of this local group. The probability for receiving a retransmission from another local group is assumed to be p l for any receiver. Such received transmissions from other local groups increase the load of a node. In our analysis we assume that transmissions from other local groups do not decrease the necessary number of local retransmis- sions, since in many cases they are received after a local retransmission have already been triggered. First we want to determine the number of group leaders. The number of nodes R in a complete tree with branching factor B and height h is: and the tree height follows to: The number of group leaders plus the sender is therefore: The number of received transmissions E(N H1 r ) from the parent node at receiver r is: E(N H1 The total number of received transmissions E(N H1 r;t ) at receiver r is now: E(N H1 r;t Finally, the bandwidth requirement W H1 R for a receiver is: r Wa (j) (60) E(W H1 r )E(Wa 4.5.3 Group leader (inner node) Since a group leader is a sender and receiver as well, the bandwidth requirement is the sum of the sender and receiver bandwidth requirements. However, Wd(1) is not considered here, since the initial transmission is sent using the multi-cast routing tree rather than the ACK tree. Furthermore, a group leader may receive additional retransmissions only from G 2 group leaders, since its parents group leader and this group leader itself have to be subtracted. e Wa | {z } as sender r | {z } as receiver E(N H1 E(W H1 E( e E(N H1 E(Wd (1)) E(M H1 )(1 qD )p l E(Wd The maximum throughput rates H1 R , H1 H for the sender, receiver and group leader are: H1 R =E(W H1 system throughput H1 is given by the minimum of the throughput rates for the sender, receiver and group leader: R g: (66) The total bandwidth consumption of protocol (H1) is then the sum of the sender's, leaf node receivers' and group lead- ers' bandwidth consumptions: (R G+ 1)W H1 4.6 Tree-Based Protocol (H2) uses selective periodical ACKs (SAKs) and NAKs with avoidance. The sender and group leaders collect all NAKs belonging to one round and send a retransmission if the waiting time has expired and at least one NAK has been received. We have to distinguish between the number of rounds and the number of transmissions. The number of rounds is equal or greater than the number of retransmis- sions, since if a sender or receiver receives no NAK within one round, no retransmission is invoked. A SAK is sent by the receiver to announce its state, i.e. its received and missed packets, after a sequence of data packets have been received. We assume that a SAK is sent after a certain period of time. Therefore, when analyzing the processing requirements for a single packet, only the proportionate requirements for sending and receiving a SAK (W ) is considered. S is assumed to be the number of SAKs received by the sender in the presence of possible SAK loss: 4.6.1 Sender (root node) e E(W H2 E( e E(M H2 ) is determined analogous to protocol (N2) (B instead of R). To determine E( e that NAKs are received from the child nodes of this local group as well as may be received from other local groups with probability p l (see Eq. 33, 34 and 59): E( e Nk , the number of NAKs sent in round k and pk , the failure probability for empty rounds are obtained analogous to Eq. 44 of (N2). G, the number of group leaders is obtained analogous to Eq. 57 of (H1). 4.6.2 Receiver (leaf node) Retransmissions are received mainly from the parent node, but may also be received from other group leaders. Analo- gous, NAKs are mainly received from other receivers in the same local group but may also be received from receivers in other local groups. The bandwidth requirement for a receiver is analogous to Eq. 47 of protocol (N2): E(W H2 r > 1)[E(O H2 r jO H2 r > 1)[E(O H2 r jO H2 | {z } from this local group | {z } from other local groups #2 and #3 can be obtained analogous to Eq. 48 and 49 of protocol (N2) with B instead of R. 4.6.3 Group leader (inner node) As the group leader role contains the sender role and the receiver role as well, the processing requirements are: E(W H2 l E(M H2 )(1 qD )E(Wd ) The second and third line in the above equation are the processing requirements for one other local group. They have to be subtracted because in contrast to the sender or receivers, a group leader has a local parent group and local child group which are already considered for the normal operations. Finally, the maximum throughput rates are: H2 R =E(W H2 R g: (75) The total bandwidth consumption of protocol (H2) is: (R G+ 1)W H2 4.7 Tree-Based Protocol (H3) We assume that the correct transmission of a data packet consists of two phases. In the rst phase, the data is transmitted and ACKs are collected until all ACKs are received, i.e. until all nodes have received the data packet. Then the second phase starts, in which the missing AAKs are col- lected. Note that most AAKs are already received in phase one, since AAKs are sent from group leaders as soon as all children have sent their AAKs. In this case, a retransmission is acknowledged with an AAK rather than an ACK. So, only nodes whose AAK is missing must be queried in phase two. Table 2: Additional notations for (H3) and (H4) Bandwidth for an AAK or AAK query packet. bandwidth for a periodical AAK or AAK query packet, respectively. Bandwidth to send a data packet per unicast or multicast, respectively. pq Probability for AAK query loss. AA Probability of a unicast AAK loss. Current number of receivers that need a re-transmission Threshold for unicast retransmission. If n k is smaller than , unicast is used for retransmission and multicast otherwise. Probability that n k is smaller than the threshold for multicast retransmissions and therefore unicast is used. Probability that a retransmission is necessary due to data or ACK loss. Probability that an AAK query fails. Mean number of sent unicast messages per packet retransmission. m Number of necessary unicast or multicast transmissions in the presence of failures, respectively aa Number of sent ACKs or AAKs. Number of received ACKs or AAKs. Number of sent AAK queries. e L w aaq Number of received AAK queries. Baa Number of receivers in a local group from which the AAK is missing when phase two starts. pc Probability that no AAK can be sent due to missing AAKs of child nodes. 4.7.1 Sender (root node) The bandwidth requirement of a sender is: e L H3 a e Waa u are the number of necessary multicast or unicast transmissions, respectively. Wd;m and W d;u determine the bandwidth requirements for a multicast or unicast packet transmission. Waa is the necessary bandwidth for an AAK and e L H3 aa is the number of received AAKs. The processing of AAKs is similar to the processing of data packets and ACKs. If AAKs are missing after a timeout has oc- curred, the sender or group leader sends unicast AAK query messages (Waaq ) to the corresponding child nodes. Note that this processing is started after all ACKs have been received and no further retransmissions due to lost data packets are necessary. L H3 aaq is the number of necessary unicast AAK queries in the presence of message loss. that unicast is used for retransmissions, the number of unicast and multicast transmissions are: Please note that the rst transmission is always sent with multicast. The probability for a retransmission due to data or ACK loss is given by: unicast z }| { multicast z }| { | {z } data loss unicast z }| { multicast z }| { pA | {z } no data loss but ACK loss E(M H3 ) is determined by instead of ~ instead of R analogous to Eq. 8 of protocol (A1). is the threshold for unicast or multicast retransmissions. If the current number of nodes nk , which need a retransmission is smaller than the threshold , then unicast is used for the retransmission. p t is the probability that the current number of nodes nk is smaller than the threshold : is used to obtain M H3 , p t can only be determined . In this case, parameter p t is unnecessary to determine M H3 . Nu is the mean number of receivers per round for which a unicast retransmission is invoked: E(N H3 r ) is the total number of transmissions that reach receiver r with unicast and multicast from its parent node in the ACK tree: E(N H3 The number of ACKs that reach the sender or group leader in the presence of ACK loss is given by: E( e a pc is the probability that no AAK can be sent due to missing AAKs of child nodes. The number of AAK query rounds L1 , is determined by the probability ^ p that a query fails: E(L1 ) can be determined analogous to M A1 of protocol (A1) (see Eq. 8) with Baa instead of R and p instead of ~ p. Baa is the number of receivers, the sender has to query when the rst AAK timeout occurs, which is equal to the number of receivers that have not already successfully sent an AAK in the rst phase: Baa X Baa E(N H3 r is the probability that no AAK can be sent in a round or that the AAK is lost. Queries are sent with unicast to the nodes whose AAK is missing. The total number of queries in all rounds are: E(L H3 The number of AAKs received at the sender is the number of AAKs in the retransmission phase plus the number of AAKs in the AAK query phase, which is exactly one AAK from every receiver in Baa (see Eq. 84). E( e S ) is entirely determined by: E(W H3 E( e a E( e aa )E(Waa 4.7.2 Receiver (leaf node) The bandwidth requirement at the receiver is given by: a Wa (j) aa e L H3 aaq Waa (l) +Waaq (l) r;t is the total number of transmission that reach receiver r. In contrast to the already obtained N H3 r , additional data retransmissions are considered from other local groups that may be received with probability E(N H3 r The number of transmissions that are acknowledged with an ACK, L H3 a , or with an AAK, L H3 aa , are: r Here we assume that only transmissions from this local group are acknowledged. e the number of AAK queries received by an receiver are: e aaq =Baa E(L H3 where 1=Baa is the probability to be a receiver that gets an AAK query. Finally, the expectation for a receiver's band-width requirements is: E(W H3 a E( e 4.7.3 Group leader (inner node) The bandwidth requirement at a group leader consists of the sender and receiver bandwidth requirements (see Eq. 64): E(W H3 E(Wd;m (1)) E(N H3 r )p l E(Wd Finally, the maximum throughput rates are: H3 R =E(W H3 R g: The total bandwidth consumption of protocol (H3) is: (R G+ 1)W H3 4.8 Tree-Based Protocol (H4) The generic denition of protocol class (H4) is given in Section 3.3. As in (H3), the correct transmission of a data packet consists of two phases. In the rst phase, the data is transmitted. If NAKs are received by the sender or group leaders, retransmissions are invoked. We assume that the re-transmission phase is nished before the second phase starts. In this phase AAKs are sent from receivers to their parent in the ACK tree. Missing AAKs are queried per unicast messages by the sender and group leaders. In a NAK-based protocol this is only reasonable if it is done after a certain number of correct data packet transmissions rather than after every transmission. Therefore, the costs for sending and receiving AAKs (Waa; ) as well as the costs for querying AAKs (Waaq; ) can be set to a proportionate cost of the other costs. 4.8.1 Sender (root node) e Wn (j) e aa E(W H4 E( e E(L H4 E( e E(M H4 ) and E( e L H4 ) are determined analogous to protocol (H2). The number of AAK queries is determined by the p that a query fails: The number of query rounds E(L1) can be determined analogous to M A1 of protocol (A1) (see Eq. 8) with Baa instead of R and p instead of ~ p. Baa is the number of receivers, the sender has to query when the rst AAK timeout at the sender occurs. Since receivers send one AAK autonomously after a certain number of successfully receptions, the number of nodes to query in phase 2 is the number of lost AAKs. Baa X Baa The total number of unicast query messages in all rounds are: E(L H4 Using unicast, only those nodes are queried whose AAK is missing. So nally, the number of received AAKs at the sender is equal to the number of child nodes in the ACK tree: E( e 4.8.2 Receiver (leaf node) E(W H4 r > 1)[E(O H4 r jO H4 E( e r > 1)[E(O H4 r jO H4 | {z } from this local group | {z } from other local groups #2 and #3 can be obtained analogous to (N2) with B instead of R. E( e aaq ), the number of received AAK queries and replied AAKs is (see Eq. 95): e aaq =Baa E(L H4 and the number of rounds O H4 is determined analogous to (N2). 4.8.3 Group leader (inner node) As the group leader role contains the sender role and the receiver role as well, the processing requirements are: E(W H4 l E(M H4 )(1 qD )E(Wd ) Finally, the maximum throughput rates at the sender, re- ceiver, group leader and overall throughput are: H4 R =E(W H4 R g: (112) The total bandwidth consumption of protocol (H4) is: (R G+ 1)W H4 5. NUMERICAL RESULTS We examine the relative performance and bandwidth consumption of the analyzed protocols by means of some numerical examples. The mean bandwidth costs are set equal to 1 for data packets (Wd , W d;u , Wd;m ), 0.1 for control packets (Wa , Wn , Waa , Waaq ) and 0.01 for periodical control packets (W , Waa; and Waaq; ). The following graphs show the throughput of the various protocol classes relative to the normalized maximum throughput of 1. Figure 1.a shows the bandwidth limited maximum through-put of the sender-initiated protocol (A1) and the receiver-initiated protocols (N1) and (N2). The loss probability for data packets as well as control packets is 0.01 for the dotted lines and 0.1 for the solid ones. The probability for simultaneous NAK sending in (N2) is set to 0.1. The results in Figure 1.a show, that a protocol based on positive acknowledgments like (A1) is not applicable for large receiver groups, since the large number of ACKs overwhelms the sender. The performance of (N1) and (N2) is much better than (A1)'s performance. Particularly, if packet loss probability is low, only few NAK messages are returned to the sender which improves the performance. (N2) with NAK avoidance scheme provides the best performance of all non-hierarchical approaches. In Figure 1.b, the results for the hierarchical protocol classes (H1), (H2), (H3) and (H4) are shown. The number of child nodes is set equal to 10 for all classes and the probability for receiving packets from other local groups, p l , is set equal to 0.001. (H3) is shown with which corresponds with protocol (H1) except for the additional aggregated ACKs of (H3). means that all retransmission are sent with multicast. All protocol classes experience a throughput degradation with increasing group sizes although the local group size 0,3 0,4 0,6 Number of Receivers Throughput (a) sender and receiver-initiated protocols0,1 0,3 0,4 0,6 Number of Receivers Throughput (b) tree-based protocols Figure 1: Throughput of reliable multicast protocols1000100000 Number of Receivers Bandwidth Consumption (a) sender and receiver-initiated protocols1000100000 Number of Receivers Bandwidth Consumption H1 (p=0,01) (b) tree-based protocols Figure 2: Bandwidth consumption of reliable multicast protocols remains constant. This results from our assumption that a packet is received with probability outside the scope of a local group. With increasing number of receivers, the number of groups increase also and therefore more packets from other local groups are received. Note that if each local group is assigned a separate multicast address for re- transmissions, no packets from other local groups are received and therefore p l has to be set equal to 0. In this case, the throughput of all hierarchical approaches remains constant. The protocols with negative acknowledgments and NAK avoidance provide again the best performance. As it can be further seen in the gure, the additional overhead for periodical aggregated acknowledgments is very low, therefore provides almost the same performance as (H2). In case of (H3), the aggregated acknowledgments are sent after every correct message transmission. Therefore, the performance reduction compared to (H1) is more signicant than between (H4) and (H2). If (H3)'s aggregated ACKs are also sent periodically as in (H4), the performance would be almost the same as (H1)'s performance. This means that the additional costs for providing reliability even in the presence of node failures are small and therefore acceptable for protocol implementations. Due to readability, the result for (H3) with shown in the gure. With only retransmissions for equal or more than 2 nodes are made using multicast and with unicast otherwise. In this case, (H3) provides better performance especially for a large number of receivers. For a packet loss probability of 0.1 the performance is equal to (H1). By comparing Figure 1.a and 1.b it can be seen that tree-based protocols are superior. Their throughput degradation with increasing multicast group size is much smaller compared to non-hierarchical approaches. Furthermore, they are more robust against high packet loss probabilities. The following gure depicts the bandwidth consumption of all analyzed protocols. Figure 2.a shows that the bandwidth consumption of (N2) for small group sizes is below (A1)'s requirements. However, for large group sizes and higher loss probability, (N2)'s bandwidth consumption is 2,5 times the bandwidth consumption of (A1). (N1) provides the lowest bandwidth consumption of the three classes. In Figure 2.b it can be seen that tree-based protocols require less overall bandwidth than non-hierarchical approaches. In tree-based protocols save in most cases about 50% of the bandwidth costs and compared to (N2) up to 85% (please note the logarithmic y-axis). The results for (H3) with are not shown in the gure, which are in any case lower compared to (H3) with In case of a packet loss probability of 0.01, (H3) with only the bandwidth of (H1). If the packet loss probability is higher than 0.01, the bandwidth consumption is even smaller than that of (H1). For example, with packet loss probability 0.1, (H3)'s bandwidth consumption with than 30% below (H1)'s requirements. In contrast to non-hierarchical approaches, i.e. if local group sizes are small, NAK with NAK avoidance protocols require low bandwidth costs. Therefore, protocols (H2) and (H4) provide the lowest bandwidth consumption. 6. We have analyzed the throughput in terms of bandwidth requirements and the overall bandwidth consumption of sender-initiated, receiver-initiated and tree-based multi-cast protocols assuming a realistic system model with data packet loss, control packet loss and asynchronous clocks. Of particular importance are the analyzed protocol classes with aggregated acknowledgments. In contrast to other hierarchical approaches, these classes provide reliability even in the presence of node failures. The results of our numerical examples show that hierarchical approaches are superior. They provide higher through-put and lower overall bandwidth consumption compared to sender-initiated or receiver-initiated protocols. The protocol classes with aggregated acknowledgments lead to only a small throughput decrease and slightly increased overall bandwidth consumption compared to the same classes without aggregated acknowledgments. This means, that the additional costs for providing a reliable multicast service even in the presence of node failures are small and therefore acceptable for reliable multicast protocol implementations. 7. --R A reliable multicast framework for light-weight sessions and application level framing A comparison of server-based and receiver-based local recovery approaches for scalable reliable multicast How bad is reliable multicast without local recovery. Reliable multicast transport protocol (RMTP). A comparison of sender-initiated and receiver-initiated reliable multicast protocols Performance comparison of sender-based and receiver-based reliable multicast protocols PGM reliable transport protocol speci An overview of the reliable multicast transport protocol II. A reliable dissemination protocol for interactive collaborative applications. --TR XTP: the Xpress Transfer Protocol A comparison of sender-initiated and receiver-initiated reliable multicast protocols A reliable dissemination protocol for interactive collaborative applications The case for reliable concurrent multicasting using shared ACK trees A reliable multicast framework for light-weight sessions and application level framing A comparison of reliable multicast protocols --CTR Thorsten Lohmar , Zhaoyi Peng , Petri Mahonen, Performance Evaluation of a File Repair Procedure Based on a Combination of MBMS and Unicast Bearers, Proceedings of the 2006 International Symposium on on World of Wireless, Mobile and Multimedia Networks, p.349-357, June 26-29, 2006

analysis;receiver-initiated;bandwidth;tree-based;reliable multicast;sender-initiated;AAK

354704

Parametric Analysis of Computer Systems.

A general parametric analysis problem which allows the use of parameter variables in both the real-time automata and the specifications is proposed and solved. The analysis algorithm is much simpler and can run more efficiently in average cases than can previous works.

Introduction A successful real-world project management relies on the satisfaction of various timing and nontiming restraints which may compete with each other for resources. Examples of such restraints include timely re- sponses, budget, domestic or international regulations, system configurations, environments, compatibilities, In this work, we define and algorithmically solves the parametric analysis problem of computer systems which allows for the formal description of system behaviors and design requirements with various timing and nontiming parameter variables and asks for a general conditions on all solutions to those parameter variables. The design of our problem was influenced by previous work of Alur et al. [AHV93] and Wang [Wang95] which will be discussed briefly later. Our parametric analysis problem is presented in two parts : an automaton with nontiming parameter variables and a specification with both timing and nontiming parameter variables. The following example is adapted from the railroad crossing example and shows how such a platform can be useful. The popular railroad crossing example consists of a train monitor and gate-controller. In figure 1, we give a parametric version of the automaton descriptions of the monitor and controller respectively. The ovals represent meta-states while arcs represent transitions. By each transition, we label the transition condition and the clocks to be reset to zero on the transition. The global state space can be calculated as the Cartesian- product of local state spaces. The safety requirement is that whenever a train is at the crossing, the gate must be in the D mode (gate is down). The more money you spend on monitor, the more precise you can tell how far away a train is approaching. Suppose we now have two monitor types, one costs 1000 dollars and can tell if a train is coming to the crossing in 290 to 300 seconds; the other type costs 500 and can tell if a train is coming to the crossing in 200 to 350 seconds. We also have two gate-controller types. One costs 900 dollars and can lower the gate in 20 to 50 seconds and skip the U mode (gate is up) when a train is coming to the crossing and the controller is in the R mode (gate-Raising mode). The other type costs 300 dollars and can lower the gate in 100 to 200 seconds and cannot skip the U mode once the controller is in the R mode. Suppose now the design of a rail-road crossing gate-controller is subjected to the budget constraint : the cost of the monitor ($ M ) and that of controller ($ C ) together cannot exceed 1500 dollars. We want to make sure under this constraint, if the safety requirement can still be satisfied. This can be expressed in our logic Monitor U Controller A R Figure 1: Railroad Gate Controller Example D). Here 82 is a modal operator from CTL [CE81, CES86] which means for all computations henceforth, the following statement must be true. k Our system behavior descriptions are given in statically parametric automata (SPA) and our specifications are given in parametric computation tree logic (PCTL). The outcome of our algorithm are Boolean expressions, whose literals are linear inequalities on the parameter variables, and can be further processed with standard techniques like simplex, simulated extract useful design feedback. In the remainder of the introduction, we shall first briefly discuss related work on the subject, and then sketch an outline of the rest of the paper. 1.1 Related work In the earliest development [CE81, CES86], people use finite-state automata to describe system behavior and check to see if they satisfy specification given in branching-time temporal logic CTL. Such a framework is usually called model-checking. A CTL (Computation Tree Logic) formula is composed of binary propositions Boolean operators (:; -), and branching-time modal operators (9U ; 9fl; 8U ; 8fl). 9 means "there exists" a computation. 8 means "for all" computations. U means something is true "until" something else is true. fl means "next state." For example, 9pUq says there exists a computation along which p is true until q is true. Since there is no notion of real-time (clock time), only ordering among events are considered. The following shorthands are generally accepted besides the usual ones in Boolean algebra. 93OE 1 is for 9true U Intuitively 3 means "eventually" while 2 means "henceforth." CTL model-checking has been used to prove the correctness of concurrent systems such as circuits and communication protocols. In 1990, the platform was extended by Alur et al. to Timed CTL (TCTL) model-checking problem to verify dense-time systems equipped with resettable clocks [ACD90]. Alur et al. also solve the problem in the same paper with an innovative state space partitioning scheme. In [CY92], the problems of deciding the earliest and latest times a target state can appear in the computation of a timed automaton was discussed. However, they did not derive the general conditions on parameter variables. In 1993, Alur et al. embark on the reachability problem of real-time automata with parameter variables [AHV93]. Particularly, they have established that in general, the problem has no algorithm when three clocks are compared with parameter variables in the automata [AHV93]. This observation greatly influences the design of our platform. In 1995, Wang propose another platform which extends the TCTL model-checking problem to allow for timing parameter variables in TCTL formulae [Wang95]. His algorithm gives back Boolean conditions whose literals are linear equalities on the timing parameter variables. He also showed that his parametric timing analysis problem is PSPACE-hard while his analysis algorithm is of double-exponential time complexity. Henzinger's HyTech system developed at Cornell also has parametric analysis power[AHV93, HHWT95]. However in their framework, they did not identify a decidable class for the parametric analysis problem and their procedure is not guaranteed to terminate. In comparison, our framework has an algorithm which can generate the semilinear description of the working solutions for the parameter variables. 1.2 Outline Section 2 presents our system behavior description language : the Statically Parametric Automaton (SPA). Section 3 defines Parametric Computation Tree Logic (PCTL) and the Parametric Analysis Problem. Section 4 presents the algorithm, proves its correctness, and analyzes its complexity. Section 5 concludes the paper. We also adopt N and R + as the sets of nonnegative integers and nonnegative reals respectively. Statically parametric automata (SPA) In an SPA, people may combine propositions, timing inequalities on clock readings, and linear inequalities of parameter variables to write the invariance and transition conditions. Such a combination is called a state predicate and is defined formally in the following. Given a set P of atomic propositions, a set C of clocks, and a set H of parameter variables, the syntax of a state predicate j of P , C, and H, has the following syntax rules. a are state predicates. Notationally, we let B(P; C; H) be the set of all state predicates on P , C, and H. Note the parameter variables considered in H are static because their value do not change with time during each computation of an automaton. A state predicate with only a literals is called static. Statically Parametric Automata A Statically Parametric Automaton (SPA) is a tuple (Q; -) with the following restrictions. ffl Q is a finite set of meta-states. is the initial meta-state. ffl P is a set of atomic propositions. ffl C is a set of clocks. ffl H is a set of parameters variables. function that labels each meta-state with a condition true in that meta-state. Q is the set of transitions. defines the set of clocks to be reset during each transition. defines the transition triggering conditions. k An SPA starts execution at its meta-state q 0 . We shall assume that initially, all clocks read zero. In between meta-state transitions, all clocks increment their readings at a uniform rate. The transitions of the SPA may be fired when the triggering conditon is satisfied. With different interpretation to the parameter variables, it may exhibit different behaviors. During a transition from meta-state q i to q j , for each x 2 ae(q reading of x will be reset to zero. There are state predicates with parameter variables on the states as well as transitions. These parameters may also appear in the specifications of the same analysis problem instance. A state s of SPA -) is a mapping from P [ C to ftrue; falseg [ R + such that for each for each x 2 C, s(x) is the set of nonnegative real numbers. k The same SPA may generate different computations under different interpretation of its parameter vari- ables. An interpretation, I, for H is a mapping from N [H to N such that for all c 2 N , -) is said to be interpreted with respect to I, when all state predicates in A have their parameter variables interpreted according to I. Satisfaction of interpreted state predicates by a state predicate j is satisfied by state s under interpretation I, written as s a a Now we are going to define the computation of SPA. For convenience, we adopt the following conventions. An -) is unambiguous iff for all states s, there is at most one q 2 Q such that for some I, s Ambiguous SPA's can be made unambiguous by incorporating meta-state names as propositional conjuncts in the conjunctive normal forms of the -state predicate of each meta-state. For convenience, from now on, we shall only talk about unambiguous SPA's. When we say an SPA, we mean an unambiguous SPA. Given an SPA - ), an interpretation I for H, and a state s, we let s Q be the meta-state in Q such that s there is no meta-state q 2 Q such that s undefined. Given two states s; s 0 , there is a meta-state transition from s to s 0 in A under interpretation I, in symbols are both defined, Also, given a state s and a ffi be the state that agrees with s in every aspect except for all of interpreted SPA Given a state s of SPA -) and an interpretation I, a computation of A starting at s is called an s-run and is a sequence ((s of pairs such that ffl for each t there is an i 2 N such that t i - t; and ffl for each integer i - 1, s Q i is defined and for each real 0 - ffl for each i - 1, A goes from s i to s i+1 because of - a meta-state transition, i.e. t 3 PCTL and parametric analysis problem Parametric Computation Tree Logic (PCTL) is used for specifying the design requirements and is defined with respect to a given SPA. Suppose we are given an SPA OE for A has the following syntax rules. Here j is a state predicate in B(P; C; H), OE 1 and OE 2 are PCTL formulae, and ' is an element in N [ H. Note that the parameter variable subscripts of modal can also be used as parameter variables in SPA. Also we adopt the following standard shorthands for 8true U-' OE 1 , 92-' OE 1 for :83-':OE 1 . With different interpretations, a PCTL formula may impose different requirements. We write in notations s I OE to mean that OE is satisfied at state s in A under interpretation I. The satisfaction relation is defined inductively as follows. ffl If OE is a state predicate, then s I OE iff OE is satisfied by s as a state predicate under I. there are an in A, an i - 1, and a s.t. - for all - for all - for all - for all Given an SPA A, a PCTL formula OE, and an interpretation I for H, we say A is a model of OE under I, written as A j= I OE, iff s I OE for all states s such that s We now formally define our problem. Statically Parametric Analysis Problem Given an SPA A and a specification (PCTL formula) OE, the parametric analysis problem instance for A and OE, denoted as PAP(A, OE), is formally defined as the problem of deriving the general condition of all interpretation I such that A j= I OE. I is called a solution to PAP(A; OE) iff A We will show that such conditions are always expressible as Boolean combinations of linear inequalities of parameter variables. 4 Parametric analysis In this section, we shall develop new data-structures, parametric region graph and conditional path graph, to solve the parametric analysis problem. Parametric region graph is similar to the region graph defined in [ACD90] but it contains parametric information. A region is a subset of the state space in which all states exhibit the same behavior with respect to the given SPA and PCTL formula. Given a parametric analysis problem for A and OE, a modal subformula OE 1 of OE, and the parametric region (v; v) (v; w) Figure 2: Railroad Gate Controller Example graph with region sets V , the conditional path graph for OE 1 is a fully connected graph of V whose arcs are labeled with sets of pairs of the form : (-; T ) where - is a static state predicate and T is an integer set. Conveniently, we call such pairs conditional time expressions (CTE). Alternatively, we can say that the conditional path graph J OE 1 for OE 1 is a mapping from V \Theta V to the power set of CTE's. For a v; v is satisfied by I, then there is a finite s-run of time t ending at an s 0 2 v 0 such that OE 1 is satisfied all the way through the run except at s 0 . In subsection 4.2, we shall show that all our modal formula evaluations can be decomposed to the computation of conditional time expressions. The kernel of this section is a Kleene's closure procedure which computes the conditional path graph. Its computation utilizes the following four types of integer set manipulations. g. g. means the addition of i consecutive T 1 . is the complement of T 1 , i.e., g. It can be shown that all integer sets resulting from such manipulations in our algorithm are semilinear. 1 Semilinear expressions are convenient notations for expressing infinite integer sets constructed regularly. They are also closed under the four manipulations. There are also algorithms to compute the manipulation results. Specifically, we know that all semilinear expressions can be represented as the union of a finite number of sets like a + c . Such a special form is called periodical normal form (PNF). It is not difficult to prove that given operands in PNF, the results of the four manipulations can all be transformed back into PNF. Due to page-limit, we shall skip the details here. The intuition behind our algorithm for computing the conditional path graph is a vertex bypassing scheme. Suppose, we have three regions u; v; w whose connections in the conditional path graph is shown in Figure 2. Then it is clear that by bypassing region v, we realized that J OE 1 should be a superset of (v; w); D ' J OE 1 (v; v)g Our conditional path graph construction algorithm utilizes a Kleene's closure framework to calculate all the arc labels. In subsection 4.1, we kind of extend the regions graph concepts in [ACD90] and define parametric region graph. In subsection 4.2, we define conditional path graph, present algorithm to compute it, and present our labelling algorithm for parametric analysis problem. In subsections 4.3 and 4.4, we briefly prove the seminlinear integer set is expressible as the union of a finite number of integer sets like for some a; algorithm's correctness and analyze its complexity. 4.1 Parametric region graph The brilliant concept of region graphs were originally discussed and used in [ACD90] for verifying dense-time systems. A region graph partitions its system state space into finitely many behavior-equivalent subspaces. Our parametric region graphs extend from Alur et al's region graph and contains information on parameter variable restrictions. Beside parameter variables, our parametric region graphs have an additional clock - which gets reset to zero once its reading reaches one. - is not used in the user-given SPA and is added when we construct the regions for the convenience of parametric timing analysis. It functions as a ticking indicator for evaluating timed modal formulae of PCTL. The reading of - is always between 0 and 1, that is, for every state s, 0 - s(- 1. The timing constants in an SPA A are the integer constants c that appear in conditions such as and x - c in A. The timing constants in a PCTL formula OE are the integer constants c that appear in subformulae like x \Gamma y - c; x - c; 9OE 1 U-c OE 2 , and 8OE 1 U-c OE 2 . Let KA:OE be the largest timing constant used in both A and OE for the given parametric analysis problem instance. For each ffi 2 R + , we define fract(ffi) as the fractional part of ffi, i.e. Regions Given an SPA -) and a PCTL formula OE for A, two states s; s 0 of A, s - =A:OE s 0 (i.e. s and s 0 are equivalent with respect to A and OE) iff the following conditions are met. ffl For each ffl For each x \Gamma y - c used in A or OE, ffl For each ffl For every x; y [s] denotes the equivalent class of A's states, with respect to relation - =A:OE , to which s belongs and it is called a region. k Note because of our assumption of unambiguous SPA's, we know that for all s Using the above definition, parametric region graph is defined as follows. Graph (PR-graph) The Parametric Region Graph (PR-graph) for an SPA -) and a PCTL formula OE is a directed graph such that the vertex set V is the set of all regions and the arc set F consists of the following two types of arcs. ffl An arc (v; v 0 transitions in A. That is, for every s 2 v, there is an s such that s ! s 0 . ffl An arc (v; v 0 ) may be a time arc and represent passage of time in the same meta-state. Formally, for every s 2 v, there is an s 0 2 v 0 such that - there is no - s and - s, and Just as in [Wang95], propositional value-changings within the same meta-states are taken care of automatically For each (v; v 0 ) in F , we let ffl(v; v 0 ) =" if going from states in v to states in v 0 , the reading of - increments from a noninteger to an going from states in v to states in v 0 , the reading of - increments from an integer to a noninteger; otherwise ffl(v; v 0 is an KClosure OE 1 /* It is assumed that for all regions v 2 V , we know the static state predicate condition L OE 1 (v) which makes 1 satisfied at v. */ f (1) For each (v; w) 2 F , if ffl(v; w) =", f (2) else let J OE 1 0)g. (2) for each v 2 V , do f (1) for each u; w (v; w); Table 1: Construction of the conditional path graph integer. Also we conveniently write v Similarly, we let v Q be the meta-state such that for all s 2 v(v Since regions have enough informations to determine the truth values all propositions and clock inequalities used in a parametric analysis problem, we can define the mapping from state predicates to static state predicates through a region. Formally, given a region v and a state predicate j, we write v(j) for the static predicate constructed according to the following rules. ffl v(false) is false. ffl v(p) is true iff 8s 2 c) is false otherwise. c) is false otherwise. a a For convenience, we let h-iv be the region in a PR-graph that agrees with v in every aspect except that for all Given a PCTL formula OE and a path is called a OE-path (OE-cycle) iff there is an interpretation I such that for each 1 - 4.2 Labeling Algorithm To compute the parametric condition for a parametric modal formula like 9OE 1 U-' OE 2 at a region, we can instead decompose the formula into a Boolean combinations of path conditions and then compute those path conditions. For example, suppose under interpretation I, we know there exists a OE 1 -path v 1 Then a sufficient condition for all states in v 1 satisfying 9OE 1 U-' OE 2 is that I(') - 5-v n Now we define our second new data structure : conditional path graph to prepare for the presentation of the algorithm. Conditional path graph Given a region graph of OE, the conditional path graph for OE 1 , denoted as is a mapping from V \Theta V to the power set of conditional time expressions such that for all v; v there is a finite s-run of time t ending at an s 0 2 v 0 such that OE 1 is satisfied all the way through the run except at s 0 . k The procedure for computing J OE 1 () is presented in table 1. Once the conditional path graph has been constructed for using KClosure OE 1 (), we can then turn to the labeling algorithm in table 2 to calculate the parametric conditions for the modal formulas properly containing OE 1 . However, there is still one thing which we should define clearly before presenting our labeling algorithm, that is : "How should we connect the conditional time expressions in the arc labels to parametric conditions ?" Suppose, we want to examine if from v to v 0 , there is a run satisfying the parametric requirement of - '. The condition can be derived as expressions T in PNF and (numerical or variable) parameter ' is calculated according to the following rewriting rules. is a new integer variable never used before. Note since we assume that the operands are in PNF, we do not have to pay attention to the case of +; ; . Table 2 presents the labeling alogrithm for L OE (v). This algorithm maps pairs of vertices and temporal logic formulas to a Boolean combination of linear inequalities with parameter variables as free variables. Also note the labeling algorithm relies on the special case of 92-0 OE j which essentially says there is an infinite computation along which OE j is always true. Also the presentation in table 2 only covers some typical cases. For the remaining cases, please check the appendix. 4.3 Correctness The following lemma establishes the correctness of our labeling algorithm. Given PAP(A; OE), an interpretation I for H, and a vertex v in GA:OE , after executing L OE (v) in our labeling algorithm, I satisfies L OE (v) iff v proof : The proof follows a standard structural induction on OE, which we often saw in related model-checking literature, and very much resembles the one in [Wang95]. Due to page-limit, we shall omit it here. k 4.4 Complexity According to our construction, the number of regions in GA:OE , denoted as jG A:OE j, is at most 3jQj coefficient 3 and constant +1 reflect the introduction of ticking indicator -. The inner loop of KClosure OE 1 will be executed for jG A:OE j 3 times. Each iteration takes time proportional to (v; w)j2 jJ OE 1 (v;v)j . The conditional path graph arc labels, i.e. J OE 1 roughly corresponds to the set of simple paths from u to v, although they utilize the succinct representation of semilinear expressions. Thus according to the complexity analysis in [Wang95], we find that procedure KClosure OE 1 () has complexity doubly exponential to the size of GA:OE , and thus triply exponential to the size of input, assuming constant time for the manipulation of semilinear expressions. We now analyze the complexity of our labeling procedure. In table 2, procedure L OE i () invokes KClosure OE j () at most once. Label(A; OE) invokes L OE i () at most jGA jjOEj times. Thus the complexity of the algorithm is roughly triply exponential to the size of A and OE, since polynomials of exponentialities are still exponential- ities. Finally, PCTL satisfiability problem is undecidable since it is no easier than TCTL satisfiability problem[ACD90]. (1) construct the PR-graph (2) for each v 2 V , recursively compute L OE (v); case (false), L false (v) := false; case (p) where case or y is zero in v, evaluate x \Gamma y - c as in the next case; else x \Gamma y - c is evaluated to the same value as it is in any region u such that (u; v) 2 F . case case ( P a a a case case case (1) KClosure OE j (V; F ); (2) let L 92-0 OE j (v) be W ii W case (1) KClosure OE j (V; F ); (2) let L 9OE j U-' OE k (v) be W ii W case These cases are treated in ways similar to the above case and are left in table ?? which is appended at the end of the paper. case (1) KClosure OE j (V; F ); (2) let L 8OE j U-' OE k (v) be case These cases are treated in ways similar to above case and are left in table ?? which is appended at the end of the paper. Table 2: Labeling algorithm 5 Conclusion With the success of CTL-based techniques in automatic verification for computer systems [Bryant86, BCMDH90, HNSY92], it would be nice if a formal theory appealing to the common practice of real-world projects can be developed. We feel hopeful that the insight and techniques used in this paper can be further applied to help verifying reactive systems in a more natural and productive way. Acknowledgements The authors would like to thank Prof. Tom Henzinger. His suggestion to use dynamic programming to solve timing analysis problem triggered the research. --R "Automata, Languages and Programming: Proceedings of the 17th ICALP," "Proceedings, 5th IEEE LICS." "Proceedings, 25th ACM STOC," Symbolic Model Checking: 10 20 States and Beyond "Proceedings, Workshop on Logic of Programs," Automatic Verification of Finite-State Concurrent Systems using Temporal-Logic Specifications "Proceedings, 3rd CAV," the next generation. Symbolic Model Checking for Real-Time Systems "Proceedings, 10th IEEE Symposium on Logic in Computer Science." --TR Automatic verification of finite-state concurrent systems using temporal logic specifications Graph-based algorithms for Boolean function manipulation Automata for modeling real-time systems Parametric real-time reasoning Model-checking in dense real-time Parametric timing analysis for real-time systems Minimum and Maximum Delay Problems in Real-Time Systems Design and Synthesis of Synchronization Skeletons Using Branching-Time Temporal Logic HYTECH --CTR Farn Wang , Hsu-Chun Yen, Reachability solution characterization of parametric real-time systems, Theoretical Computer Science, v.328 n.1-2, p.187-201, 29 November 2004

real-time systems;parameters;verification;model-checking

354705

Modelling IP Mobility.

We study a highly simplified version of the proposed mobility support in version 6 of Internet Protocols (IP). We concentrate on the issue of ensuring that messages to and from mobile agents are delivered without loss of connectivity. We provide three models, of increasingly complex nature, of a network of routers and computing agents that are interconnected via the routers: the first is without mobile agents and is treated as a specification for the next two&semi; the second supports mobile agents, and the third additionally allows correspondent agents to cache the current location of a mobile agent. Following a detailed analysis of the three models to extract invariant properties, we show that the three models are related by a suitable notion of equivalence based on barbed bisimulation. Finally, we report on some experiments in simulating and verifying finite state versions of our model.

Introduction We study the modelling of mobile hosts on a network using a simple process description language, with the intention of being able to prove properties about a protocol for supporting mobility. The present case study grew out of our interest in understanding the essential aspects of some extant mechanisms providing mobility support. Indeed, the model we study may be considered an extreme simplification of proposals for mobility support in version 6 of Internet Protocols (IP) [IDM91, This work was supported under the aegis of IFCPAR 1502-1. An extended abstract of this paper appears in the Proceedings of CONCUR 98. y Corresponding author. CMI (LIM), 39 rue Joliot-Curie, F-13453, Marseille, France. amadio@gyptis.univ-mrs.fr. Partly supported by CTI-CNET 95-1B-182, Action Incitative IN- z IIT Delhi, New Delhi 110016, India. sanjiva@cse.iitd.ernet.in. Partly supported by AICTE 1-52/CD/CA(08)/96-97. TUSM94, JP96] 1 . IPv6 and similar mobile internetworking protocols enable messages to be transparently routed between hosts, even when these hosts may change their location in the network. The architecture of the model underlying these solutions may be described as follows: A network consists of several subnetworks, each interfaced to the rest of the network via a router. Each node has a globally unique permanent identification and a router address for routing messages to it, with a mapping associating a node's identifying name to its current router address. The router associated by default with a node is called its "home router". When a mobile node moves to a different subnet, it registers with a "foreign" router administering that subnet, and arranges for a router in its home subnet to act as a "home" proxy that will forward messages to it at its new "care-of address". Thus any message sent to a node at its home router can eventually get delivered to it at its current care-of address. In addition, a mobile node may inform several correspondent nodes of its current location (router), thus relaxing the necessity of routing messages via its home subnet. This model, being fairly general, also applies to several mobile software architectures. The particular issue we explore here, which is a key property desired of most mobility protocols, is whether messages to and from mobile agents are delivered without loss of connectivity during and after an agent's move. Although IP does not guarantee that messages do not get lost, we model an idealized form of Mobile IPv6 without message loss, since the analysis presented here subsumes that required for Mobile IP with possible loss of messages. We should clarify at the outset that we are not presenting a new architecture for mobility support; nor are we presenting a new framework or calculus for mo- bility. Rather, our work may be classified as protocol modelling and analysis: We take an informal description of an existing protocol, idealize it and abstract away aspects that seem irrelevant to the properties we wish to check or which are details for providing a particular functionality, then make a model of the simplified protocol and apply mathematical techniques to discover the system structure and its behavioral properties. We believe that this approach constitutes a useful way of understanding such protocols, and may assist in the formulation and revision of real-world protocols for mobile systems. From the informal descriptions of mobility protocols in the literature, it is difficult to assure oneself of their correctness. As borne out by our work, the specification and combinatorial analysis of such protocols is too complicated to rely on an informal justification. The literature contains various related and other proposals for mobility sup- port, for example, in descriptions of kernel support for process migration [PM83, runtime systems for migrant code [BKT92, JLHB88]. A significant body of work concerns object mobility support in various object-based software architectures, see, e.g., [Dec86, Piq96, VRHB While these studies address several other issues relevant to software mobility (e.g., garbage collection), 1 We do not model various aspects of network protocols for mobility. In particular, we totally ignore security and authentication issues, as well as representational formats and conventions in network packets, e.g., encapsulation/decapsulation of messages, and tunneling. Broadcast is not dealt with at all. we are not aware of any complete modelling and analysis in those settings that subsumes our work. We have recently learnt of a "light-weight" formal analysis of the IPv6 protocol [JNW97]. In that work, Nitpick, a tool that checks properties of finite binary relations and generates counter-examples, is applied to a finite instance of an abstract version of the IPv6, to verify that messages do not travel indefinitely in cycles 2 . The approach is quite similar in spirit to our work summarized in Appendix B, and uses inductive invariants to verify a cache acyclicity property. This property is one of those that arise in our analysis presented in x4.3. The Nitpick analysis focuses on the particular cache management policy suggested for IPv6 together with timestamping of messages (which requires a protocol for approximating a global clock), whereas our analysis shows that any cache management policy that satisfies a particular invariant will ensure correctness. The literature also contains a number of frameworks for describing mobility protocols, such as the extension of Unity called Mobile Unity [MR97, PRM97, RMP97]. One should distinguish the "analysis of protocols for mobility" from the "definition of models or calculi for mobility"; mobility protocols provide an implementation basis for the latter, just as, e.g., garbage collection algorithms provide an implementation basis for functional programming. While we agree that frameworks including the dynamic generation of names and processes as primitive operations, such as the -calculus and related formalisms [MPW92, AMST97], may be suitable for describing mobile systems, we believe that our work provides some evidence for the assertion that these primitives are not always necessary nor always appropriate for describing and analyzing mobility protocols. Further, there is no need to develop an "ad hoc" (in the original, non-pejorative sense) formalism for analyzing mobility protocols. The structure of the paper reflects our analysis methodology. After introducing our language for modelling the protocol (x2), we present a model of the protocol (x3), and then look for its essential structure in the form of a big invariant (x4). From this analysis, we gather some insight on why the protocol works correctly, and suggest some variations. We present the protocol model in stages, giving three models of increasingly complex nature of a network of routers and computing agents that are interconnected via the routers. In the first (x3.1), which we call Stat, computing agents are not mobile. We extend Stat to a system Mob where agents can move from a router to another (x3.2). Finally, to reduce indirection and to avoid excessive centralization and traffic congestion, we extend the system Mob to a system CMob, where the current router of an agent may be cached by its correspondent agents (x3.3). In x4, we analyze these three different models, and establish the correspondence between them by showing that systems Stat, Mob and CMob are barbed bisimilar, with respect to a suitable notion of observation. We conclude in x5, by summarizing our contributions, recalling the simplifications we have made, and reporting on some simulation and automatic verification experiments on a finite-state formulation of the protocol. The paper need not be read sequentially. By reading x3.1, x3.2 and glancing In IPv6, messages can in fact travel in cycles. Consider the case where a node keeps moving in a ring and a message is always being forwarded to it one step behind. at x4.1, x4.2 the reader will have a general idea of the basic systems Stat and Mob (in particular Figure 4 should provide a good operational intuition) and of the techniques we apply in their analysis. The more challenging system CMob, whose size is about twice the size of Mob, is described in x3.3 and analyzed in x4.3. The reader motivated by the formal analysis, will have to take a closer look at the invariants described in Figures 10, 12. Understanding the invariants is the demanding part, the related proofs (which are in Appendix are basically large case analyses that require little mathematical sophistication. The footnotes comment on the relationship between our model and the IP informal specification. They are addressed mainly to the reader familiar with IP, and they can be skipped at a first reading. Finally, the reader interested in the finite state version of the protocol, can get a general overview of the issues in x5, and more details in Appendix B. 2 The Process Description Language We describe the systems in a standard process description language. The notation we use is intended to be accessible to a general reader, but can be considered an extension of a name-passing process calculus with syntactic sugar. A system consists of some asynchronous processes that interact by exchanging messages over channels with unlimited capacity (thus sending is a non-blocking operation, whereas attempting to receive on an empty channel causes the process to block). Messages in the channels can be reordered in arbitrary ways. Processes are described as a system of parametric equations. The basic action performed by a process in a certain state is: (i) (possibly) receiving a message, (ii) (possibly) performing some internal computation, (iii) (possibly) emitting a multiset of messages, and (iv) going to another state (possibly the same). There are several possible notations for describing these actions; we follow the notation of CCS with value passing. We assume a collection of basic sorts, and allow functions between basic sorts to represent cache memories abstractly. The functions we actually need have a default value almost everywhere and represent finite tables. Let names, values of basic sorts, and functions from basic sorts to sorts. x stands for a tuple x . The expressions T; over name equality tests; are process identifiers, and V; value domains. Processes are typically denoted by are specified by the following grammar: Here, 0 is the terminated process, x(x):p is the input prefix, xx is a message, j is the (asynchronous) parallel composition operator, and [T ]p; p 0 is a case statement. We write \Pi i2I p i to denote an indexed parallel composition of processes. X(x) is a process identifier applied to its actual parameters; as usual for every process identifier X, there is a unique defining equation such that all variables are contained in fxg. \Phi is the internal choice operation, where in p we substitute a non-deterministically chosen tuple of values (from appropriate domains, possibly infinite) for the specified tuple of variables. We use internal choice to abstract from control details (note that internal choice can be defined in CCS as the sum of processes guarded by a - action). Being based on "asynchronous message passing over channels", our process description language could be regarded as a fragment of an Actor language [AMST97] or of an asynchronous (polyadic) -calculus [HT91]. The main feature missing is the dynamic generation of names. As we will see in x3.2, it is possible to foresee the patterns of generation, and thus model the system by a static network of processes. We define a structural equivalence j on processes as the least equivalence relation that includes: ff-renaming of bound names, associativity and commutativity of j, the equation unfolding and: Reduction is up to structural equivalence and is defined by the following rules: These rules represent communication, internal choice and compatibility of reduction with parallel contexts, respectively. Using these rules, we can reduce a process if and only if (i) employing structural equivalence we can bring the process to the form p j q, and (ii) the first or second rule applies to p. We introduce the following abbreviations: if 3 The Model We describe three systems Stat , Mob, and CMob. All three consist of a collection of communicating agents that may interact with one another over the network. Each agent is attached to a router, with possibly many agents attached to the same router. We assume that each entity, agent or router, has a globally unique identifying name. For simplicity, we assume a very elementary functionality for the agents - they can only communicate with other agents, sending or receiving messages via the routers to which they are attached. The agents cannot communicate directly between themselves, all communication being mediated by the routers. We assume that routers may directly communicate with one another, abstracting away the details of message delivery across the network. The communication mechanism we assume is an asynchronous one, involving unbounded buffers and allowing message overtaking. We assume a collection of names defined as the union of pairwise disjoint sets Agent Names The set CN of Control Directives has the following elements (the Control Directives that we consider in Stat consist of exactly msg, to indicate a data message; the directives fwdd and upd will be used only in CMob): msg message regd registered infmd informed fwdd forwarded immig immigrating repat repatriating mig migrating upd update The sets AN and DN are assumed to be non-empty. The elements of RN and LAN are channel names that can carry values of the following domain (note that the sort corresponding to the set RN is recursively defined): AN \Theta RN \Theta AN \Theta RN \Theta DN Elements of this domain may be interpreted as: [control directive; to agent ; at router ; from agent ; from router ; data ] We often write x to stand for the tuple [x indicates that the name is irrelevant ("don't care''). The tables L and H are used for the address translation necessary to route a message to its destination. L is an injective function that gives the local address for an agent at a given router, H computes the "home router" of an agent. Tables We denote with obs(x) an atomic observation. If z[x] is a message, we call the triple [x its observable content (original sender, addressee and data). We assume a distinguished channel name o on which we can observe either the reception of a message or anomalous behavior, represented by a special value ffl. 3.1 The system without mobility Stat In Figure 1, we present (formally) the system Stat . Agents An agent A(a) either receives a message from its home router on its local address and observes it, or it generates a message to a correspondent agent that it gives to its home router for delivery to the correspondent agent via the latter's AN \Theta RN \Theta AN \Theta RN \Theta DN in (a) in r 0 [msg; A in (a) in Router in lx j Router (r) Figure 1: System without mobility home router. L(H(a); a) represents the local address of the agent a in its home subnet 3 . Router The router examines an incoming message, and if it is the destination router mentioned in the message, accordingly delivers it to the corresponding agent. Otherwise it sends it to the appropriate router. L(r; x 2 ) is the local address of x 2 , the addressee of the message, whereas x 3 is the destination router. 3.2 The system with mobility Mob We now allow agents to migrate from one router (i.e., subnet) to another. While doing so, the agents and routers engage in a handover protocol [JP96]. When an agent moves to another router, a proxy "home agent" at its home router 4 forwards messages intended for the mobile agent to a "care-of address" 5 , the agent's current router. To avoid message loss, the forwarding home agent should have an up-to-date idea of the current router of the mobile agent. Hence when a mobile agent moves, it must inform the home agent of its new coordinates. In the first approximation, we model all messages addressed to a mobile agent being 3 We have used nondeterminism to model actions arising from the transport or higher layers corresponding to processing a received message, or generating a message to a correspondent agent. Communication on the LAN channels abstracts link-level communication between the router and the agent. We have a simplifying assumption that a node can be on-link to only one router. 4 For simplicity, we identify the routers serving as mobility agents / proxies with the routers administering a subnet. We also assume that each router is always capable of acting as a home or foreign agent. 5 We model only what are called "foreign agent care-of addresses" and not "co-located care-of addresses" in IPv6 parlance. forwarded via the home agent; later we will consider correspondent agents caching the current router of a mobile agent. The router description remains unchanged. We observe that the migration of a mobile agent from one router to another can be modelled "statically": for each router, for each agent, we have a process that represents the behavior of a mobile agent either being present there or absent there, or that of a router enacting the role of a forwarder for the agent, routing messages addressed to that agent to its current router. Migration may now be described in terms of a coordinated state change by processes at each of the locations involved 6 . Although the model involves a matrix of shadow agents running at each router, it has the advantage of being static, in terms of processes and channels, requiring neither dynamic name generation nor dynamic process generation. The conceptual simplicity of the model is a clear advantage when carrying on proofs which have a high combinatorial complexity, as well as when attempting verification by automated or semi-automated means. For instance, the only aspect of the modelling that brings us outside the realm of finite control systems is the fact that channels have an infinite capacity, and there is no bound on the number of messages generated. Starting from this observation, it is possible to consider a revised protocol which relies on bounded channels (see Appendix B). In the commentary below, we refer to various processes as agents. Note that only the agents Ah, Ah in , Ma and Ma in correspond to "real" agents, i.e., the behavior of mobile nodes. The others may be regarded as roles played by a router on a mobile node's behalf. Their analogues in IPv6 are implemented as routers' procedures that use certain tables. States of the agent at home We describe an agent at its home router in Figure 2. Ah The mobile agent is at its home base. It can receive and send messages, as in the definition of A(a) in Figure 1, and can also move to another router. When the agent "emigrates", say, to router u, it changes state to Ham(a). We model the migration by the agent intimating its "shadow" at router u that it is "immigrating" there, and to prepare to commence operation 7 . Ham The mobile agent during emigration. We model the agent during "emigra- tion" by the state Ham(a). During migration, messages addressed to the agent may continue to arrive; eventually, these messages should be received and handled by the mobile agent. The emigration completes when the shadow agent at the target site registers (by sending control message regd) its new care-of router at the home base. The agent is ready to operate at that foreign subnet once it 6 Thus, our formalization of the migration of an agent, involving the small coordinated state change protocol, may be considered an abstraction (rather than a faithful representation) of some of the actions performed when a mobile node attaches itself to a new router and disengages itself from an old one. 7 Registration is treated in a simple fashion using the immig and repat messages, ignoring details of Agent Discovery, Advertisement, Solicitation, and protocols for obtaining care-of ad- dresses. Deregistration is automatic rather than explicit. The issue of re-registration is totally ignored. in r 0 [msg; in if Ah in in in Haf (a; in Router(r) (as in Figure 1) Figure 2: States of the agent at home receives an acknowledgement from the home agent (control message infmd). The control messages (regd and infmd) are required to model the coordinated change of state at the two sites 8 . The home agent filters messages while waiting for the regd message; this filtration can be expressed in our asynchronous communication model by having other messages "put back" into the message buffer, and remaining in state Ham(a). Haf The home agent as a forwarder. The home agent forwards messages to the mobile agent at its current router 9 (via the routers of course), unless informed by the mobile agent that it is moving from that router. There are two cases we consider: either the mobile agent is coming home ("repatriation") or it is migrating elsewhere. States of the agent away from home We describe the agents at a foreign router in Figure 3. Idle If the agent has never visited. The Idle state captures the behavior of the shadow of an agent at a router it has never visited. If the agent moves to that 8 These messages may be likened to the "binding update for home registration" and "binding acknowledgement from home". 9 This is the primary care-of address. in in in \Phi c2fin;out;mvg;y2AN;w2DN ;u2RN in r[msg; in if if u 6= Ma in (a; in Figure 3: States of the agent away from home router, indicated by the control message immig, then the shadow agent changes state to Bma(a; r), from where it will take on the behavior of mobile agent a at the foreign router r. Any other message is ignored, and indeed it should be erroneous to receive any other message in this state. Fwd If the agent is not at foreign router r, but has been there earlier. This state is similar to Idle , except that any delayed messages that had been routed to the agent while it was at r previously are re-routed via the home router 10 . This state may be compared to Haf , except that it does not have to concern itself with the agent migrating elsewhere. Bma Becoming a foreign mobile agent. Once the protocol for establishing movement to the current router is complete, the agent becomes a foreign mobile agent. Messages are filtered looking for an acknowledgement from the home agent that it is aware of the mobile agent's new current router. Once the home agent has acknowledged that it has noted the new coordinates, the mobile agent may become operational 11 . Ma The mobile agent at a foreign router. As with the mobile agent at its home base Ah(a), the mobile agent may receive messages, send messages, or move away. The behavior of the mobile agent in state Ma is similar to that of Ah except that during movement, different control messages need to be sent to the target site depending on whether it is home or another site. If the target site is the home base, then a repat message is sent. Otherwise the target site is intimated of the wish to "immigrate". The agent goes into the state Fwd . In the upper part of Figure 4 we describe the possible transitions that relate to control messages, not including filtering, forwarding, and erroneous situations. We decorate the transitions with the control messages that are received (-) and emitted (+). In the lower part of Figure 4 we outline the three basic movements of an agent a: leaving the home router, coming back to the home router, and moving between routers different from the home router. messages forwarded by the home agent may get arbitrarily delayed in transit, it is important that the mobile agent, in addition to informing its home agent of its current router, arrange for a forwarder at its prior router to handle such delayed messages. This point is the only major difference between our model and the IPv6 proposal. In order not to lose messages, we require a forwarder at any router where the mobile agent has previously visited. The default target for forwarding is the home router. In the Mobile IPv6 proposal, however, it is not mandatory for the mobile agent to arrange for a forwarder at the previous router, and if a message reaches a router that had previously served as a foreign agent, the message may be dropped. This is permissible in the context of IP since dealing with lost messages is left to the transport and higher layers. Our analysis shows that Mobile IP can use our default policy of forwarding to the home router, without messages traversing cycles indefinitely, but at the cost of some increase in the number of hops for a message. The need for forwarders is, of course, well known in the folklore regarding implementation of process migration. In IPv6 a mobile node may begin operation even before it has registered its new location with the home agent or received an acknowledgement from the home router. Correct updates of the primary care-of address at the home router are achieved using time-stamping of messages, which in turn requires synchronized clocks. In contrast, our asynchronous communication model makes no timeliness assumptions and permits message overtaking. Hence our protocol requires an acknowledgement from home before permitting further migrations. (1) Ah(a) +immig \Gamma! Ham(a) \Gammaregd +infmd \Gamma! Haf (a; r) \Gammaimmig +mig \Gamma! Bma(a; r) \Gammainfmd \Gamma! Ma(a; r) \Gammaimmig +regd \Gamma! Bma(a; r) \Gammainfmd \Gamma! Ma(a; r) \Gamma! Fwd(a; r) +repat \Gamma! Fwd(a; r) (4:1) Haf (a; r) \Gammamig +infmd \Gamma! Haf (a; r 0 ) (4:2) Haf (a; r) \Gammarepat \Gamma! Ah(a) I-Leaving home Ham(a) Idle=Fwd(a; r) immig Ham(a) Bma(a; r) regd Haf (a; r) Bma(a; r) infmd Haf (a; r) Ma(a; r) II-Coming home Ma(a; r) Haf (a; r) Fwd(a; r) Haf (a; r) repat Fwd(a; r) Ah(a) III-Moving between routers different from the home router Ma(a; r) Haf (a; r) Idle=Fwd(a; r 0 ) Figure 4: Control transitions We briefly describe how our "static" description that requires a thread for each agent at each router relates to a more "dynamic" model that is more natural from a programming viewpoint. First we observe that an agent's name is obtained by combining a router's name with a local identifying name. The computation of function H is distributed, in that an agent's name contains sufficient information for computing its home router's name. Further, our infinite name space of agents is a virtual representation of a finite location name space with dynamic generation of names at each location. As observed earlier, the only actual processes are Ah, Ah in , Ma and Ma in , which run in parallel with the routers. The other "agents" are threads run on the router. The Ham thread is spawned on the home router when Ah wishes to move this thread becomes Haf, a thread that forwards messages to the mobile agent and terminates when the agent returns. Each router maintains a list of agents for whom it serves as a home router, with their current locations as well as a list of mobile agents currently visiting. The default policy of a router is to deliver messages to agents actually present there, to forward messages to mobile agents for whom it serves as a home router, and to otherwise forward the message to the target agent's home router. Messages to a non-existent agent trigger an error. As our analysis will show, the only message an Idle thread can receive is an immig message. So this thread need not exist. Instead, on receiving an immig message, the router spawns a Bma thread, updating the list of agents actually present there. When Ma moves away from a router, it notifies the router to spawn a Fwd thread. In practice, the Fwd thread will synchronize with the router to empty the buffer of messages left behind by the agent and then terminate. Following this implementation, the number of threads running at a router r is proportional to the number of agents whose home is r or who are currently visiting r. 3.3 The system with caching CMob The previous system suffers from overcentralization. All traffic to an agent is routed through its home router, thus creating inefficiencies as well as poor fault tolerance. So, correspondent agents can cache the current router of a mobile agent [JP96]. The agents' definitions are parametric in a function f : AN ! RN , which represents their current cache. The cache is used to approximate knowledge of the current location of an agent; this function parameter can be implemented by associating a list with each agent 12 . We now use control directives fwdd and upd; the former indicates that the current data message has been forwarded thus pointing out a "cache miss", the latter suggests an update of a cache entry, following a cache miss 13 . An agent may also decide to reset a cache entry to the home router 14 . Note that the protocol does not require the coherence of the caches. In case of cache miss, we may forward the message either to the home router (which, as in the previous protocol, maintains an up-to-date view of the current router) or to the router to which the agent has moved. We present in Figure 5, the new definitions of the agent at home. Note the use of the directives fwdd and upd to update the cache and to suggest cache updates. In Figures 6,7 we present the modified definitions for the agent away from home. We note the introduction of two extra states: Fwd in (a; To model timing out of cached entries by a forwarder, an extra state Fwd in (a; is introduced. Non-determinism is used in Fwd and Ma in to model possible resetting or updating cache entries 15 . Mam(a; r) is an extra state that we need when an agent moves from a router different from the home router. Before becoming When moving to another router, we could deliver the current cache with the message immig. In the presented version we always re-start with the default cache H. 13 A fwdd message can be regarded as having been tunnelled, while a upd message is a binding update. 14 In IPv6, the validity of a cache entry may expire. In the informal description of the protocol, the update and deletion of a cache entry are often optional operations. We model this by using internal choice. No messages to reset a cache entry (binding deletion updates) are ever sent out, nor are negative acknowledgements sent out. We also note that maintaining the "binding update list" is not essential to the protocol, but is only a pragmatic design choice. Instead, an agent may non-deterministically decide to reset a cache entry, thus abstracting from a particular cache management mechanism. Timing out of cache entries is modelled using non-determinism, rather than by explicit representation of time stamps in a message. Note that in the Mobile IP protocol no hypothesis is made regarding the coordination of the clocks of the agents, so it seems an overkill to introduce time to speak about these time stamps. in r 0 [msg; in if in Ah(a; f [r 0 =y]) Ah in (a; f; c let r in let r in Haf (a; let r in Router(r) (as in Figure 1) Figure 5: Modified control for agent at home with caching in \Phi c2f0;1g let r in Fwd(a; Fwd in (a; in in Figure Modified control for agent away from home with caching, part I a forwarder to the router to which the agent moved, we have to make sure that the agent has arrived there, otherwise we may forward messages to an Idle(a; r 0 ) process, thus producing a run-time error (this situation does not arise in system Mob because we always forward to the home router). 4 Analysis We now analyze the three different systems Stat, Mob, and CMob. In each case, the first step is to provide a schematic description of the reachable configurations, and to show that they satisfy certain desirable properties. Technically, we introduce a notion of admissible configuration, i.e., a configuration with certain properties, and go on to show that the initial configuration is admissible, and that admissibility is preserved by reduction. A crucial property of admissible configurations for Mob and CMob is control stabilization. This means that it is always possible to bring these systems to a situation where all migrations have been completed (we can give precise bounds on the number of steps needed to achieve this). We call these states stable. Other interesting properties we show relate to the integrity and delivery of messages. The Ma(a; in r[msg; in if if in if y Ma in (a; in in Figure 7: Modified control for agent away from home with caching, part II control stabilization property of admissible configurations is also exploited to build (barbed) bisimulation relations, with respect to a suitable notion of observation, between Stat and Mob, and between Stat and CMob. 4.1 Analysis of Stat Figure 8 presents the notion of admissible configuration for Stat. We will write s:Rt, s:Ob, s:Ag, and s:Ms to denote the state of the routers, atomic observa- tions, agents, and data messages, respectively, in configuration s. We will abuse notation, and regard products of messages as multisets, justified since parallel composition is associative and commutative. When working with multisets we will use standard set-theoretic notation, though operations such as union and difference are intended to take multiplicity of the occurrences into account. We assume that ]RN - 3, to avoid considering degenerate cases when es- Ag j A in (a); Figure 8: Admissible configurations for Stat tablishing the correspondence between Stat and Mob (if ]RN - 2, then the transitions III in Figure 4 cannot arise). Proposition 4.1 The initial configuration Stat is admissible, and admissible configurations are closed under reduction. By the definition of admissible configuration, we can conclude that the error message offl is never generated (a similar remark can be made for the systems Mob and CMob, applying theorems 4.5 and 4.13, respectively). Messages do not get lost or tampered with. Corollary 4.2 (message integrity) Let s be an admissible configuration for Stat, let zx 2 s:Ms and suppose s when the message gets received by its intended addressee. Corollary 4.3 (message delivery) Let s be an admissible configuration for Stat such that zx 2 s:Ms. Then the data message can be observed in at most 4 reductions 4.2 Analysis of Mob The table in Figure 9 lists the situations that can arise during the migration of an agent from a router to another. k is the case number, Ag(a; k; denotes the shadow agents of a not in a Fwd or Idle state in situation k, CMs(a; k; the migration protocol control messages at z involving at most the sites H(a); that situation, and R(a; k; denotes the routers involved in situation k of the protocol at which a's shadow is not in a Fwd or Idle state. Relying on this table, we define in Figure 10 a notion of admissible function fl. Intuitively, the function fl associates with each agent a its current migration control (the state and the protocol messages), the routers already visited, and the data messages in transit that are addressed to a. We denote with P fin (X) and M fin (X) the finite parts, and finite multisets of X, respectively, and with fl(a) i the i-th projection of the tuple fl(a). Then Act(a; fl) denotes the routers where a has visited, which are not in an Idle state. Condition states that at most finitely many agents can be on the move ("deranged") at 2 Ah in (a) 0 fr 0 g 3 Ham(a) z[immig; a; 5 Haf (a; r) j Bma(a; r) z[infmd; a; 8 Haf (a; r) j Ma in (a; r) 0 fr 9 Haf (a; r) z[immig; a; r Figure 9: Control migration (r M fin ((RN [ LAN ) \Theta RN \Theta AN \Theta RN \Theta DN ) (data messages) Admissibility conditions on fl: (CMs(a; k; Figure 10: Admissible configurations for Mob any instant. (C 2 ) is a hygiene condition on migration control messages, indicating that they may be at exactly one of three positions, and that if an agent is on the move (cases 7 and 9), the home forwarder always points to an active router, where a proxy agent will return delayed data messages back to a's home; after receiving the pending control message, the home forwarder will deliver the data message to the current (correct) location of the mobile agent. Thus, although there may apparently be forwarding cycles, these will always involve the home forwarder and will be broken immediately on receipt of the pending control message. Condition explicitly indicates where a control message involving a may be. Definition 4.4 (admissible configuration) An admissible configuration for Mob, m, is generated by a pair (fl; Ob) comprising an admissible function and a process as follows: where Rt,Ob are as in Figure 8 and Ag(a; k; Let m be an admissible configuration for Mob, generated by (fl; Ob). Further, let m:DMs(a) denote \Pi (z;r 1 ;a 2 ;r 2 ;d)2fl(a) 6 d], the data messages in state m addressed to a, and let m:DMs denote all data messages in state m. We will write m:CMs and m:Ob to denote the state of the control messages and atomic observations, respectively, in configuration m. Theorem 4.5 The initial configuration Mob is admissible, and admissible configurations are closed under reduction. From this result, it is possible to derive an important property of system Mob: it is always possible to bring the system to a stable state. Corollary 4.6 (control stabilization) Let m be an admissible configuration for Mob generated by (fl; Ob) and let g. that m 0 is determined by (fl In particular, if fl(a) 6g. The analogies of corollaries 4.2 and 4.3 can be stated as follows. Messages neither get lost nor is their observable content tampered with. Corollary 4.7 (message integrity) Let m be an admissible configuration for Mob, generated by (fl; Ob), and suppose either or else there exists a z 0 l x 0 that obs(x 0 l Corollary 4.8 (message delivery) Let a 2 AN and let m be an admissible configuration for Mob generated by (fl; Ob) such that fl(a) 1 2 K s and (z; r 1 ; a fl(a) 6 . Then this data message can be observed in at most 10 reductions. We now introduce a notion of what is observable of a process and a related notion of barbed bisimulation (cf. [Par81, Pnu85]). Definition 4.9 Let p be a process. Then O(p) is the following multiset (y can be ffl): We note that on an admissible configuration s, can be applied to an admissible configuration m or c (cf. following definition 4.12). Definition 4.10 A binary relation on processes R is a barbed bisimulation if whenever p R q then the following conditions hold: and symmetrically. Two processes p; q are barbed bisimilar, written p ffl they are related by a barbed bisimulation. We use the notion of barbed bisimulation to relate the simple system Stat (viewed as a specification) to the more complex systems Mob and CMob. Note that each process p has a unique commitment O(p). Taking as commitments the atomic observations would lead to a strictly weaker equivalence. Theorem 4.11 Stat ffl Mob. Proofhint. We define the relevant observable content of data messages in an admissible configuration s for Stat and m for Mob as the following multisets, respectively: O O Next, we introduce a relation S between admissible configurations for Stat and Mob as: We show that S is a barbed bisimulation. 2 4.3 Analysis of CMob The analysis of CMob follows the pattern presented above for Mob. The statement of the invariant however is considerably more complicated. The table in Figure 11 lists the situations that can arise during the migration of an agent from a router to another. Relying on this table, we define in Figure 12 a notion of admissible function fl. Intuitively, the function fl associates with each agent a its current migration control (state and protocol messages), the routers already visited (either Fwd's or Mam's), and the data messages and update messages in transit which are addressed to a. 3 Ham(a) z[immig; a; 5 Haf (a; r) j Bma(a; r) z[infmd; a; 6 Haf (a; r) j Ma(a; 7 Haf (a; r) j Ma in (a; 9 Haf (a; r) j Mam(a; r) z[immig; a; r Figure 11: Control migration with caching (r Again, Act(a; fl) denotes the routers where a has visited, which are not in an Idle state. Condition (C 1 ), as before, states that at most finitely many agents can be on the move ("deranged") at any instant. (C 2 ) is, as before, an invariant on control messages and the forwarder caches, indicating that there are no forwarding cycles and the cached entries for each agent a always point to routers where the mobile agent has visited. M serves to indicate the router at which there is a Mam(a) when there is a pending regd message whose current location is indicated using Z. (C 3 ) is an invariant dealing with data messages or forwarded data mes- sages, which indicates that such messages may never arise from, be addressed to, or be present at agents located at as yet unvisited routers. (C 4 ) is a condition on update messages, stating that such messages are only sent between shadow agents of two different agents, and that they may only originate, be present at and be targetted to routers where the two agents have been active. Definition 4.12 (admissible configuration with caching) An admissible configuration with caching cm is generated by a pair (fl; Ob), consisting of an admissible function with caching and a process, as follows: where Rt and Ob are as in Figure 8, and Ag(a; k; (K \Theta RN \Theta RN \Theta (AN ! RN ) \Theta (RN [ LAN ) \Theta f0; 1g 2 (control migration) (RN * (RN \Theta fin; ning))\Theta (RN * (RN \Theta (RN [ LAN )))\Theta (Mam's) M fin (fmsg; fwddg \Theta (RN [ LAN ) \Theta RN \Theta AN \Theta RN \Theta DN )\Theta (data messages) Admissibility conditions on fl: (CMs(a; k; dom(F Acyclic(F; M); and defined where Acyclic(F; M) means: Figure 12: Admissible configurations with caching Let c be an admissible configuration with caching, generated by (fl; Ob). Fur- ther, let c:DMs(a) denote \Pi (ddir ;z;r 1 ;a 2 ;r d], the data messages addressed to a in configuration c, and let c:DMs stand for all data messages in configuration c. For convenience, we will write c:CMs and c:Ob to denote the state of the control messages and atomic observations, respectively, in configuration c. Theorem 4.13 The initial configuration CMob is admissible, and admissible configurations with caching are closed under reduction. As in Mob, it is possible to bring CMob to a stable state. Corollary 4.14 (control stabilization) Let c be an admissible configuration with caching generated by (fl; Ob) and let g. Then c ! -(10\Lambdan) c 0 such that c 0 is generated by (fl In particular, if fl(a) 6g. As in Stat and Mob, it is easy to derive corollaries concerning message integrity and message delivery. Corollary 4.15 (message integrity) Messages do not get lost nor is their observable content tampered with. Let c be an admissible configuration with caching, generated by (fl; Ob), and suppose c ! c 0 . Then for all z or else there exists a z 0 l x 0 l 2 c 0 :DMs such that l Corollary 4.16 (message delivery) Let a 2 AN and let c be an admissible configuration with caching generated by (fl; Ob) such that (ddir ; z; r 1 ; a Then the data message can be "delivered" in a number of reductions proportional to the length of the longest forwarding chain. The analysis of the invariant allows us to extract some general principles for the correct definition of the protocol (note that these principles are an output of the analysis of our protocol model, they are not explicitly stated in the informal description of the protocol). ffl Cache entries and Fwd 's always point to routers which have been visited by the agent. Any message from agent a to agent a 0 comes from a router r and is directed to a router r 0 , which have been visited, respectively, by agent a and a 0 . ffl Agent a never sends update messages to its own shadow agents. ffl The protocol for moving an agent a from a router to another terminates in a fixed number steps. ffl Given an agent a, the forwarding proxy agents never form forwarding cycles. This ensures that once the agent a has settled in one router, data messages and update messages in transit can reach it in a number of steps which is proportional to the length of the longest chain of Fwd 's. The bottom line of our analysis for system CMob is the analogue of theorem 4.11. Theorem 4.17 Stat ffl Conclusions We have described in a standard process description language a simplified version of the Mobile IP protocol. We believe that a precise yet abstract model is useful in establishing the correctness of the protocol, as well as providing a basis for simulation and experimentation. Our modelling uses non-determinism and asynchronous communication (with unbounded and unordered buffers). Non-determinism serves as a powerful abstraction mechanism, assuring us of the correctness of the protocol for arbitrary behaviors of the processes, even if we try different instances of particular management policies (e.g., routing and cache management policies) provided they maintain the same invariants as in the non-deterministic model. Asynchronous communication makes minimal assumptions on the properties of the communication channels and timeliness of messages. All we require is that messages are not lost and in particular we assume there is a mechanism for avoiding store-and-forward deadlocks. Our analysis shows that message loss can be avoided by a router forwarding messages addressed to a mobile agent that is no longer present in that subnet to its home router or to a router to which it has moved. Moreover, these forwarding links never form cycles. Control Stabilization is a key property, since cycles that a message may potentially traverse are broken on stabilization. Furthermore, any (reasonable) cache update policy can be used provided messages to an agent are forwarded to routers it has previously visited. Our model allows mobility protocol designers explore alternative policies and mechanisms for message forwarding and cache management. A concrete suggestion is that rather than dropping a data message (delayed in transit) for an agent that has moved away from a router, IPv6 designers could examine the tradeoff between increased traffic and employing a default policy of tunneling the message to the home subnet of the agent - particularly for applications where message loss is costly, or in the context of multi-layer protocols. Other concrete applications include designing mobility protocols where losing messages may be unacceptable, e.g., forwarding signals in process migration mechanisms. In our modelling, we have greatly simplified various details. On the one hand, this simplification is useful, since it again serves as a way of abstracting from particular protocols for establishing connections (e.g., Neighbor Discovery, etc. On the other hand, we have assumed that our so-called "control messages" eventually reach their destination without getting lost or corrupted. A future direction of work may be to model protocols that cope with failures, or to model security and authentication issues. By concentrating on an abstract and simple model we have been able to specify the protocol and by a process of analysis to discover and explicate some of its organizing principles. The specification and combinatorial analysis of the protocol is sufficiently complicated to preclude leaving it "implicit" in the informal protocol description. By a careful analysis we have been able to carry out a hand proof. A direction for further research is the formal development of the proof using a proof assistant. Finally, we report on a finite state formulation of the protocol for which automatic simulation and verification tools are available. The sets RN ; AN ; DN are assumed finite, so that there are finitely many entities in the systems. Ensuring that the number of messages does not grow in an unbounded manner also requires that communication is over bounded capacity channels. In particular we will consider the limit case where all communications are synchronous (we expect that a protocol which works with synchronous communication can be easily adapted to a situation where additional buffers are added). The main difficulty lies in understanding how to transform asynchronous communication into synchronous communication without introducing deadlocks. The synchronous version seems to require a finer, more detailed description of the protocol and makes the proof of correctness much more complicated. In retro- spect, this fact justifies the use of an asynchronous communication model with unbounded and unordered buffers. The systems FStat and FMob with synchronous communication are described in Appendix B. We have compiled these descriptions in the modelling language Promela of the simulation and verification tool SPIN [Hol91]. Extensive simulations on configurations including three routers and three agents have revealed no errors. We have been able to complete a verification for the FStat system with two routers and two agents. The size of the verification task and the complexity of the system FMob make verification of larger systems difficult. The Promela sources for FMob are available at URL http://protis.univ- mrs.fr/-amadio/fmob. --R A foundation for actor computation. Orca: a language for parallel programming of distributed systems. Design of a distributed object manager for Smalltalk-80 system Design and validation of computer protocols. An object calculus for asynchronous communication. A nitpick analysis of mobile IPv6. Mobility support in IPv6 (RFC). A Calculus of Mobile Process Mobile Unity coordination constructs applied to packet forwarding. The sprite network operating system. Concurrency and automata on infinite sequences. Indirect distributed garbage collection: handling object migration. Process migration in demos/mp. Linear and branching systems in the semantics and logics of reactive systems. Expressing code mobility in mobile UNITY. The Locus Distributed System Architecture. Mobile UNITY: reasoning and specificaton in mobile computing. Vip: a protocol providing host mobility. Mobile objects in distributed Oz. --TR The LOCUS distributed system architecture Design of a distributed object manager for the Smalltalk-80 system Fine-grained mobility in the Emerald system The Sprite Network Operating System Design and validation of computer protocols IP-based protocols for mobile internetworking Orca A calculus of mobile processes, I VIP: a protocol providing host mobility Indirect distributed garbage collection Mobile UNITY Mobile objects in distributed Oz Expressing code mobility in mobile UNITY An Object Calculus for Asynchronous Communication Linear and Branching Structures in the Semantics and Logics of Reactive Systems Modelling IP Mobility Mobile UNITY Coordination Constructs Applied to Packet Forwarding for Mobile Hosts Concurrency and Automata on Infinite Sequences Process migration in DEMOS/MP A foundation for actor computation

internet protocols;formal methods;mobility;bisimulation;modelling;process description languages;protocol analysis;verification

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Design and Evaluation of a Switch Cache Architecture for CC-NUMA Multiprocessors.

AbstractCache coherent nonuniform memory access (CC-NUMA) multiprocessors provide a scalable design for shared memory. But, they continue to suffer from large remote memory access latencies due to comparatively slow memory technology and large data transfer latencies in the interconnection network. In this paper, we propose a novel hardware caching technique, called switch cache, to improve the remote memory access performance of CC-NUMA multiprocessors. The main idea is to implement small fast caches in crossbar switches of the interconnect medium to capture and store shared data as they flow from the memory module to the requesting processor. This stored data acts as a cache for subsequent requests, thus reducing the need for remote memory accesses tremendously. The implementation of a cache in a crossbar switch needs to be efficient and robust, yet flexible for changes in the caching protocol. The design and implementation details of a CAche Embedded Switch ARchitecture, CAESAR, using wormhole routing with virtual channels is presented. We explore the design space of switch caches by modeling CAESAR in a detailed execution driven simulator and analyze the performance benefits. Our results show that the CAESAR switch cache is capable of improving the performance of CC-NUMA multiprocessors by up to 45 percent reduction in remote memory accesses for some applications. By serving remote read requests at various stages in the interconnect, we observe improvements in execution time as high as 20 percent for these applications. We conclude that switch caches provide a cost-effective solution for designing high performance CC-NUMA multiprocessors.

Introduction To alleviate the problem of high memory access latencies, shared memory multiprocessors employ processors with small fast on-chip caches and additionally larger off-chip caches. Symmetric multiprocessor (SMP) systems are usually built using a shared global bus. However the contention on the bus and memory heavily constrains the number of processors that can be connected to the bus. To build high performance systems that are scalable, several current systems [1, 10, 12, 13] employ the cache coherent non-uniform memory access (CC-NUMA) architecture. In such a system, the shared memory is distributed among all the nodes in the system to provide a closer local memory and several remote memories. While local memory access latencies can be tolerated, the remote memory accesses generated during the execution can bring down the performance of applications drastically. To reduce the impact of remote memory access latencies, researchers have proposed improved caching strategies [14, 18, 27] within each cluster of the multiprocessor. These caching techniques are primarily based on data sharing among multiple processors within the same cluster. Nayfeh et al. [18] explore the use of shared L2 caches to benefit from the shared working set between the processors within a cluster. Another alternative is the use of network caches or remote data caches [14, 27]. Network caches reduce the remote access penalty by serving capacity misses of L2 caches and by providing an additional layer of shared cache to processors within a cluster. The HP Exemplar [1] implements the network cache as a configurable partition of the local memory. Sequent's NUMA-Q [13] dedicates a 32MB DRAM memory for the network cache. The DASH multiprocessor [12] has provision for a network cache called the remote access cache. A recent proposal by Moga et al.[14] explores the use of small SRAM (instead of DRAM) network caches integrated with a page cache. The use of 32KB SRAM chips reduces the access latency of network caches tremendously. Our goal is to reduce remote memory access latencies by implementing a global shared cache abstraction central to all processors in the CC-NUMA system. We observe that network caches provide such an abstraction limited to the processors within a cluster. We explore the implementation issues and performance benefits of a multi-level caching scheme that can be incorporated into current CC-NUMA systems. By embedding a small fast SRAM cache within each switch in the interconnection network, called switch cache, we capture shared data as it flows through the interconnect and provide it to future accesses from processors that re-use this data. Such a scheme can be considered as a multi-level caching scheme, but without inclusion property. Our studies on application behavior indicate that there is enough spatial and temporal locality between requests from processors to benefit from small switch caches. Recently, a study by Mohapatra et al. [15] used synthetic workloads and showed that increasing the buffer size in a crossbar switch beyond a certain value does not have much impact on network performance. Our application-based study [4] confirms that this observation holds true for the CC-NUMA environment running several scientific applications. Thus we think that the large amount of buffers in current switches, such as SPIDER [7], is an overkill. A better utilization of these buffers can be accomplished by organizing them as a switch cache. There are several issues to be considered while designing such a caching technique. These include cache design issues such as technology & organization, cache coherence issues, switch design issues such as arbitration, and message flow control issues such as appropriate routing strategy, message layout etc. The first issue is to design and analyze a cache organization that is large enough to hold the reusable data, yet fast enough to operate during the time a request passes through the switch. The second issue involves modifying the directory-based cache protocol to handle an additional caching layer at the switching elements, so that all the cache blocks in the system are properly invalidated on a write. The third issue is to design buffers and arbitration in the switch which will guarantee certain cache actions within the switch delay period. For example, when a read request travels through a switch cache, it must not incur additional delays. Even in the case of a switch cache hit, the request must pass on to the home node to update the directory, but not generate a memory read operation. The final issue deals with message header design to enable request encoding, network routing etc. The contribution of this paper is the detailed design and performance evaluation of a switch cache interconnect employing CAESAR, a CAche Embedded Switch ARchitecture. The CAESAR switch cache is made up of a small SRAM cache operating at the same speed as a wormhole routed cross-bar switch with virtual channels. The switch design is optimized to maintain crossbar bandwidth and throughput, while at the same time providing sufficient switch cache throughput and improved remote access performance. The performance evaluation of the switch cache interconnect is conducted using six scientific applications. We present several sensitivity studies to cover the entire design space and to identify the important parameters. Our experiments show that switch caches offer a great potential for use in future CC-NUMA interconnects for some of these applications. The rest of the paper is organized as follows. Section 2 provides a background on the remote access characteristics of several applications in a CC-NUMA environment and builds the motivation behind our proposed global caching approach. The switch cache framework and the caching protocol are presented in Section 3. Section 4 covers the design and implementation of the crossbar switch cache architecture called CAESAR. Performance benefits of the switch cache framework are evaluated and analyzed in great detail in Section 5. Sensitivity studies over various design parameters are also presented in Section 5. Finally, Section 6 summarizes and concludes the paper. Memory Interconnection Network Memory Network Interface/Router Network Interface/Router Remote Memory Node X's Figure 1: CC-NUMA system & memory hierarchy Application Characteristics and Motivation Several current distributed shared memory multiprocessors have adopted the CC-NUMA architecture since it provides transparent access to data. Figure 1 shows the disparities in proximity and access time in the CC-NUMA memory hierarchy of such systems. A load or store issued by processor X can be served in a few cycles upon L1 or L2 cache hits, in less than a hundred cycles for local memory access or incurs few hundreds of cycles due to a remote memory access. While the latency for stores to the memory (write transactions) can be hidden by the use of weak consistency models, the stall time due to loads (read transactions) to memory can severely degrade application performance. 2.1 Global Cache Benefits: A Trace Analysis To reduce the impact of remote read transactions, we would like to exploit the locality in sharing patterns between the processors. Figure 2 plots the read sharing pattern for six applications with processors using a cache line size of 32 bytes. The six applications used in this evaluation are the Floyd-Warshall's Algorithm (FWA), Gaussian Elimination (GAUSS), Gram-Schmidt (GS), Matrix Multiplication (MM), Successive Over Relaxation (SOR) and the Fast Fourier Transform (FFT). The x-axis represents the number of sharing processors (X) while the y-axis denotes the number of accesses to blocks shared by X number of processors. From the figure, we observe that for four out of the six applications (FWA, GAUSS, GS, MM), multiple processors read the same block between two consecutive writes to that block. These shared reads form a major portion (35 to 80%) of the application's read misses. To take advantage of such read-sharing patterns across processors, we (a) FWA (b) SOR (c) GAUSS (d) GS (e) FFT (f) MM Figure 2: Application read sharing characteristics introduce the concept of an ideal global cache that is centrally accessible to all processors. When the first request is served at the memory, the data sent back in the reply message is stored in a global cache. Since the cache is accessible by all processors, subsequent requests to the data item can be satisfied by the global cache at low latencies. There are two questions that arise here: ffl What is the time lag between two accesses from different processors to the same block? We define this as temporal read sharing locality between the processors, somewhat equivalent to temporal locality in a uniprocessor system. The question raised here particularly relates to the size and organization of the global cache. In general, it would translate to: the longer the time lag, the bigger the size of the required global cache. ffl Given that a block can be held in a central location, how many requests can be satisfied by this cached block? We call this term attainable read sharing to estimate the performance improvement by employing a global cache. Again, this metric will give an indication of the required size for the global cache. To answer these questions, we instrumented a simulator to generate an execution trace with information regarding each cache miss. We then fed these traces through a trace analysis tool, Sharing Identifier and Locality Analyzer. In order to evaluate the potential of a global QWHU#DUULYDO#7LPH#SF\FOHV# (a) FWA (b) GS (c) GAUSS (d) SOR (e) FFT (f) MM Figure 3: The temporal locality of shared accesses cache, SILA generates two different sets of data: temporal read shared locality (Figure 3), and attainable sharing (Figure 4). The data sets can be interpreted as follows. Temporal Read Sharing Locality: Figure 3 depicts the temporal read sharing locality as a function of different block sizes. A point fX,Yg from this data set indicates that Y is the probability that two read transactions (from different processors) to the same block occur within a time distance of X or lower. (i.e. is the average inter-arrival time between two consecutive read requests to the same block). As seen in the figure, most applications have an inherent temporal re-use of the cached block by other processors. The inter-arrival time between two consecutive shared read transactions from different processors to the same block is found to be less than 500 processor cycles (pcycles) for 60-80% of the shared read transactions for all applications. Ideally, this indicates a potential for atleast one extra request to be satisfied per globally cached block. Attainable Read Sharing: Figure 4 explores this probability of multiple requests satisfied by the global caching technique termed as attainable sharing degree. A point fX,Yg in this data set indicates that if each block can be held for X cycles in the global cache, the average number of subsequent requests per block that can be served is Y . The figure depicts the attainable read sharing degree for each application based on the residence time for the block in the global cache. The residence time of a cache block is defined as the amount of time the block is held in the cache before it is replaced or invalidated. While invalidations cannot be avoided, note that the residence time directly relates to several cache design issues such as cache size, block size and associativity. QWHU#DUULYDO#7LPH#SF\FOHV# (a) FWA (b) GS (c) GAUSS (d) SOR (e) FFT (f) MM Figure 4: The attainable sharing degree From Figure 2 we observed that FWA, GS and GAUSS have high read sharing degrees close to the number of processors in the system (in this case, 16 processors). However, it is found that the attainable sharing degree varies according to the temporal locality of each application. While GAUSS can attain a sharing degree of 10 in the global cache with a residence time of 2000 processor cycles, GS requires that the residence time be 5000 and FWA requires that this time be 7000. The MM application has a sharing degree of approximately four to five, whereas the attainable sharing degree is much lower. SOR and FFT are not of much interest since they have a very low percentage (1-2%) of shared block accesses (see Figure 2). 2.2 The Interconnect as a Global Cache In section 2.1, we identified the need for global caching to improve the remote access performance of CC-NUMA systems. In this section, we explore the possible use of the interconnect medium as a central caching location: ffl What makes the interconnect medium a suitable candidate for central caching? Which interconnect topology is beneficial for incorporating a central caching scheme? Communication in a shared memory system occurs in transactions that consist of requests and replies. The requests and replies traverse from one node to another through the network. The Node A Node C Interconnect Overlapping Path Caching Potential Figure 5: The caching potential of the interconnect medium interconnection network becomes the only global yet distributed medium in the entire system that can keep track of all the transactions that occur between all nodes. A global sharing framework can efficiently employ the network elements for coherent caching of data. The potential for such a network caching strategy is illustrated in Figure 5. The path of two read transactions to the same memory module that emerge from different processors overlap at some point in the network. The common elements in the network can act as small caches for replies from memory. A later request to the recently accessed block X can potentially find the block cached in the common routing element (illustrated by a shaded circle). The benefit of such a scheme is two-fold. From the processor point of view, the read request gets serviced at a latency much lower than the remote memory access latency. Secondly, the memory is relieved of servicing the requests that hit in the global interconnect cache, thus improving the access times of other requests that are sent to the same memory module. From the example, we observe that incorporating such schemes in the interconnect requires a significant amount of path overlap between processor to memory requests and that replies follow the same path as requests in order to provide an intersection. The routing and flow of requests and replies depends entirely upon the topology of the interconnect. The ideal topology for such a system is the global bus. However, the global bus is not a scalable medium and bus contention severely affects the performance when the number of processors increases beyond a certain threshold. Con- sequently, multiple bus hierarchies and fat-tree structures have been considered effective solutions to the scalability problem. The tree structure provides the next best alternative to hierarchical caching schemes. 3 The Switch Cache Framework In this section, we present a new hardware realization of the ideal global caching solution for improving the remote access performance of shared memory multiprocessors. 3.1 The Switch Cache Interconnect Network topology plays an important role in determining the paths from a source to a destination in the system. Tree-based networks like the fat tree [11], the heirarchical bus network [23, 3] and the multistage interconnection network (MIN) [17] provide hierarchical topologies suitable for global caching. In addition, the MIN is highly scalable and it provides a bisection bandwidth that scales linearly with the number of nodes in the system. These features of the MIN make it very attractive as scalable high performance interconnects for commercial systems. Existing systems such as Butterfly [2], CM-5 [11] and IBM SP2 [21] employ a bidirectional MIN. The Illinois Cedar multiprocessor [22] employs two separate uni-directional MINs (one for requests and one for replies). In this paper, the switch cache interconnect is a bidirectional MIN to take advantage of the inherent tree structure. Note, however, that logical trees can also be embedded on other popular direct networks like the mesh and the hypercube [10, 12]. The baseline topology of the 16-node bi-directional MIN (BMIN) is shown in Figure 6(a). In general, an N-node system using a BMIN comprises of N=k switching elements (a 2k \Theta 2k crossbar) in each of the log k N stages connected by bidirectional links. We chose wormhole routing with virtual channels as the switching technique because it is prevalent in current systems such as the SGI Origin[10]. In a shared memory system, communication between nodes is accomplished via read/write transactions and coherence requests/acknowledgments. The read/write requests and coherence replies from the processor to the memory use forward links to traverse through the switches. Similarly, read/write replies with data and coherence requests from memory to the processor traverse the backward path, as shown in bold in Figure 6(a). Separating the paths enables separate resources and reduces the possibility of deadlocks in the network. At the same time, routing them through the same switches provides identical paths for requests and replies for a processor-memory pair that is essential to develop a switch cache hierarchy. The BMIN tree structure that enables this hierarchy is shown in Figure 6(b). The basic idea of our caching strategy is to utilize the tree structure in the BMIN and the path overlap of requests, replies, and coherence messages to provide coherent shared data in the intercon- nect. By incorporating small fast caches in the switching elements of the BMIN, we can serve the sharing processors at these switching elements. We use the term switch cache to differentiate these caches from the processor caches. An example of a BMIN employing switch caches that can serve Invalidation Response Ack, WriteBack Request (a) Bidirectional MIN - Structure and Routing Processor/Cache Interface Memory Interface (b) Multiple Tree Structures in the BMIN Processor/Cache Interface Memory Interface (c) Switch Caching in the BMIN serviced by memory Switch Cache serviced by switch caches at different levels Figure The Switch Cache Interconnect multiple processors is shown in Figure 6(c). An initial shared read request from processor P i to a block is served at the remote memory M j . When the reply data flows through the interconnection network, the block is captured and saved in the switch cache at each switching element along the path. Subsequent requests to the same block from sharing processors, such as P j and P k , take advantage of the data blocks in the switch cache at different stages, thus incurring reduced read latencies. 3.2 The Caching Protocol The incorporation of processor caches in a multiprocessor introduces the well-known cache coherence problem. Many hardware cache-coherent systems employ a full-map directory scheme [6], In this scheme, each node maintains a bit vector to keep track of all the sharers of each block in its local shared memory space. On every write, an ownership request is sent to the home node, invalidations are sent to all the sharers of the block and the ownership is granted only when all the corresponding acknowledgments are received. At the processing node, each cache employs a three-state (MSI) protocol to keep track of the state of each cache line. Incorporating switch caches comes with the requirement that these caches remain coherent, and data access remain consistent with the system consistency model. Cannot reach these 2 states Write Initiated by another processor Expected State SHARED DIRTY UNCACHED ReadReply* ReadReply* InvType INVALID SHARED ReadReply ReadReply (a) Switch Cache State Diagram Add sharer to directory vector Send invalidation to requestor Increment ack_counter to wait for additional ack Protocol (b) Change in Directory Protocol Switch Hit Read Figure 7: Switch Cache Protocol Execution We adopt a sequential consistency model in this paper. Our basic caching scheme can be represented by the state diagram in Figure 7 and explained as follows: The switch cache stores only blocks in a shared state in the system. When a block is read to the processor cache in a dirty state, it is not cached in the switch. Effectively, the switch cache needs to employ only a 2-state protocol where the state of a block can be either SHARED or NOT VALID. The transitions of blocks from one state to another is shown in Figure 7a. To illustrate the difference between block invalidations (Inv Type) and block replacements (ReadReply ), the figure shows the NOT VALID state conceptually separated into two states INVALID and NOT PRESENT respectively. Read Requests: Each read request message that enters the interconnect checks the switch caches along its path. In the event of a switch cache hit , the switch cache is responsible for providing the data and sending the reply to the requestor. The original message is marked as switch hit and it continues to the destination memory (ignoring subsequent switch caches along its path) with the sole purpose of informing the home node that another sharer just read the block. Such a message is called a marked read request. This request is necessary to maintain the full-map directory protocol. Note that memory access is not needed for these marked requests and no reply is generated. At the destination memory, a marked read request can find the block in only two states, SHARED, or TRANSIENT (see Figure 7b). If the directory state is SHARED, then this request only updates the sharing vector in the directory. However, it is possible that a write has been initiated to the same cache line by a different processor and the invalidation for this write has not yet propagated to the switch cache. This can only be present due to false sharing or an application that allows data race conditions to exist. If this occurs, then the marked read request observes a TRANSIENT state at Physical Address Swc Hit Dest bits Flits Req Age 4bits 2bits1 Addr (contd) Src 4bits 6bits 6bits 8bits Figure 8: Message Header Format the directory. In such an event, the directory sends an invalidation to the processor that requested the read and waits for this additional acknowledgment before committing the write. Read Replies: Read replies can originate from two different sources, namely, the memory node or an owner's cache in dirty state. Both read replies enter the interconnect following the backward path. The read reply originating from the memory node should check the switch cache along its path. If the line is not present in a switch cache, the data carried by the read reply is written into the cache. The state of the cache line is marked SHARED. For messages originating from an owner's cache, only those replies whose home node and requester node are not identical can be allowed to check the switch caches along the way. Those replies that find the line absent in the cache will enter the data in the cache. If the home node and the requester are the same, the reply should ignore the switch cache. This reply is not allowed to enter the switch cache because subsequent modification of the block by the owner will not be visible to the interconnection network. Thus the path from the owner to the requester is not coherent and does not overlap with the write request or the invalidation request responsible for coherence (as explained next). In summary, only read replies with non-identical home node and requester node should enter data into the switch cache, if not already present. Writes, Write-backs and Coherence Messages: These requests flow through the switches, check the switch cache and invalidate the cache line if present in the cache. By doing so, the switch cache coherence is maintained somewhat transparently. Message Format: The messages are formatted at the network interface where requests, replies and coherence messages are prepared to be sent over the network. In a wormhole-routed network, messages are made up of flow control digits or flits. Each flit is 8 bytes as in Cavallino [5]. The message header contains the routing information, while data flits follow the path created by the header. The format of the message header is shown in Figure 8. To implement the caching technique, we require that the header consists of 3 additional bits of information. Two bits (Reqtype are encoded to denote the switch cache access type as follows: cache line from the switch cache. If present, mark read header and generate reply message. 4w 4w 4w Forward Link Inputs Backward Link Inputs Forward Link Outputs Link Outputs Backward Input Block Input Block Input Block Input Block Arbiter Crossbar 4w Routing Tables 4w 4w 4w 4w Figure 9: A conventional crossbar switch line into the switch cache. Invalidate the cache line, if present in the cache. switch cache, no processing required. Note from the above description and the caching protocol that read requests are encoded as sc read requests. Read replies whose home node and requestor id are different are encoded as sc write. Coherence messages, write ownership requests and write-back requests are encoded as sc inv requests. All other requests can be encoded as sc ignore requests. An additional bit is required to mark sc read requests as earlier switch cache hit. Such a request is called a marked read request. This is used to avoid multiple caches servicing the same request. As discussed, such a marked request only updates the directory and avoids a memory access. 4 Switch Cache Design and Implementation Crossbar switches provide an excellent building block for scalable high performance interconnects. Crossbar switches mainly differ in two design issues: switching technique and buffer management. We use wormhole routing as the switching technique and input buffering with virtual channels since these are prevalent in current commercial crossbar switches [5, 7]. We begin by presenting the organization and design alternatives for a 4 \Theta 4 crossbar switch cache. In a later subsection, the extensions required for incorporating a switch cache module into a larger (8 \Theta 8) crossbar switch are presented. 4w 4w 4w Forward Link Inputs Backward Link Inputs Forward Link Outputs Link Outputs Backward Input Block Input Block Input Block Input Block Routing Tables Crossbar 4w Arbiter Input Block Switch Cache Module 4w 4w 4w 4w 4w 4w 4w Forward Link Inputs Backward Link Inputs Forward Link Outputs Link Outputs Backward Input Block Input Block Input Block Input Block Crossbar 4w Routing Tables 4w 4w 4w 4w Switch Cache Module Arbiter Input Block (a) Arbitration-Independent (b) Arbitration-Dependent Figure 10: Crossbar Switch Cache Organization 4.1 Switch Cache Organization Our base bi-directional crossbar switch has four inputs and four outputs as shown in Figure 9. Each input link in the crossbar switch has two virtual channels thus providing 8 possible input candidates for arbitration. The arbitration process is the age technique, similar to that employed in the SGI Spider Switch [7]. At each arbitration cycle, a maximum of 4 highest age flits are selected from 8 possible arbitration candidates. The flit size is chosen to be 8 bytes (4w) with links similar to the Intel Cavallino [5]. Thus it takes four link cycles to transmit a flit from output of one switch to the input of the next. Buffering in the crossbar switch is provided at the input block at each link. The input block is organized as a fixed size FIFO buffer for each virtual channel that stores flits belonging to a single message at a time. The virtual channels are also partitioned based on the destination node. This avoids out-of-order arrival of messages originating from the same source to the same destination. We also provide a bypass path for the incoming flits that can be directly transmitted to the output if the input buffer is empty. While organizing the switch cache, we are particularly interested in maximizing performance by serving flits within the cycles required for the operation of the base crossbar switch. Thus the organization depends highly on the delay experienced for link transmission and crossbar switch processing. Here we present two different alternatives for organizing the switch cache module within conventional crossbar switches. The arbitration-independent organization is based on a crossbar switch operation similar to the SGI Spider [7]. The arbitration-dependent organization is based on a crossbar switch operation similar to the Intel Cavallino [5]. Arbitration-Independent Organization: This switch cache organization is based on link and switch processing delays similar to those experienced in the SGI Spider. The internal switch core runs at 100 MHz, while the link transmission operates at 400MHz. The switch takes four 100MHz clocks to move flits from the input to the link transmitter at the output. The link, on the other hand, can transmit a 8byte flit in a single 100 MHz clock (four 400 MHz clocks). Figure 10a shows the arbitration-independent organization of the switch cache. The organization is arbitration independent because the switch cache performs the critical operations in parallel with the arbitration operation. At the beginning of each arbitration cycle, a maximum of four input flits stored in the input link registers are transmitted to the switch cache module. In order to maintain flow control, all required switch cache processing should be performed in parallel with the arbitration and transmission delay of the switch (4 cycles). Arbitration-Dependent Organization: This switch cache organization is based on link and switch processing delays similar to those experienced in the Intel Cavallino [5]. The internal switch core and link transmission operate at 200MHz. It takes 4 cycles for the crossbar switch to pass the flit from its input to the output transmitter and 4 cycles for the link to transmit one flit from one switch to another. Figure 10b shows the arbitration-dependent organization of the switch cache. The organization is arbitration dependent because it performs the critical operations at the end of the arbitration cycle and in parallel with the output link transmission. At the end of every arbitration cycle, a maximum of four flits passed through the crossbar from input buffers to the output link transmitters are also transmitted to the switch cache. Since the output transmission takes four 200MHz cycles, the switch cache needs to process a maximum of four flits within 4 cycles. Each organization has its advantages/disadvantages. In the arbitration-independent organization, the cache operates at the switch core frequency and remains independent of the link speed. On the other hand, this organization lacks arbitration information which could be useful for completing operations in an orderly manner. While this issue does not affect 4 \Theta 4 switches, the drawback will be evident when we study the design of larger crossbar switches. The arbitration-dependent organization benefits from the completion of the arbitration phase and can use the resultant information for timely completion of switch cache processing. However in this organization, the cache is required to run at link transmission frequency in order to complete critical switch cache operations. As in the Intel Cavallino, it is possible to run the switch core, the switch cache and the link transmission at the same speed. Finally, note that, in both cases the reply messages from the switch cache module are stored in another input block as shown in Figure 10. With two virtual channels per input block, the crossbar switch size expands to 4. Also, in both cases, the maximum number of switch cache inputs are 4 requests and processing time is limited to 4 switch cache cycles. Cache Size (in bytes) Cache Access Time (in Direct Cache Size (in bytes) Cache Access Time (in Direct (a) 32-byte cache lines (b) 64-byte cache lines Figure 11: Cache Access Time (in FO4) 4.2 Cache Design: Area and Access Time Issues The access time and area occupied by an SRAM cache depends on several factors such as asso- ciativity, cache size, number of wordlines and number of bitlines[25, 16]. In this section, we study the impact cache size and associativity on access time and area constraints. Our aim is to find the appropriate design parameters for our crossbar switch cache. Cache Access Time: The CACTI model [25] quantifies the relationship between cache size, associativity and cache access time. We ran the CACTI model to measure the switch cache access time for different cache sizes and set associativity values. In order to use the measurements in a technology independent manner, we present the results using the delay measurement technique known as the fan-out-of-four One FO4 is the delay of a signal passing through an inverter driving a load capacitance that is four times larger than its input. It is known that a 8 Kbyte data cache has a cycle time of 25 FO4 [9]. Figure 11 shows the results obtained in FO4 units. In Figure 11, the x-axis denotes the size of the cache, and each curve represents the access time for a particular set associativity. We find that direct mapped caches have the lowest access time since a direct indexing method is used to locate the line. However, a direct mapped cache may exhibit poor hit ratios due to mapping conflicts in the switch cache. Set-associative caches can provide improved cache hit ratios, but have a longer cycle time due to a higher data array decode delay. Most current processors employ multi-ported set associative L1 caches operating within a single processor cycle. We have chosen a 2-way set associative design for our base crossbar switch cache to maintain a balance between access time and hit ratio. However, we also study the effect of varied set associativity on switch cache performance. Cache output width is also an important issue that primarily affects the data read/write delay. As studied by Wilton et al. [25], the increase in data array width increases the number of sense amplifiers required. The organization of the cache can also make a significant difference in terms of (a) 32-byte cache lines (b) 64-byte cache lines Figure 12: Cache Relative Area chip area. Narrower caches provide data in multiple cycles, thus increasing the cache access time for an average read request. For example, a cache with 32-byte blocks and a width of 64 bits decreases the cache throughput to one read in four cycles. Within the range of 64 to 256 bits of data output width, we know that 64 bits will provide the worst possible performance scenario. We designed our switch cache using a 64-bit output width and show that the overall performance is not affected by this parameter. Cache Relative Area: In order to determine the impact of cache size and set associativity on the area occupied by an on-chip cache, we use the area model proposed by Mulder et al.[16]. The area model incorporates overhead area such as drivers, sense amplifiers, tags and control logic to compare data buffers of different organizations in a technology independent manner using register- bit equivalent or rbe. One rbe equals the area of a bit storage cell. We used this area model and obtained measurements for different cache sizes and associativities. Figure 12 shows the results obtained in relative area. The x-axis denotes the size of the cache and each curve represents different set associativity values. For small cache sizes ranging from 512 bytes to 4KB, we find that the amount of area occupied by the direct mapped cache is much lower than that for an 8-way set associative cache. From the figure, we find that an increase in associativity from 1 to 2 has a lower impact on cache area than an increase from 2 to 4. From this observation, we think that a 2-way set associative cache design would be the most suitable organization in terms of cache area and cache access time, as measured earlier. 4.3 CAche Embedded Switch ARchitecture (CAESAR) In this section, we present a hardware design for our crossbar switch cache called CAESAR, (CAche Embedded Switch ARchitecture). A block diagram of the CAESAR implementation is shown in Figure 13. For a 4 \Theta 4 switch, a maximum of 4 flits are latched into switch cache registers at each 4w 4w F 4w F 4w 4w R Y U Switch Cache Module Cache Access Control Switch Cache Module Arbiter Crossbar Select Source Link/Buffer Information Process Incoming Flits Flits Transmitted from Crossbar header vector To Input Block Blocking Info Update Read Header Cache Data Unit Cache Tag Unit Snoop Registers RI Buffer WR Buffers Figure 13: Implementation of CAESAR arbitration cycle. The operation of the CAESAR switch cache can be divided into (1) process incoming flits, (2) switch cache access, (3) switch cache reply generation, and (4) switch cache feedback. In this section, we cover the design and implementation issues for each of these operations in detail. Process Incoming Flits: Incoming flits stored into the registers can belong to different request types. The request type of the flit is identified based on the 2 bits (R 1 R 0 ) stored in the header. Header flits of each request contain the relevant information including memory address required for processing reads and invalidations. Subsequent flits belonging to these messages carry additional information not essential for the switch cache processing. Write requests to the switch cache require both the header flit for address information and the data flits to be written into the cache line. Finally ignore requests need to be discarded since they do not require any switch cache processing. An additional type of request that does not require processing is the marked read request. This read request has the swc hit bit set in the header to inform switch caches that it has been served at a previous switch cache. Having classified the types of flits entering the cache, the switch cache processing can be broken into two basic operations. The first operation performed by the flit processing unit is that of propagating the appropriate flits to the switch cache. As mentioned earlier, the flits that need to enter the cache are read headers, invalidation headers and all write flits. Thus, the processing unit masks out ignore flits, marked read flits and the data flits of invalidation and read requests. This is done by reading the R 1 R 0 bits from the header vector and the swc hit bit. To utilize this header information for the subsequent data flits of the message, the switch cache maintains a register that stores these bits. Flits requiring cache processing are passed to the request queue one in every cycle. The request queue is organized as two buffers, the RI buffer and the set of WR buffers shown in Figure 13. The RI buffer holds the header flits of read and invalidation requests. The WR buffers store all write flits and are organized as num vc \Theta k=2 different buffers. Here multiple buffers are required to associate data flits with the corresponding headers. When all data flits of a write request have accumulated into a buffer, the request is ready to initiate the cache line fill operation. The second operation to complete the processing of incoming flits is as follows. All unmarked read header flits need to snoop the cache to gather hit=miss information. This information is needed within the 4 cycles of switch delay or link transmission to be able mark the header by setting the last bit (swc hit). To perform this snooping operation on the cache tag, the read headers are also copied to the snoop registers (shown in Figure 13). We require two snoop registers because a maximum of two read requests can enter the cache in a single arbitration cycle. Switch Cache Access: Figure 14 illustrates the design of the cache subsystem. The cache module shown in the figure is that of a 2-way set associative SRAM cache The cache operates at the same frequency as the internal switch core. The set associative cache is organized using two sub-arrays for tag and data. The cache output width is 64 bits, thus requiring 4 cycles of data transfer for reading a line. The tag array is dual ported to allow two independent requests to access the tag at the same time. We now describe the switch cache access operations and their associated access delays. Requests to the switch cache can be broken into two types of requests: snoop requests and regular requests. Snoop Requests: Read requests are required to snoop the cache to determine hit or miss before the outgoing flit is transmitted to the next switch. For the arbitration independent switch cache organization (Figure 10a), it takes a minimum of four cycles for moving the flit from the switch input to the output. Thus we need the snoop operation within the last cycle to mark the message before link transmission. Similarly, for the arbitration dependent organization (Figure 10b), it takes 4 cycles to transmit a 64-bit header on a 16-bit output link after the header is loaded into the 64-bit (4w) output register. From the message format in Figure 8, the phit containing the swc hit bit to be marked is transmitted in the fourth cycle. Thus it is required that the cache access be completed within a maximum of 3 cycles. From Figure 13, copying the first read to the snoop registers is performed by the flit processing unit and is completed in one cycle. By dedicating one of the ports of the tag array primarily for snoop requests, each snoop in the cache takes only an additional cycle to complete. Since a maximum of 2 read headers can arrive to the switch cache in a single arbitration cycle, we can complete the snoop operation in the cache within 3 cycles. Note from Figure 14 that the snoop operation is done in parallel with the pending requests in the RI buffer Qsize Gen. Reply Data Buffer Header Bit Directory 0 Queue Status Send ReplyREPLY UNIT Request Address Address Write Data Data In/Out Data In/Out Hit/Miss BLOCKING INFO TO INPUT BLOCK Figure 14: Design of the CAESAR Cache Module and the WR buffers. When the snoop operation completes, the hit/miss information is propagated to the output transmitter to update the read header in the output register. If the snoop operation results in a switch cache miss, the request is also dequeued from the RI buffer. Regular Requests: A regular request is a request chosen from either the RI buffer or the WR buffers. Such a request is processed in a maximum of 4 cycles in the absence of contention. Requests from the RI buffer are handled on a FCFS basis. This avoids any dependency violation between read and invalidation requests in that buffer. However, we can have a candidate for cache operation from the RI buffer as well as from one or more of the WR buffers. In the absence of address dependencies, the requests from these buffers can progress in any order to the switch cache. When a dependency exists between two requests, we need to make sure that cache state correctness is preserved. We identify two types of dependencies between a request from the RI buffer and a request from the WR buffer: ffl An invalidation (from the RI buffer) to a cache line X and a write (from the the WR buffer) to the same cache line X . To preserve consistency, the simplest method is to discard the write to the cache line, thus avoiding incorrectness in the cache state. Thus, when invalidations enter the switch cache, write addresses of pending write requests in the WR buffer are compared and invalidated in parallel with the cache line invalidation. ffl A read (from the RI buffer) to a cache line X and a write (from the WR buffer) to a cache line Y that map on to the same cache entry. If the write occurs first, then cache line X will be replaced. In such an event, the read request cannot be served. Since such an occurrence is rare, the remedy is to send the read request back to the home node destined to be satisfied as a typical remote memory read request. Switch Cache Reply Generation: While invalidations and writes to the cache do not generate any replies from the switch cache, read requests need to be serviced by reading the cache line from the cache and sending the reply message to the requesting processor. The read header contains all the information required to send out the reply to the requester. The read header and cache line data are directly fed into the reply unit shown in Figure 14. The reply unit gathers the header at the beginning of the cache access and modifies the source/destination and request/reply information in the header in parallel with the cache access. When the entire cache line has been read, the reply packet is generated and sent to switch cache output block. The reply message from the switch cache now acts as any other message entering a switch in the form of flits and gets arbitrated to the appropriate output link and progresses using the backward path to the requesting processor. Switch Cache Feedback: Another design issue for the CAESAR switch is the selection of queue sizes. In this section, we identify methods to preserve crossbar switch throughput by blocking only those requests that violate correctness. As shown in Figure 13 and 14, finite size queues exist at the input of the switch cache (RI buffer and WR buffer) and at the reply unit (virtual channel queues in the switch cache output block). When any limited size buffer gets full, we have two options for the processing of read/write requests. The first option is to block the requests until a space is available in the buffer. The second option, probably the wiser one, is to allow the request to continue on its path to memory. The performance of the switch cache will be dependent on the chosen scheme only when buffer sizes are extremely limited. Finally, invalidate messages have to be processed through the switch cache since they are required to maintain coherence. These messages need to be blocked only when the RI buffer gets full. The modification required to the arbiter to make this possible is quite simple. To implement the blocking of flits at the input, the switch cache needs to inform the arbiter of the status of all its queues. At the end of each cycle, the switch cache informs the crossbar about the status of its queues in the form of free space available in each queue. The modification to the arbiter to perform the required blocking is minor. Depending on the free space of each queue, appropriate requests (based on R 1 R 0 ) will be blocked while others will traverse through the switch in a normal fashion. 4.4 Design of a 8 \Theta 8 Crossbar Switch Cache In the previous sections, we presented the design and implementation of a 4 \Theta 4 cache embedded crossbar switch. Current commercial switches such as SGI Spider and Intel Cavallino have six Hit/Miss Way BankBank Select Addr/Data Pair (1) Way Way Way BankBankBankQsize Gen. Reply REPLY UNIT Queue Status Reply Addr/Data Pair Address (1) Address Figure 15: Design of the CAESAR bidirectional inputs, while the IBM SP2 switch has 8 inputs and 8 outputs. In this section, we present extensions to the earlier design to incorporate a switch cache module in a 8 \Theta 8 switch. We maintain the same base parameters for switch core frequency, core delay, link speed, link width and flit size. The main issue when expanding the switch is that the number of inputs to the switch cache module doubles from 4 to 8. Thus, in each arbitration cycle, a maximum of four read requests can come into the switch. They require snoop operation on the switch cache within 4 cycles of switch core delay or link transmission depending on the switch cache organization shown in Figure 10. As shown in Figure 13, it takes one cycle to move the flits to the snoop registers. Thus we require to complete the snoop operation for 4 requests within 2 cycles and mark the header flit in the last cycle depending on the snoop result. In order to perform four snoops in two cycles, we propose to use a multiported CAESAR cache module. Multiporting can be implemented either by duplicating the caches or interleaving it into two independent banks. Since duplicating the cache consumes tremendous amount of on-chip area, we propose to use a 2-way interleaved CAESAR (CAESAR as shown in Figure 15. Interleaving splits the cache organization into two independent banks. Current processors such as MIPS R10000 [26] use even and odd addressed banked caches. However, the problem remains that four even addressed or four odd addressed requests will still require four cycles for snooping due to bank conflicts. We propose to interleave the banks based on the destination memory by using the outgoing link ids. In a 8 \Theta 8 switch there are four outgoing links that transmit flits from the switch towards the destination memory and vice-versa. Each cache bank will serve requests flowing through two of these links, thus partitioning the requests based on the destination memory. In an arbitration-independent organization (Figure 10a), it is possible that four incoming read requests are directed to the same memory module and result in bank conflicts. However, in the arbitration- dependent organization (Figure 10b), the conflict gets resolved during the arbitration phase. This guarantees that the arbitrated flits flow through different links. For 8 \Theta 8 switches, it would be more advantageous to use an arbitration dependent organization, thus assuring a maximum of 2 requests per bank in each arbitration cycle. As a result, the snoop operation of four requests can be completed in the required two cycles. Finally, note that only a few bits from the routing tag are needed to identify the bank in the cache. Such an interleaved organization changes the aspect ratio of the cache [25], and may affect the cycle time of the cache. Wilson et al.[24] showed that the increase in cycle time measured using the fan-out-of-four banked or interleaved caches over single ported caches was minimal. The 2-way interleaved implementation also doubles the cache throughput. Since two requests can simultaneously access the switch cache, the reply unit needs to provide twice the buffer space for storing the data from the cache. Similarly the header flit of the two read requests also need to be stored. As shown in Figure 15, the buffers are connected to outputs from different banks to gather the cache line data. Performance Evaluation In this section, we present a detailed performance evaluation of the switch cache multiprocessor based on execution-driven simulation. 5.1 Simulation Methodology To evaluate the performance impact of switch caches on the application performance of CC-NUMA multiprocessors, we use a modified version of the Rice Simulator for ILP Multiprocessors (RSIM) [19]. RSIM is an execution driven simulator for shared memory multiprocessors with accurate models of current processors that exploit instruction-level parallelism. In this section, we present the various system configurations and the corresponding modifications to RSIM for conducting simulation runs. The base system configuration consists of 16 nodes. Each node consists of a 200MHz processor Multiprocessor System - processors Processor Memory Speed 200MHz Access time 40 Issue 4-way Interleaving 4 Cache Network L1 Cache 16KB Switch Size 8x8 line size 32bytes Core delay 4 set size 2 Core Freq 200MHz access time 1 Link width 16 bits L2 Cache 128KB Xfer Freq 200MHz line 32bytes Flit length 8bytes set size 4 Virtual Chs. 2 access time 8 Buf. Length 4 flits Switch/Network Caches Switch Cache 128bytes-8KB Network Cache 4KB Application Workload FWA 128x128 GE 128x128 GS 96x128 MM 128x128 Table 1: Simulation parameters capable of issuing 4 instructions per cycle, a 16KB L1 cache a 128KB L2 cache, a portion of the local memory, directory storage and a local bus interconnecting these components. The L1 cache is 2-way set associative and an access time of a single cycle. The L2 cache is 4-way set associative and has an access time of 8 cycles. The raw memory access time is 40 cycles, but it takes more than 50 cycles to submit the request to the memory subsystem and read the data over the memory bus. The system employs the full-map three-state directory protocol [6] and the MSI cache protocol to maintain cache coherence. The system uses a release consistency model. We modified RSIM to employ a wormhole routed bidirectional MIN using 8 \Theta 8 switches organized in 2 stages as shown earlier in Figure 6. Virtual channels were also added to the switching elements to simulate the behavior of commercial switches like Cavallino and Spider. Each input link to the switch is provided with 2 virtual channel buffers capable of storing a maximum of 4 flits from a single message. The crossbar switch operation is similar to the description in Section 4.1. A detailed list of simulation parameters is also shown in Table 1. Figure Percentage Reduction in Memory Reads To evaluate switch caches, we further modified the simulator to incorporate switch caches into each switching element in the IN. The switch cache system improves on the base system in the following respects. Each switching element of the bidirectional MIN employs a variable size cache that models the functionality of the CAESAR switch cache presented in Section 4. Several parameters such as cache size and set associativity are varied for evaluating the design space of the switch cache. We have selected some numerical applications to investigate the potential performance benefits of the switch cache interconnect. These applications are Floyd-Warshall's all-pair-shortest-path algorithm, Gaussian elimination (GE), QR factorization using the Gram-Schmidt Algorithm (GS) and the multiplication of 2D matrices (MM), successive over-relaxation of a grid (SOR) and the six-step 1D fast fourier transform (FFT) from SPLASH [20]. The input data sizes are shown in Table 1 and the sharing characteristics were discussed in Section 2.1. 5.2 Base Simulation Results In this subsection, we present and analyze the results obtained through extensive simulation runs to compare three systems: the base system (Base), network cache (NC) and switch cache (SC). The Base system does not employ any caching technique beyond the L1 and L2 caches. We simulate a system with NC by enabling 4KB switch caches in all the switching elements of stage 0 in the MIN. Note that stage 0 is the stage close to the processor, while stage 1 is the stage close to the remote memory as shown in Figure 6. The SC system employs switch caches in all the switching elements of the MIN. The main purpose of switch caches in the interconnect is to serve read requests as they traverse to memory. This enhances the performance by reducing the number of read misses served at the remote memory. Figure 16 presents the reduction in the number of read misses to memory by Appl Hit Distribution Sharing GS 69.02 30.98 1.94 2.37 GE 59.55 40.45 1.58 2.66 Table 2: Distribution of switch cache accesses employing network caches (NC) and switch caches (SC) over the base system (Base). In order to perform a fair comparison, here we compare a SC system with 2KB switch caches at both stages (overall 4KB cache space) to a NC system with 4KB network caches. Figure 16 shows that network caches reduce remote read misses by 6-20% for all the applications, except FFT. The multiple layers of switch caches are capable of reducing the number of memory reads requests by upto 45% for FWA, GS and GE applications. Table 2 shows the distribution of switch cache hits across the two stages (St0 and St1) of the network. From the table, we note that a high percentage of requests get satisfied in the switch caches present at the lowest stage in the interconnect. Note however, that for three of the six applications, roughly 30-40% of the requests are switch cache hits in the stage close to the memory (St1). It is also interesting to note the number of requests satisfied by storing each block in the switch cache. Table 2 presents this data as sharing, which is defined as the number of different processor requests served after a block is encached in the switch cache. We find that this sharing degree ranges from 1.0 to 2.7 across all applications. For applications with high overall read sharing degrees (FWA, GS and GE), we find that the the degree of sharing is approximately 1:7 in the stage closer to the processor. With only 4 of 16 processors connected to each switch, many read requests do not find the block in the first stage but get satisfied in the subsequent stage. Thus we find a higher (approximately 2:5) read sharing degree for the stage closer to the remote memory for these applications. The MM application has an overall sharing degree of approximately 4 (see Figure 2). The data is typically shared by four processors physically connected to the same switch in the first stage of the network. Thus most of the requests (88:2%) get satisfied in the first stage and attain a read sharing degree of 1:8. Finally, the SOR and FFT applications have very few read shared requests, most of which are satisfied in the first stage of the network. Figure 17: Impact on average read latency Figure Application execution time improvements Figure 17 shows the improvement in average memory access latency for reads for each application by using switch caching in the interconnect. For each application, the figure consists of three bars corresponding to the Base, NC and SC systems. The average read latency comprises of processor cache delay, bus delay, network data transfer delay, memory service time and queueing delays at the network and memory module. As shown in the figure, by employing network caches, we can improve the average read latency by atmost 15% for most of the applications. With switch caches in multiple stages of the interconnect, we find that the average read latency can be improved by as high as 35% for FWA, GS and GE applications. The read latency reduces by about 7% for the MATMUL application. Again, SOR and FFT are unaffected by network caches or switch caches due to negligible read sharing. The ultimate parameter for performance is execution time. Figure 18 shows the execution time improvement. Each bar in the figure is divided into computation and synchronization time, read stall time and write stall time. In a release consistent system, we find that the write stall time is negligible. However, the read stall time in the base system comprises as high as 50% of the overall Figure 19: Impact of cache size on the number of memory reads Figure 20: Impact of cache size on execution time execution time. Using network caches, we find that the read stall time reduces by a maximum of 20% (for the FWA, GS and GE applications) and thus translates to an improvement in execution time by up to 10%. Using switch caches over multiple stages in the interconnect, we observe execution time improvements as high as 20% in the same three applications. The execution time of the MM application is comparable to that with network caches. SOR and FFT are unaffected by switch caches. 5.3 Sensitivity Studies Sensitivity to Cache Size In order to determine the effect of cache size on performance, we varied the switch cache size from a mere 128 bytes to a large 8KB. Figures 19 & 20 show the impact of switch cache size on the number of memory reads and the overall execution time. As the cache size is increased, we find that a switch cache size of 512 bytes provides the maximum performance improvement (up to 45% reads 5HSO ,QY Figure 21: Effect of cache size on the eviction rate Figure 22: Effect of cache size on switch cache hits across stages and 20% execution time) for three of the six applications. The MM and SOR applications require larger caches for additional improvement. The MM application attains a performance improvement of 7% in execution time at a switch cache size of 2KB. Increasing the cache size further has negligible impact on performance. For SOR, we found some reduction in the number of memory reads, contrary to the negligible amount of sharing in the application (shown in Figure 2). Upon investigation, we found that the switch cache hits come from replacements in the L2 caches. In other words, blocks in the switch cache are accessed highly by the same processor whose initial request entered the block into the switch cache. The switch cache acts as a victim cache for this application. The use of switch caches does not affect the performance of the FFT application. Figure 21 investigates the impact of cache size on the eviction rate and type in the switch cache for the FWA application. The x-axis in the figure represents the size of the cache in bytes. A block in the switch cache can be evicted either due to replacement or due to invalidation. Each bar in the figure is divided into two portions to represent the amount of replacements versus invalidations in the switch cache. The figures are normalized to the number of evictions for a system with 128 byte Figure 23: Effect of line size on the number of memory reads switch caches. The first observation from the figure is the reduction in the number of evictions as the cache size increases. Note that the number of evictions remains constant beyond a cache size of 1KB. With small caches, we also observe that roughly 10-20% of the blocks in the switch cache are invalidated while all others are replaced. In other words, for most blocks, invalidations are not processed through the switch cache since they have already been evicted through replacements due to small capacity. As the cache size increases, we find the fraction of invalidations increase, since fewer replacements occur in larger caches. For the 8KB switch cache, we find that roughly 50% of the blocks are invalidated from the cache. We next look at the impact of cache size on the amount of sharing across stages. Figure 22 shows the amount of hits obtained in each stage of the network for the FWA application. Each bar is divided into two segments, representing each stage of switch caches, denoted by the stage number. Note that Stage0 is the stage closest to the processor interface. From the figure, it is interesting to note that for small caches, an equal amount of hits are obtained from each stage in the network. On the other hand, as the cache size increases, we find that a higher fraction of the hits are due to switch caches closer to the processor interface (60-70% from St0). This is beneficial, because fewer hops are required in the network to access the data, thereby reducing the read latency considerably. Sensitivity to Cache Line Size In the earlier sections, we analyzed data with lines. In this section, we vary the cache line size to study its impact on switch cache performance. Figures 23 & 24 show the impact of a larger cache line (64 bytes) on the switch cache performance for three applications and GE). We vary the cache size from 256 bytes to 16KB and compare the performance to the base system with lines and 64 byte cache lines. Note that the results are normalized to the base system with 64 byte cache lines. We found that the number of memory reads were Figure 24: Effect of line size on execution time Figure 25: Effect of associativity on switch cache hits reduced by 37 to 45% when we increase the cache line size in the base system. However, the use of switch caches still has significant impact on application performance. With 1KB switch caches, we can reduce the number of read requests served at the remote memory by as high as 45% and the execution time by as high as 20%. In summary, the switch cache performance does not depend highly on cache line size for highly read shared applications with good spatial locality. Sensitivity to Set Associativity In this section, we study the impact of cache set associativity on application performance. Figure 25 shows the percentage of switch cache hits as the cache size and associativity are varied. We find that set associativity has no impact on switch cache performance. We believe that frequently accessed blocks need to reside in the switch cache only for a short amount of time, as we observed earlier from our trace analysis. A higher degree of associativity tries to prolong the residence time by reducing cache conflicts. Since we do not require a higher residence time in the switch cache, the performance is neither improved nor hindered. Figure 26: Effect of application size on execution time Sensitivity to Application Size Another concern for the performance of switch caches is the relatively small data set that we have used for faster simulation. In order to verify that the switch cache performance does not change drastically for larger data sets, we used the FWA application and increased the number of vertices from 128 to 192 and 256. Note that the data set size increases by a square of the number of vertices. The base system execution time increases by a factor of 2.3 and 4.6 respectively. With 512 byte switch caches, the execution time reduces by 17% for 128 vertices, 13% for 192 vertices and 10% for 256 vertices. In summary, we believe that switch caches require small cache capacity and can provide sufficient performance improvements for large applications with frequently accessed read shared data. 6 Conclusions In this paper, we presented a novel hardware caching technique, called switch cache, to improve the remote memory access performance of CC-NUMA multiprocessors. A detailed trace analysis of several applications showed that accesses to shared blocks have a great deal of temporal locality. Thus remote memory access performance can be greatly improved by caching shared data in a global cache. To make the global cache accessible to all the processors in the system, the interconnect seems to be the best location since it has the ability to monitor all inter-node transactions in the system in an efficient, yet distributed fashion. By incorporating small caches within each switching element of the MIN, shared data was captured as they flowed from the memory to the processor. In designing such a switch caching framework, several issues were dealt with. The main hindrance to global caching techniques is that of maintaining cache coherence. We organized the caching technique in a hierarchical fashion by utilizing the inherent tree structure of the BMIN. By doing so, caches were kept coherent in a transparent fashion by the regular processor invalidations sent by the home node and other such control infor- mation. To maintain full-map directory information, read requests that hit in the switch cache were marked and allowed to continue on their path to the memory for the sole purpose of updating the directory. The caching technique was also kept non-inclusive and thus devoid of the size problem in a multi-level inclusion property. The most important issue while designing switch caches was that of incorporating a cache within typical crossbar switches (such as SPIDER and CAVALLINO) in a manner such that requests are not delayed at the switching elements. A detailed design of a cache embedded switch architecture (CAESAR) was presented and analyzed. The size and organization of the cache depends heavily on the switch transmission latency. We presented a dual-ported 2-way set associative SRAM cache organization for a 4 \Theta 4 crossbar switch cache. We also proposed a link-based interleaved cache organization to scale the size of the CAESAR module for 8 \Theta 8 crossbar switches. Our simulation results indicate that a small cache of size 1 KB bytes is sufficient to provide up to 45% reduction in memory service and thus a 20% improvement in execution time for some applications. This relates to the fact that applications have a lot of temporal locality in their shared accesses. Current switches such as SPIDER maintain large buffers that are under-utilized in shared memory multiprocessors. It seems that by organizing these buffers as a switch cache, the improvement in performance can be realized. In this paper, we studied the use of switch caches to store recently accessed data in the shared state to be re-used by subsequent requests from any processor in the system. In addition to these requests, applications also have a significant amount of accesses to blocks in the dirty state. To improve the performance of such requests, directories have to be embedded within the switching elements. By providing shared data through switch caches and ownership information through switch directories, the performance of the CC-NUMA multiprocessor can be significantly improved. Latency hiding techniques such as data prefetching or forwarding can also utilize the switch cache and reduce the risk of processor cache pollution. The use of switch caches along with the above latency hiding techniques can further improve the application performance on CC-NUMA multiprocessors tremendously. --R "An Overview of the HP/Convex Exemplar Hardware," "Butterfly Parallel Processor Overview, version 1," "Performance of the Multistage Bus Networks for a Distributed Shared Memory Multiprocessor," "The Impact of Switch Design on the Application Performance of Shared Memory Multiprocessors," "Cavallino: The Teraflops Router and NIC," "A New Solution to Coherence Problems in Multicache Sys- tems," "Scalable Pipelined Interconnect for Distributed Endpoint Routing: The SGI SPIDER Chip," "Tutorial on Recent Trends in Processor Design: Reclimbing the Complexity Curve," "High Frequency Clock Distribution," "The SGI Origin: A ccNUMA Highly Scalable Server," "The Network Architecture of the Connection Machine CM-5," "The Stanford DASH Multiprocessor," "STiNG: A CC-NUMA Computer System for the Commercial Mar- ketplace.," "The Effectiveness of SRAM Network Caches on Clustered DSMs," "A Performance Model for Finite-Buffered Multistage Interconnection Networks," "An Area Model for On-Chip Memories and its Applica- tion," "Design and Analysis of Cache Coherent Multistage Interconnection Networks," "The Impact of Shared-Cache Clustering in Small-Scale Shared-Memory Mul- tiprocessors," "RSIM Reference Manual. Version 1.0," "SPLASH: Stanford Parallel Applications for Shared- Memory," "The SP2 High Performance Switch," "The Performance of the Cedar Multistage Switching Network," "Hierarchical cache/bus architecture for shared memory multiprocessors," "Designing High Bandwidth On-Chip Caches," "An Enhanced Access and Cycle Time Model for On-Chip Caches," "The MIPS R10000 Superscalar Microprocessor," "Reducing Remote Conflict Misses: NUMA with Remote Cache versus COMA," --TR --CTR Takashi Midorikawa , Daisuke Shiraishi , Masayoshi Shigeno , Yasuki Tanabe , Toshihiro Hanawa , Hideharu Amano, The performance of SNAIL-2 (a SSS-MIN connected multiprocessor with cache coherent mechanism), Parallel Computing, v.31 n.3+4, p.352-370, March/April 2005

shared memory multiprocessors;wormhole routing;crossbar switches;scalable interconnects;cache architectures;execution-driven simulation

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Properties of Rescheduling Size Invariance for Dynamic Rescheduling-Based VLIW Cross-Generation Compatibility.

AbstractThe object-code compatibility problem in VLIW architectures stems from their statically scheduled nature. Dynamic rescheduling (DR) [1] is a technique to solve the compatibility problem in VLIWs. DR reschedules program code pages at first-time page faults, i.e., when the code pages are accessed for the first time during execution. Treating a page of code as the unit of rescheduling makes it susceptible to the hazards of changes in the page size during the process of rescheduling. This paper shows that the changes in the page size are only due to insertion and/or deletion of NOPs in the code. Further, it presents an ISA encoding, called list encoding, which does not require explicit encoding of the NOPs in the code. Algorithms to perform rescheduling on acyclic code and cyclic code are presented, followed by the discussion of the property of rescheduling-size invariance (RSI) satisfied by list encoding.

Introduction The object-code compatibility problem in VLIW architectures stems from their statically scheduled nature. The compiler for a VLIW machine schedules code for a specific machine model (or a machine generation), for precise, cycle-by-cycle execution at run-time. The machine model assumptions for a given code schedule are unique, and so are its semantics. Thus, code scheduled for one VLIW is not guaranteed to execute correctly on a different VLIW model. This is a characteristic of VLIWs often cited as an impediment to VLIWs becoming a general-purpose computing paradigm [2]. An example to illustrate this is shown in Figures 1, 2, and 3. Figure 1 shows an example VLIW schedule for a machine model which has two IALUs, one Load unit, one Multiply unit, and one Store unit. Execution latencies of these units are as indicated. Let this machine generation be known as generation X. Figure 2 shows the next-generation (generation where the Multiply and Load d s s s s cycle latency 1 cycle latency IALU IALU MUL G:R8 R4 R3 1 cycle latency 1 cycle latency 3 cycle latency Figure 1: Scheduled code for VLIW Generation X. latencies have changed to 4 and 3 cycles respectively. The generation X schedule will not execute correctly on this machine due to the flow dependence between operations B and C, between D and H, and between E and F. Figure 3 shows the schedule for a generation X+n machine which includes an additional multiplier. The latencies of all FUs remain as shown in Figure 1. Code scheduled for this new machine will not execute correctly on the older machines because the code has been moved in order to take advantage of the additional multiplier. (In particular, E and F have been moved.) There is no trivial way to adapt this schedule to the older machines. This is the case of downward incompatibility between generations. In this situation, if different generations of machines share binaries (e.g., via a file server), compatibility requires either a mechanism to adjust the schedule or a different set of binaries for each generation. One way to avoid the compatibility problem would be to maintain binary executables customized to run on each new VLIW generation. But this would not only violate the copy-protection rules, but would also increase the disk-space usage. Alternatively, program executables may be translated or rescheduled for the target machine model to achieve compatibility. This can be done in hardware or in software. The hardware approach adds superscalar-style run-time scheduling hardware to a VLIW [3], [4], [5], [6], [7]. The principle disadvantage of this approach is that it adds to the complexity of the hardware and may potentially stretch cycle time of the machine if the rescheduling hardware falls in the critical path. The software approach is to perform off-line compilation and scheduling of the program from the source code or from decorated object modules (.o files). Code rescheduled in this manner yields better relative speedups, but the technique is cumbersome to use due to its off-line nature. It could also imply violation of copy protection. Dynamic Rescheduling (DR) [1], is a third alternative to solve the compatibility problem. Under dynamic rescheduling, a program binary compiled for a given VLIW generation machine model is allowed to run on a different VLIW generation. At each first-time page fault (a page-fault that occurs when a code page is accessed for the first time during program d s s s R4 ready s cycle latency 1 cycle latency IALU IALU MUL G:R8 R4 R3 R6 ready R7 ready 1 cycle latency 1 cycle latency 4(3) cycle latency Figure 2: Generation Incompatibility due to changes in functional unit latencies (shown by arrows). The old latencies are shown in parentheses. Operations now produce incorrect results because of the new latencies for operations B, D, and E. s s s R4 ready s cycle latency 1 cycle latency IALU IALU MUL G:R8 R4 R3 R6 ready 1 cycle latency 1 cycle latency 3 cycle latency MUL R7 ready 3 cycle latency Figure 3: Generation X+n schedule: downward incompatibility due to change in the VLIW machine organization. No trivial way to translate new schedule to older generation. execution), the page fault handler invokes a module called the rescheduler, to reschedule the page for that host. Rescheduled code pages are cached in a special area of the file system for future use to avoid repeated translations. Since the dynamic rescheduling technique translates the code on a per-page basis, it is susceptible to the hazard of changes in the page-size due to the process of reschedul- ing. If the changes in the machine model across the generation warrant addition and/or deletion of NOPs to/from the page, it would lead to page overflow or an underflow. This paper discusses a technique called list encoding for the ISA and proves the property of rescheduling-size invariance (RSI), which guarantees that there is no code-size change due to dynamic rescheduling. The organization of this paper is as follows. Section 2 presents the terminology used in this paper. Section 3 briefly describes dynamic rescheduling and demonstrates the problem of code-size change with an example. Section 4 introduces the concept of rescheduling-size invariance (RSI), presents the list encoding, and then proves the RSI properties of list encoding. Section 5 presents concluding remarks and directions for future research. The terminology used in this paper is originally from Rau [3] [8], and is introduced here for the discussion that follows. Each wide instruction-word, or MultiOp, in a VLIW schedule, consists of several operations, or Ops. All Ops in a MultiOp are issued in the same cycle. VLIW programs are latency-cognizant, meaning that they are scheduled with the knowledge of functional unit latencies. An architecture which runs latency-cognizant programs is termed a Non-Unit Assumed Latency (NUAL) architecture. A Unit Assumed Latency (UAL) architecture assumes unit latencies for all functional units. Most superscalar architectures are UAL, whereas most VLIWs are NUAL. The machine models discussed in this paper are NUAL. There are two scheduling models for latency-cognizant programs: the Equals model and the Less-Than-or-Equals (LTE) model [9]. Under the equals model, schedules are built such that each operation takes exactly as much as its specified execution latency. In contrast, under the LTE model an operation may take less than or equal to its specified latency. In general, the equals model produces slightly shorter schedules than the LTE model; this is mainly due to register re-use possible in the equals model. However, the LTE model simplifies the implementation of precise interrupts and provides binary compatibility when latencies are reduced. Both the scheduler (in the back-end of the compiler) and the dynamic rescheduler (in the page-fault handler) presented in this paper follow the LTE scheduling model. For the purposes of this paper, it is assumed that all program codes can be classified into two broad categories: acyclic code and cyclic code. Cyclic code consists of short inner loops in the program which typically are amenable to software pipelining [10]. On the other hand, acyclic code contains a relatively large number of conditional branches, and typically has large loop bodies. This makes the acyclic code un-amenable to software pipelining. Instead, the body of the loop is treated as a piece of acyclic code, surrounded by the loop control Ops. Examples of cyclic code are the inner loops like counted DO-loops found in scientific code. Examples of acyclic code are non-numeric programs, and interactive programs. This distinction between the types of code is made because the scheduling and rescheduling algorithms for cyclic and acyclic code differ considerably, because of which the dynamic rescheduling technique treats each separately. It is also assumed that the program code is structured in the form of superblocks [11] or the hyperblocks [12]. Hyperblocks are constructed by if-conversion of code using predication [13], [14]. Support for predicated execution of Ops is also assumed. Both superblocks and hyperblocks have a single entry point into the block (at the beginning of the block) and may have multiple side-exits. This property is useful in bypassing the problems introduced by speculative code motion in DR, discussion of which can be found elsewhere (see [15]). Overview Reschedule Buffer cache Resume Execution First-time page fault Context switch Figure 4: Dynamic Rescheduling: Sequence of events. The technique of dynamic rescheduling performs translation of code pages at first-time page faults and stores the translated pages for subsequent use. Figure 4 shows the sequence of events that take place in Dynamic Rescheduling. Event 1 indicates a first-time page-fault. On a page-fault, the OS switches context and fetches the requested page from the next level of the memory hierarchy; this is shown shown as Events 2 & 3 respectively. Events 1, 2, 3 are standard in the case of every page fault encountered by the OS. What is different in the case of DR is the invocation of a module called the rescheduler at each first-time page fault. The rescheduler operates on the newly fetched page to reschedule it to execute correctly on the host machine. This is shown as event 4. Event 5 shows that the rescheduled page is written to an area of the file system for future use, and in event 6, the execution resumes. To facilitate the detection of a VLIW generation mismatch at a first-time page fault, each program binary holds a generation-id in its header. The machine model for which the binary was originally scheduled and the boundaries to identify the pieces of cyclic code in the program are also stored in the program binary. This information is made available to the rescheduler it while performs rescheduling. A page of the rescheduled code remains in the main memory until it is displaced (as any other page in the memory), at which time it is written to a special area of the file system called text swap. All subsequent accesses to the page during the lifetime of the program are fulfilled from the text swap. Text swap may be allocated on a per-executable basis at compile time, or be allocated by the OS as a system-wide global area shared by all the active processes. The overhead of rescheduling can be quantitatively expressed in terms of the following factors: (1) the time spent at the first-time page-faults to reschedule the page, (2) the time spent in writing the rescheduled pages to the text swap area, and, (3) the amount of disk space used to store the translated pages. Further discussion of the overhead introduced by DR and an investigation of trade-offs involved in the design of the text-swap used to reduce the overhead are beyond the scope of this paper (see [16] for more details). 3.1 Insertion and deletions of NOPs When the compiler schedules code for a VLIW, independent Ops which can start execution in the same machine cycle are grouped together to form a single MultiOp; each Op in a MultiOp is bound to execute on a specific functional unit. Often, however, the compiler cannot find enough Ops to keep all the FUs busy in a given cycle. These empty slots in a MultiOp are filled with NOPs. In some machine cycles the compiler cannot schedule even a single Op for execution; NOPs are scheduled for all FUs in such a cycle and the instruction is called as an empty MultiOp. Logically, the rescheduler in DR can be thought of as performing the following steps to generate the new code 1 , no matter what the type of code. First, it breaks down each MultiOp into individual Ops, to create an ordered set of Ops. Second, it discards the NOPs from this set. The ordered set of Ops thus obtained is a UAL schedule. In the third step, depending upon the resource constraints and the data dependence constraints, it re-arranges the Ops in the UAL schedule to create the new, NUAL schedule. In the fourth and last step, new NOPs and empty MultiOps are inserted as required to preserve the semantics of the computation. Note that the number of NOPs and the empty MultiOps that are newly inserted may not be the same as that in the old code, which may lead to the problem that the size of the code may change due to rescheduling. It is important to note at this time that the change in the code-size, if any, is only due to the NOPs and the empty MultiOps. An example of changes in code-size is illustrated in Figure 5. In the left portion of the Figure, the old code is shown. Assume that the execution latency of Ops A; D; E; F; G; H is 1-cycle each; that of Op B is 3-cycles and of the LOAD Op C is 2-cycles. Further, Ops E; F are dependent on the result of Op C, hence should not begin execution before Op C finishes execution. In a newer generation of the architecture shown on right, one IALU is removed from the machine, while increasing the latency of the LOADs by 1 (to 3-cycles). When the old code is executed on the newer generation, DR invokes the rescheduler, which generates the new code as shown. To account for the new, longer latency of the LOAD unit, it inserts 1 The terms "new code" and "old code" do not necessarily mean that the code input to the rescheduler belongs to the older machine generation, or similar for the counterpart. "Old code" as used here means any code input to the rescheduler, and new code means the code output by the rescheduler. IALU IALU FPAdd FPMul Load Store Cmpp Br nop nop nop nop nop nop nop nop G nop nop nop nop nop nop H nop Load latency increases, one less IALU F dependent on C, C takes 2 cycles bytes total FPAdd FPMul Load Store Cmpp Br IALU nop nop nop nop nop nop nop nop nop nop nop nop nop nop nop nop nop nop nop A nop F nop nop nop nop nop nop nop G nop nop nop nop nop H nop 336 bytes total (10 extra nops) Figure 5: Example illustrating the insertion/deletion of NOPs and empty MultiOps due to Dynamic Rescheduling. an empty MultiOp in the third cycle. Also, the old MultiOp consisting of operations E and F is broken into two consecutive MultiOps due to the reduction in the number of IALUs. Observe that the new code is bigger than the old code. Assuming all the Ops are 64-bits each, the net increase in the size of the code is 80 bytes, corresponding to the 10-extra NOPs inserted during rescheduling. The page size with which a computer system operates is usually dictated by the hardware or the OS or both. It is non-trivial for the OS to handle any changes in the page sizes at run- time. Previous work done in this area by Talluri and Hill attempts to support multiple page- sizes, where each page-size is an integral multiple of a base page-size [17] [18] [19]. Enhanced VM hardware (the Translation-Lookaside Buffer (TLB), and an enhanced VM management policy must be available in the to support the proposed technique. It is possible that with the help of this extra hardware, multiple code page sizes can be used to handle variations in page-size due to DR, but this would lead to a multitude of problems. The first problem is that of inefficient memory usage: if a new page is created to accommodate the "spill-over" generated by the rescheduler, the remainder of the new page remains unused. On the other hand, if the code in a page shrinks due to DR, that leads to a hole in the memory. The second problem arises due to control restructuring: when a new page is inserted, it must be placed at the end of the code address space. The last MultiOp in the original page must then be modified to jump to the new page, and the last MultiOp on the new page must be modified to jump to the page which lies after the original page. Now, if a code positioning optimization was performed on the old code in order to optimize for I-Cache accesses, this process could violate the ordering, potentially leading to performance degradation. Perhaps the most serious problem is that the code movement within the old page or into the new page could alter branch target addresses (merge points) in the old code, leading to incorrect code. It may not even be possible to repair this code, because the code which jumps to the altered branch targets may not be visible to the rescheduler at rescheduling time. One solution to avoid the problem of code-size change is to use a specialized ISA encoding which "hides" the NOPs and the empty MultiOps in the code. Since all code-size changes in DR are due to the addition/deletion of NOPs, such an encoding circumvents the problem. An encoding called the list encoding which has this ability is discussed in detail in Section 4, along with the rescheduling algorithms for cyclic and acyclic code. Rescheduling Size-Invariance empty MultiOps in the object code, and hence it is a zero-NOP encoding. This property of List encoding is used to support DR. This section presents a formal definition of list encoding, followed by an introduction to the concept of Rescheduling Size-Invariance (RSI). It will also be shown that any list-encoded schedule of code is rescheduling size-invariant . 4.1 List encoding and RSI Operation (Op) is defined by a 6-tuple: fH, p n , s pred , FU- type, opcode, operandsg, where, H 2 f0; 1g is a 1-bit field called header-bit, p n is an n-bit field called pause, s.t. pred is a stage predicate (discussed further in Section 4.3), FUtype uniquely identifies the FU instance where the Op must execute, opcode uniquely identifies the task of the Op, and operands is the set of valid operands defined for Op. All Ops have a constant width. 2 O, is a Header Op iff the value of the header-bit field in O is 1. 2 Definition 3 (VLIW MultiOp) A VLIW MultiOp, M , is defined as an unordered sequence of Ops fO is the number of hardware resources to which the Ops are issued concurrently, and O 1 is a Header Op. 2 Definition 4 (VLIW Schedule) A VLIW schedule, S, is defined as an ordered sequence of MultiOps fM A discussion of the list encoding is now in order. All Ops in this scheme of encoding are fixed-width. In a given VLIW schedule, a new MultiOp begins at a Header Op and ends exactly at the Op before the next Header Op; the MultiOp fetch hardware uses this rule to identify and fetch the next MultiOp. The value of the p n field in an Op is referred to as the pause, because it is used by the fetch hardware to stop MultiOp fetch for the number of machine cycles indicated by p n . This is a mechanism devised to eliminate the explicit encoding of empty MultiOps in the schedule. The FUtype field indicates the functional unit where the Op will execute. The FUtype field allows the elimination of NOPs inserted by the compiler in an arbitrary MultiOp. Prior to the execution of a MultiOp, its member Ops are routed to their appropriate functional units based on the value of their FUtype field. This scheme of encoding the components of a VLIW schedule is termed as List encoding. Since the size of every Op is the same, the size of a given list encoded schedule, S, can be expressed in terms of the number of Ops in it: O j where i is the number of MultiOps in S, O j is an Op, and j is the number of Ops in a given MultiOp. Definition 5 (VLIW Generation) A VLIW generation G is defined by the set fR; Lg, where R is a set of hardware resources in G, and L 2 is the set of execution latencies of all the Ops in the operation set of G. R itself is a set consisting of pairs where r is a resource type and n r is the number of instances of r. This definition of a VLIW generation does not model complex resource usage patterns for each Op, as used in [20], [21] and [22]. Instead, each member of the set of machine resources R, presents a higher-level abstraction of the "functional units" found in modern processors. Under this abstraction, the low-level machine resources such as the register-file ports and operand/result busses required for the execution of an Op on each functional unit are bundled with the resource itself. All the resources indicated in this manner are assumed to be busy through the period of time equal to the latency of the executing Op, indicated by the appropriate member of set L. Definition 6 (Rescheduling Size-Invariance (RSI)) A VLIW schedule S is said to satisfy the RSI property iff sizeof(S Gn are the versions of the original schedule S prepared for execution on arbitrary machine generations G n and Gm re- spectively. Further, schedule S is said to be rescheduling size-invariant iff it satisfies the RSI property. 2 The proof that list encoding is RSI will be presented in two parts. First, it will be shown that acyclic code in the program is RSI when list encoded, followed by the proof that the cyclic code is RSI when list encoded. Since all code is assumed to be either acyclic or cyclic, the result that list encoding makes it RSI will follow. In the remainder of this section, algorithms to reschedule each of these types of codes are presented, followed by the proofs themselves. 4.2 Rescheduling Size-Invariant Acyclic Code The algorithm to reschedule acyclic code from VLIW generation G old to generation G new is shown in Algorithm Reschedule Acyclic Code. It is assumed that both the old and new schedules are LTE schedules (see Section 2), and that both have the same register file architecture and compiler register usage convention. Algorithm Reschedule Acyclic Code input old , the old schedule, (assumed no more than n old cycles long); old g, the machine model description for the old VLIW; new g, the machine model description for the new VLIW; output new , the new schedule; var old , the length of S old ; new , the length of S new ; Scoreboard[number of registers], to flag the registers "in-use" in S old . RU [n new ][ the resource usage matrix, where: r represents all the resource types in G new , and, n r is the number of instances of each resource type r in G new ; UseInfo[n new ][number of registers], to mark the register usage in S new ; the cycle in which an Op can be scheduled while satisfying the data dependence constraints; the cycle in which an Op can be scheduled while satisfying the data dependence and resource constraints; functions RU loopkup (O (T ffi )) returns the earliest cycle, later than cycle T ffi , in which Op O can be scheduled after satisfying the data dependence and resource constraints; RU update (oe; O) marks the resources used by Op O in cycle oe of S new ; dest register (O) returns the destination register of Op O; source register (O) returns a list of all source registers of Op O; latest use time (OE) returns the latest cycle in S new that register OE was used in; most recent writer (ae) returns the id of the Op which modified register ae latest in S old ; begin for each MultiOp M old [c] 2 S old , 0 - c - n old do begin - resource constraint check: for each Op Ow 2 S old that completes in cycle c do begin new [new cycle] /M new [new cycle] j Ow ; RU update (T rc+ffi ; Ow ); - update the scoreboard: for each Op O i 2 S old which is unfinished in cycle c do begin Scoreboard[dest register (O r )] / reserved; do data dependence checks: for each Op O r 2 M old [c] do begin O r (T ffi ) / 0; anti-dependence: for each OE 2 dest register (O r ) do O latest use time (OE)); pure dependence: for each OE 2 source register (O r ) do O r (T ffi - output dependence: for each OE 2 dest register (O r ) do O r (T ffi The RSI property for list encoded acyclic code schedule will now be proved. Theorem 1 An arbitrary list encoded schedule of acyclic code is RSI. Proof: The proof will be presented using induction over the number of Ops in an arbitrary list encoded schedule. Let L i be an arbitrary, ordered sequence of i Ops (i - 1) that occur in a piece of acyclic code. Let F i denote a directed dependence graph for the Ops in L i , i.e. each Op in L i is a node in F i , and the data- and control-dependences between the Ops are indicated by directed arcs in F i . Let SGn be the list encoded schedule for L i generated using the dependence graph F and designed to execute on a certain VLIW generation G n . Also, let Gm denote another VLIW generation which is the target of rescheduling under DR. Induction Basis. L 1 is an Op sequence of length 1. In this case, sizeof (S Gn the dependence graph has a single node. It is trivial in this case that SGn is RSI, because after rescheduling to generation Gm , the number of Ops in the schedule will remain 1, or, Induction Step. L p is an Op sequence of length p, where p ? 1. Assume that SGn is RSI. In other words, Now consider the Op sequence L p+1 , which is of length p+ 1, such that to L p+1 was obtained from L p by adding one Op from the original program fragment. Let this additional Op be denoted by z. Op z can be thought of as borrowed from the original program, such that the correctness of the computation is not compromised. L p is an ordered sequence of Ops, and Op z must then be either a prefix of L p , or a suffix to it. Also, let TGn denote the list encoded schedule for sequence L p+1 , which means sizeof (T Gn 1. In order to prove the current theorem, it must now be proved that TGn is RSI if SGn is RSI. The addition of Op z to L p may change the structure of the dependence graph F p in two ways: (1) if the Op z adds one or more data dependence arcs to F p , or (2) the Op z does not add any data dependence arcs to F p . ffl Op z adds dependence(s): This case corresponds to the fact that Op z is control- and/or data-dependent on one or more of the Ops in L p , or vice versa. Following are the two sub-cases in which a schedule will be constructed which includes the Op z: (1) construction of TGn using the dependence graph F , and, (2) rescheduling of TGn to TGm . In both these cases, all the dependences introduced by Op z must be honored. Further, any resource constraints must be satisfied as well. This is done using the well-known list scheduling algorithm (in the first sub-case), and the Reschedule Acyclic Code algorithm (in the second sub- case). Appropriate NOPs and empty MultiOps will be inserted in the schedule by both these algorithms. However, when the schedules TGn and TGm are list encoded, the empty MultiOps will be made implicit using the pause field in the Header Op of the previous MultiOp, and the NOPs in a MultiOp will be made implicit via the FUtype field in the Ops. Thus, the only source of size increase in schedules TGn and TGm is due to the newly added Op z. ffl Op z does not add any dependences: In this case, only the resource constraints, if any, would warrant the insertion of empty MultiOps. By an argument similar to that in the previous case, it is trivial to see that the only source of size increase in schedules TGn and TGm is the newly added Op z. Thus, in both the cases, sizeof (T Gn 1, from which and from Equation 2, it Similarly, for both the cases, sizeof (T Gm From Equations 3 and 4, and by induction, it is proved that an arbitrary list encoded schedule of acyclic code is RSI. An example of the transition of the code previously shown in Figure 5, by application of algorithm Reschedule Acyclic Code is shown in Figure 6 (assuming that the original schedule belonged to the acyclic category). It can be observed that the size of the original code (on the left) is the same as that of the rescheduled code (on the right). The NOPs and the empty MultiOps have been eliminated in the list encoded schedules; the rescheduling algorithm merely re-arranged the Ops, and adjusted the values of the H and the p n (pause) fields within the Ops to ensure the correctness of execution on G new . 4.3 Rescheduling Size-Invariant Cyclic Code Most programs spend a great deal of time executing the inner loops, and hence the study of scheduling strategies for inner loops has attracted great attention in literature [23], [24], [25], [8], [26], [27], [28], [29]. Inner loops typically have small bodies (relatively fewer Ops) which header bit pause optype rest of the Op Dynamic Rescheduling Figure Example: List encoded schedule for acyclic code is RSI. makes it hard to find ILP within these loop-bodies. Software pipelining is a well-understood scheduling strategy used to expose the ILP across multiple iterations of the loop [30], [25]. There are two ways to perform software pipelining. The first one uses loop unrolling , in which the loop body is unrolled a fixed number of times before scheduling. Loop bodies scheduled via unrolling can be subjected to rescheduling via the Reschedule Acyclic Code algorithm described in Section 4.2. The code expansion introduced due to unrolling, however, is often unacceptable, and hence the second technique, Modulo Scheduling [30], is employed. Modulo-scheduled loops have very little code expansion (as the prologue and epilogue of the loop) which makes it very attractive. In this paper, only modulo-scheduled loops are examined for the RSI property; unrolled-and-scheduled loops are covered by the acyclic RSI techniques presented previously. First, some discussion of the structure of modulo-scheduled loops in presented, followed by an algorithm to reschedule modulo scheduled code. The section ends with a formal treatment to show the list-encoded modulo-scheduled cyclic code is RSI. Concepts from Rau [29] are used as a vehicle for the discussion in this section. Assumptions about the hardware support for execution of modulo scheduled loops are as follows. In some loops, a datum generated in one iteration of the loop is consumed in one of the successive iterations (an inter-iteration data dependence). Also, if there is any conditional code in the loop body, multiple, data-dependent paths of execution exist. Modulo-scheduling such loops is non-trivial 2 . This paper assumes three forms of hardware support to circumvent these problems. First, register renaming via rotating registers [29] in order to handle the inter-iteration data dependencies in loops is assumed. Second, to convert the control dependencies within a loop body to data dependencies, support for predicated 2 See [31] and [32] for some of the work in this area. execution [14] is assumed. Third, support for sentinel scheduling [33] to ensure correct handling of exceptions in speculative execution is assumed. Also, the pre-conditioning [29] of counted-DO loops is presumed to have been performed by the modulo scheduler when necessary. A modulo scheduled loop,\Omega Gn , consists of three parts: a prologue (- Gn ), a kernel (- Gn ), and an epilogue (" Gn ), where G n is the machine generation for which the loop was scheduled. The prologue initiates a new iteration every II cycles, where II is known as the initiation interval. Each slice of II cycles during the execution of the loop is called a stage. In the last stage of the first iteration, execution of the kernel begins. More iterations are in various stages of their execution at this point in time. Once inside the kernel, the loop executes in a steady state (so called because the kernel code branches back to itself). In the kernel, multiple iterations are simultaneously in progress, each in a different stage of execution. A single iteration completes at the end of each stage. The branch Ops used to support the modulo scheduling of loops have special semantics, by which the branch updates the loop counts and enables/disables the execution of further iterations. When the loop condition becomes false, the kernel falls through to the epilogue, which allows for the completion of the stages of the unfinished iterations. Figure 7 shows an example modulo schedule for a loop and identifies the prologue, kernel, and the epilogue. Each row in the schedule describes a cycle of execution. Each box represents a set of Ops that execute in a same resource (e.g. functional unit) in one stage. The height of the box is the II of the loop. All stages belonging to a given iteration are marked with a unique alphabet 2 fA; B; C; D; E; Fg. Figure 7 also shows the loop in a different form: the kernel-only (KO) loop [26] [29]. In a kernel-only loop, the prologue and the epilogue of the loop "collapse" into the kernel, without changing the semantics of execution of the loop. This is achieved by predicating the execution of each distinct stage in a modulo scheduled loop on a distinct predicate called a stage predicate. A new stage predicate is asserted by the loop-back branch. Execution of the stage predicated on the newly asserted predicate is enabled in the future executions of the kernel. When the loop execution begins, stages are incrementally enabled, accounting for the loop prologue. When all the stages are enabled, the loop kernel is in execution and the loop is in the steady state. When the loop condition becomes false, the predicates for the stages are reset, thus disabling the stages one by one. This accounts for the iteration of the epilogue of the loop. A modulo scheduled loop can be represented in the KO form, if adequate hardware (predicated execution) and software (a modulo scheduler to predicate the stages of the loop) support is assumed. Further discussion of KO loop schedules can be found in [29]. All modulo-scheduled loops can be represented in the KO form. The KO form thus has the potential to encode modulo schedules for all classes of loops, a property which is useful in the study of dynamic rescheduling of loops, as will be shown shortly. The size of a modulo scheduled loop is larger than the original size of the loop, if the modulo schedule has an explicit prologue, a kernel, and an epilogue. In contrast, a KO loop schedule has exactly one copy of each stage in the original loop body, and hence has the same size as the original loop body, provided the original loop was completely if-converted 3 . This property of the KO loops is useful in performing dynamic rescheduling of modulo scheduled 3 For any pre-conditioned counted DO-loops, the size is the same as the size of loop body after pre-conditioning A if (p1) if (p2) if (p3) if (p4) if (p5) F if (p6) A A F A F A F A F A F Resources Cycle Prologue Epilogue Kernel1111 Cycle Figure 7: A Modulo scheduled loop: on the left, a modulo scheduled loop (with the prologue, kernel , and the epilogue marked) is shown. The same schedule is shown on the right, but in the "collapsed" kernel-only (KO) form. Stage predicates are used to turn the execution of the Ops ON or OFF in a given stage. The table shows values that the stage predicates would take for this loop. loops. Algorithm Reschedule KO Loop details the steps. The input to the algorithm is the modulo scheduled KO loop, and the machine models for the old and the new generations G old and G new . Briefly, the algorithm works as follows: identification of the predicates that enable individual stages is performed first. An order is imposed on them, which then allows for the derivation of the order of execution of stages in a single iteration. The ordering on the predicates may be implicit in the predicate-id used for a given stage (increasing order of predicate-ids). Alternatively, the order information could be stored in the object file and made available at the time DR is invoked, without substantial overhead. Once the order of execution of the stages of the loop is obtained, the reconstruction of the loop in its original, unscheduled form is performed. At this time, the modulo scheduler is invoked on it to arrive at the new KO schedule for the new generation. Algorithm Reschedule KO Loop input \Omega old =the KO (kernel-only) modulo schedule such that: number of stages old g, the machine model description for the old VLIW; new g, the machine model description for the new VLIW; output schedule for G new ; var old ], the table of n old buckets each holding the Ops from a unique stage, such that the relative ordering of Ops in the bucket is retained; functions FindStagePred (O) returns the stage predicate on which Op O is enabled or disabled; puts the Op O into the bucket B [p]; OrderBuckets (B; func) sorts the table of buckets B according to the ordering function func; StagePredOrdering () describes the statically imposed order on the stage predicates; begin - unscramble the old modulo schedule: for all MultiOps M 2\Omega old do for each Op O 2 M begin FindStagePred (O); - order the buckets: OrderBuckets (B; StagePredOrdering ()); - perform modulo scheduling: Perform modulo scheduling on the sorted table of buckets B, using the algorithm described by Rau in [30] to generate KO schedule\Omega new ; The RSI nature of List encoded modulo scheduled KO loop will now be proved. Theorem 2 An arbitrary List encoded Kernel-Only modulo schedule of a loop is RSI. Proof: Let L i be an arbitrary, ordered sequence of i Ops (i - 1) that represents the loop body. Let F i denote a directed dependence graph for the Ops in L i , i.e. each Op in L i is a node in F i , and the data- and control-dependences between the Ops are indicated by directed arcs in F i . Note that the inter-iteration data dependences are also indicated in F i . Let\Omega Gn denote a list encoded KO modulo schedule for generation G n . Also, let Gm denote the VLIW generation for which rescheduling is performed. Induction Basis. L 1 is a loop body of length 1. In this case, and the dependence graph has a single node. It is trivial in this case that\Omega Gn is RSI, because after rescheduling to generation Gm , the number of Ops in the schedule will remain 1, or, (Note that a loop where is the degenerate case). Induction Step. L p is a loop body of length p, where p ? 1. Assume that\Omega Gn is RSI. In other words, Now consider another loop body L p+1 , which is of length p + 1. Let (p st Op be denoted by z. Also, let \Theta Gn denote the list encoded KO modulo schedule for L p+1 , which means 1. In order to prove the theorem at hand, it must now be proved that \Theta Gn is RSI if\Omega Gn is RSI. It is possible that due to Op z in L p+1 , the nature of the graph F p could be different from that of the graph F p+1 in two ways: (1) Op z is data dependent on one or more Ops in L p+1 or vice versa, or (2) the Op z is independent of all the Ops in L p+1 . In both of these cases, the data dependences and the resource constraints are honored by the modulo scheduling algorithm via appropriate use of NOPs and/or empty MultiOps within the schedule. When this schedule is list encoded, the NOPs and the empty MultiOps are made implicit via the use of pause and the FUtype fields within the Ops. Hence, In other words, From this result and from Equation 6, it follows that: Similarly, for both the cases, sizeof (\Theta Gm From Equations 9 and 10, and by induction, it is proved that an arbitrary list encoded KO modulo schedule is RSI. Corollary 1 A List encoded schedule is RSI. Proof: All program codes can be divided into the two categories: the acyclic code and cyclic code as defined in Section 2. Hence, It follows from Theorem 1 and Theorem 2 that a list encoded schedule is RSI. Conclusions This paper has presented the highlights of a solution for the cross-generation compatibility problem in VLIW architectures. The solution, called Dynamic Rescheduling, performs rescheduling of program code pages at first-time page faults. Assistance from the compiler, the ISA, and the OS is required for dynamic rescheduling. During the process of reschedul- ing, NOPs must be added to/deleted from the page to ensure the correctness of the schedule. Such additions/deletions could lead to changes in the page size. The code-size changes are hard to handle at run-time, would and require extra support in hardware (TLB extensions) and software (VM management extension). An ISA encoding called List Encoding, which encodes the NOPs in the program implicitly, was presented. The list encoded ISA has fixed-width Ops. The Header Op (first Op) in a MultiOp indicates the number of empty MultiOps (if any) following it. This information eliminates the need to explicitly encode the empty MultiOps in the schedule. The OpType field encoded in each Op eliminates the need of explicitly encoding the NOPs within the MultiOp, because the decode hardware can use this information to expand and route the Op to appropriate execution resource. A property of the list encoding called Rescheduling-Size Invariance (RSI) was proved for the acyclic and cyclic (for kernel-only modulo-scheduled) codes. A schedule of code is RSI iff the code size remains constant across the dynamic rescheduling transformation. A study of the instruction fetch hardware and I-Cache organizations required to support the list encoding has previously been studied [34]. The work presented in this paper can be extended with a study of other encoding techniques which may not be rescheduling-size invariant (non-RSI encodings). Also, a study of rescheduling algorithms which operate on non-RSI encodings can be conducted. These topics are currently being investigated by the authors. --R "Dynamic rescheduling: A technique for object code compatibility in VLIW architectures," "Dynamically scheduled VLIW processors," "Hardware support for large atomic units in dynamically scheduled machines," "An architectural framework for supporting heterege- neous instruction-set architectures," "A fill-unit approach to multiple instruction issue," "An architectural framework for migration from CISC to higher performance platforms," "The Cydra 5 departmental supercomputer," "HPL PlayDoh architecture specification: version 1.0," "An approach to scientific array processing: the architectural design of the AP-120B/FPS-164 family," "The Superblock: An effective structure for VLIW and superscalar compilation," "Effective compiler support for predicated execution using the Hyperblock," "Conversion of control dependence to data dependence," "On predicated execution," "Optimization of VLIW compatibility systems employing dynamic rescheduling." "A Persistent Rescheduled-Page Cache for low-overhead object-code compatibility in VLIW architectures," "Tradeoffs in supporting two page sizes," "Virtual memory support for multiple page sizes," "Surpassing the TLB performance of superpages with less operating system support," "A reduced multipipeline machine description that preserves scheduling constraints," "Optimization of machine descriptions for efficient use," "Efficient instruction scheduling using finite state automata," "Some scheduling techniques and an easily schedulable horizontal architecture for high performance scientific computing," "Efficient code generation for horizontal architec- tures: Compiler techniques and architectural support," "Software pipelining: An effective scheduling technique for VLIW machines," "Overlapped loop support in the Cydra 5," "Realistic scheduling: compaction for pipelined architec- tures," new global software pipelining algorithm," "Code generation schemas for modulo scheduled DO-loop and WHILE-loops," "Iterative modulo scheduling: An algorithm for software pipelining loops," Modulo Scheduling with Isomorphic Control Transformations. "Software pipelining of loops with conditional branches," "Sentinel scheduling: A model for compiler-controlled speculative execution," "Instruction fetch mechanisms for VLIW architectures with compressed encodings," --TR --CTR Masahiro Sowa , Ben A. Abderazek , Tsutomu Yoshinaga, Parallel Queue Processor Architecture Based on Produced Order Computation Model, The Journal of Supercomputing, v.32 n.3, p.217-229, June 2005 Jun Yan , Wei Zhang, Hybrid multi-core architecture for boosting single-threaded performance, ACM SIGARCH Computer Architecture News, v.35 n.1, p.141-148, March 2007

list encoding;instruction cache;VLIW;microarchitecture;processor architecture;instruction-set encoding

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Computing Orthogonal Drawings with the Minimum Number of Bends.

AbstractWe describe a branch-and-bound algorithm for computing an orthogonal grid drawing with the minimum number of bends of a biconnected planar graph. Such an algorithm is based on an efficient enumeration schema of the embeddings of a planar graph and on several new methods for computing lower bounds of the number of bends. We experiment with such algorithm on a large test suite and compare the results with the state-of-the-art. The experiments show the feasibility of the approach and also its limitations. Further, the experiments show how minimizing the number of bends has positive effects on other quality measures of the effectiveness of the drawing. We also present a new method for dealing with vertices of degree larger than four.

Introduction Various graphic standards have been proposed to draw graphs, each one devoted to a specific class of applications. An extensive literature on the subject can be found in [4, 23, 2]. In particular, an orthogonal drawing maps each edge into a chain of horizontal and vertical segments and an orthogonal grid drawing is an orthogonal drawing such that vertices and bends along the edges have integer coordinates. Orthogonal grid drawings are widely used for graph visualization in many applications including database systems (entity-relationship diagrams), software engineering (data-flow diagrams), and circuit design (circuit schemat- ics). Research supported in part by the ESPRIT LTR Project no. 20244 - ALCOM-IT. y IASI, CNR, viale Manzoni 30, 00185 Roma Italy. z Dipartimento di Informatica e Automazione, Universit'a di Roma Tre, via della Vasca Navale 84, 00146 Roma, Italy. x Dipartimento di Informatica e Automazione, Universit'a di Roma Tre, via della Vasca Navale 84, 00146 Roma, Italy. Many algorithms for constructing orthogonal grid drawings have been proposed in the literature and implemented into industrial tools. They can be roughly classified according to two main approaches: the topology-shape-metrics approach determines the final drawing through an intermediate step in which a planar embedding of the graph is constructed (see, e.g., [21, 19, 22, 20]); the draw-and-adjust approach reaches the final drawing by working directly on its geometry (see, e.g., [1, 18]). Since a planar graph has an orthogonal grid drawing iff its vertices have degree at most 4, both the approaches assume that vertices have degree at most 4. Such a limitation is usually removed by "expanding" higher degree vertices into two or more vertices. Examples of expansion techniques can be found in [20]. In the topology-shape-metrics approach, the drawing is incrementally specified in three phases. The first phase, planarization, determines a planar embedding of the graph, by possibly adding ficticious vertices that represent crossings. The second phase, orthogonalization, receives as input a planar embedding and computes an orthogonal drawing. The third phase, compaction, produces the final orthogonal grid drawing trying to minimize the area. The orthogonalization step is crucial for the effectiveness of the drawing and has been extensively investigated. A very elegant O(n 2 log n) time algorithm for constructing an orthogonal drawing with the minimum number of bends of an n-vertex embedded planar graph has been presented by Tamassia in [19]. It is based on a minimum cost network flow problem that considers bends along edges as units of flow. However, the algorithm in [19] minimizes the number of bends only within the given planar embedding. Observe that a planar graph can have an exponential number of planar embeddings and it has been shown [6] that the choice of the embedding can deeply affect the number of bends of the drawing. Namely, there exist graphs that, for a certain embedding have a linear number of bends, and for another embedding have only a constant number. Unfortunately, the problem of minimizing the number of bends in a variable embedding setting is NP-complete [10, 11]. Optimal polynomial-time algorithms for subclasses of graphs are shown in [6]. Because of the tight interaction between the graph drawing area and applica- tions, the attention to experimental work on graph drawing is rapidly increasing. In [5] it is presented an experimental study comparing topology-shape-metrics and draw-and-adjust algorithms for orthogonal grid drawings. The test graphs were generated from a core set of 112 graphs used in "real-life" software engineering and database applications with number of vertices ranging from 10 to 100. The experiments provide a detailed quantitative evaluation of the performance of seven algorithms, and show that they exhibit trade-offs between "aesthetic" properties (e.g., crossings, bends, edge length) and running time. The algorithm GIOTTO (topology-shape-metrics with the Tamassia's algorithm in the orthogonalization step) performs better than the others at the expenses of a worst time performance. Other examples of experimental work in graph drawing follow. The performance of four planar straight-line drawing algorithms is compared in [14]. Himsolt [12] presents a comparative study of twelve graph drawings algorithms; the algorithms selected are based on various approaches (e.g., force-directed, lay- ering, and planarization) and use a variety of graphic standards (e.g., orthogonal, straight-line, polyline). The experiments are conducted with the graph drawing system GraphEd [12]. The test suite consists of about 100 graphs. Brandenburg and Rohrer [3] compare five "force-directed" methods for constructing straight-line drawings of general undirected graphs. Juenger and Mutzel [15] investigate crossing minization strategies for straight-line drawings of 2-layer graphs, and compare the performance of popular heuristics for this problem. In this paper, we present the following results. Let G be a biconnected planar graph such that each vertex has degree at most 4 (4-planar graph). ffl We describe a branch-and-bound algorithm called BB-ORTH that computes an orthogonal grid drawing of G with the minimum number of bends in the variable embedding setting. The algorithm is based on: several new methods for computing lower bounds on the number of bends of a planar graph (Section 3). Such methods give new insights on the relationships between the structure of the triconnected components and the number of bends; a new enumeration schema that allows to enumerate without repetitions all the planar embeddings of G (Section 4). Such enumeration schema exploits the capability of SPQR-trees [7, 8] in implicitely representing the embeddings of a planar graph. ffl We present a system that implements BB-ORTH. Such a system is provided with a graphical interface that animates all the phases of the algorithm and displays partial results, exploiting existing graph drawing algorithms to represent all the graphs involved in the computation. The interaction with the system allows to stop the computation once a sufficiently good orthogonal drawing is displayed. (Section 5). ffl We test BB-ORTH against a large test suite of randomly generated graphs with up to 60 vertices and compare the experimental results with the best state-of-the-art results (algoritm GIOTTO) (Section 6). Our experiments show: an improvement in the number of bends of 20-30%; an improvement of several other quality measures of the drawing that are affected from the number of bends. For example the length of the longest edge of the drawings obtained with BB-ORTH is about 50% smaller than the length of the longest edge of the drawings obtained with GIOTTO; a sensible increasing of the CPU-time that however is perfectly affordable within the typical size of graphs of real-life applications [5]. ffl Also, BB-ORTH can be easily applied on all biconnected components of graphs. This yields a powerful heuristic for reducing the number of bends in connected graphs. Further, the limitation on the degree of the vertices can be easily removed by using the expansion techniques cited above. Preliminaries We assume familiarity with planarity and connectivity of graphs [9, 17]. Since we consider only planar graphs, we use the term embedding instead of planar embedding. An orthogonal drawing of a 4-planar graph G is optimal if it has the minimum number of bends among all the possible orthogonal drawings of G. When this is not ambiguous, given an embedding OE of G, we also call optimal the porthogonal drawing of G with the minimum number of bends that preserves the embedding OE. Let G be a biconnected graph. A split pair of G is either a separation-pair or a pair of adjacent vertices. A split component of a split pair fu; vg is either an edge (u; v) or a maximal subgraph C of G such that C contains u and v, and fu; vg is not a split pair of C. A vertex w distinct from u and v belongs to exactly one split component of fu,vg. are some pairwise edge disjoint split components of G with split pairs respectively. The graph G 0 obtained by substituting each of G partial graph of G. We denote E virt (E nonvirt ) the set of (non-)virtual edges of G 0 . We say that G i is the pertinent graph of e is the representative edge of G i . Let OE be an embedding of G and let OE 0 be an embedding of G 0 . We say that OE preserves OE 0 if G 0 can be obtained from G OE by substituting each component G i with its representative edge. In the following we summarize SPQR-trees. For more details, see [7, 8]. SPQR-trees are closely related to the classical decomposition of biconnected graphs into triconnected components [13]. Let fs; tg be a split pair of G. A maximal split pair fu; vg of G with respect to fs; tg is a split pair of G distinct from fs; tg such that for any other split pair of G, there exists a split component of fu containing vertices u, v, s, and t. be an edge of G, called reference edge. The SPQR-tree T of G with respect to e describes a recursive decomposition of G induced by its split pairs. Tree T is a rooted ordered tree whose nodes are of four types: S, P, Q, and R. Each node of T has an associated biconnected multigraph, called the skeleton of , and denoted by skeleton(). Also, it is associated with an edge of the skeleton of the parent of , called the virtual edge of in skeleton(). Tree T is recursively defined as follows. If G consists of exactly two parallel edges between s and t, then T consists of a single Q-node whose skeleton is G itself. If the split pair fs; tg has at least three split components G the root of T is a P-node . Graph skeleton() consists of k parallel edges between s and t, denoted e Otherwise, the split pair fs; tg has exactly two split components, one of them is the reference edge e, and we denote with G 0 the other split component. If G 0 has cutvertices c that partition G into its blocks G in this order from s to t, the root of T is an S-node . Graph skeleton() is the cycle t, and e i connects c i\Gamma1 with c i If none of the above cases applies, let fs be the maximal split pairs of G with respect to fs; tg (k 1), and for be the union of all the split components of fs but the one containing the reference edge e. The root of T is an R-node . Graph skeleton() is obtained from G by replacing each subgraph G i with the edge e i between s i and t i . Except for the trivial case, has children k in this order, such that i is the root of the SPQR-tree of graph G i [ e i with respect to reference edge said to be the virtual edge of node i in skeleton() and of node in skeleton( i ). Graph G i is called the pertinent graph of node and of edge e i . The tree T so obtained has a Q-node associated with each edge of G, except the reference edge e. We complete the SPQR-tree by adding another Q-node, representing the reference edge e, and making it the parent of so that it becomes the root. Let be a node of T . We have: if is an R-node, then skeleton() is a triconnected if is an S-node, then skeleton() is a cycle; if is a P-node, then skeleton() is a triconnected multigraph consisting of a bundle of multiple edges; and if is a Q-node, then skeleton() is a biconnected multigraph consisting of two multiple edges. The skeletons of the nodes of T are homeomorphic to subgraphs of G. The SPQR-trees of G with respect to different reference edges are isomorphic and are obtained one from the other by selecting a different Q-node as the root. Hence, we can define the unrooted SPQR-tree of G without ambiguity. The SPQR-tree T of a graph G with n vertices and m edges has m Q-nodes and O(n) S-, P-, and R-nodes. Also, the total number of vertices of the skeletons stored at the nodes of T is O(n). A graph G is planar if and only if the skeletons of all the nodes of the SPQR- tree T of G are planar. An SPQR-tree T rooted at a given Q-node represents all the planar embeddings of G having the reference edge (associated to the Q-node at the root) on the external face. 3 Lower Bounds for Orthogonal Drawings In this section we propose some new lower bounds on the number of bends of a biconnected planar graph. The proofs of the theorems are omitted due to space limitation. E) be a biconnected 4-planar graph and H be an orthogonal drawing of G, we denote by b(H) the total number of bends of H and by b E 0 the number of bends along the edges of E 0 subgraphs of G such that be an optimal orthogonal drawing of G i and let H be an orthogonal drawing of G. Then, b(H) be an embedded partial graph of G and let H 0 be an orthogonal drawing of G 0 . Suppose H 0 is such that b E nonvirt (H 0 an embedding OE of G that preserves OE 0 and an optimal orthogonal drawing H OE of G OE . From Property 1 and Lemma 1 it follows a first lower bound. Theorem 1 Let E) be a biconnected 4-planar graph and G 0 an embedded partial graph of G. For each virtual edge e i of G 0 , a lower bound on the number of bends of the pertinent graph G i of e i . Consider an orthogonal drawing H 0 of G 0 such that b E nonvirt (H 0 be an embedding of G that preserves OE 0 , consider any orthogonal drawing H OE of G OE . We have that: Remark 1 An orthogonal drawing H 0 of G 0 such that b E nonvirt(HOE 0 can be easily obtained by using the Tamassia's algorithm [19]. Namely, when two faces f and g share a virtual edge, the corresponding edge of the dual graph in the minimum cost flow problem of [19] is set to zero. A second lower bound is described in the following theorem. Theorem 2 Let G OE be an embedded biconnected 4-planar graph and G 0 an embedded partial graph of G, such that OE preserves OE 0 . Derive from G 0 the embedded graph G OE 0 by substituting each virtual edge e i of G 0 , any simple path from u i to v i in G i . Consider an optimal orthogonal drawing and an orthogonal drawing H OE of G OE . Then we have that: The next corollary allows us to combine the above lower bounds into a hybrid technique. Corollary 1 Let G OE be an embedded biconnected 4-planar graph and G 0 an embedded partial graph of G. Consider a subset F virt of the set of the virtual edges of G 0 and derive from G 0 the graph G 00 by substituting each edge e i 2 F virt with any path from u i to v i in G i . Denote by E p the set of edges of G introduced by such substitution. For each e be a lower bound on the number of bends of the pertinent graph G j of e j . Consider an orthogonal drawing H 00 of G 00 , such that b E nonvirt(HOE 0 Let H OE be an orthogonal drawing of G OE , we have that: 4 A Branch and Bound Strategy Let G be a biconnected 4-planar graph. In this section we describe a technique for enumerating all the possible orthogonal drawings of G and rules to avoid examining all of them in the computation of the optimum. The enumeration exploits the SPQR-tree T of G. Namely, we enumerate all the orthogonal drawings of G with edge e on the external face by rooting T at e and exploiting the capacity of T rooted at e in representing all the embeddings having e on the external face. A complete enumeration is done by rooting T at all the possible edges. We encode all the possible embeddings of G, implicitely represented by T rooted at e, as follows. We visit the SPQR-tree such that a node is visited after its parent, e.g. depth first or breadth first. This induces a numbering r of the P- and R-nodes of T . We define a r-uple of variables r that are in one-to-one correspondence with the P- and R-nodes of T . Each variable x i of X corresponding to a R-node i can be set to three values corresponding to two swaps of the pertinent graph of i plus one unknown value. Each variable x j of X corresponding to a P-node j can be set to up to seven values corresponding to the possible permutations of the pertinent graphs of the children of j plus one unknown value. Unknown values represent portions of the embedding that are not yet specified. A search tree B is defined as follows. Each node fi of B corresponds to a different setting X fi of X . Such setting is partitioned into two contiguous (one of them possibly empty) subsets x Elements of the first subset contain values specifying embeddings while elements in the second subset contain unknown values. Leaves of B are in correspondence with settings of X without unknown values. Internal nodes of B are in correspondence with settings of X with at least one unknown value. The setting of the root of B consists of unknown values only. The children of fi (with subsets x and x child for each possible value of x h+1 . Observe that there is a mapping between embedded partial graphs of G and nodes of B. Namely, the embedded partial graph G fi of G associated to node fi of B with subsets x obtained as follows. First, set G fi to skeleton( 1 ) embedded according to x 1 . Second, substitute each virtual edge e i of 1 with the skeleton of the child i of 1 , embedded according to x i , only for 2 i h. Then, recursively substitute virtual edges with embedded skeletons until all the skeletons in fskeleton( 1 are used. We visit B breadth-first starting from the root. At each node fi of B with setting X fi we compute a lower bound and an upper bound of the number of bends of any orthogonal drawing of G such that its embedding is (partially) specified by X fi . The current optimal solution is updated accordingly. Children are not visited if the lower bound is greater than the current optimum. For each fi, lower bounds and upper bounds are computed as follows by using the results presented in Section 3. 1. We construct G fi using an array of pointers to the nodes of T . Observe that at each substitution of a virtual edge with a skeleton a reverse of the adjacency lists might be needed. 2. We compute lower bounds. G fi is a partial graph of G, with embedding derived from X fi . Let E virt be the set of virtual edges of G fi . For each consider the pertinent graph G i and compute a lower bound b i on the number of bends of G i using Theorem 1. For each e i such that the value of such lower bound is zero we substitute in G fi edge e i with any simple path from u i to v i in G i , so deriving a new graph G 0 fi . Denote by F virt the set of such edges and by E p the set of edges of G introduced by the substitution (observe that E p may be empty). By Remark 1 we apply Tamassia's algorithm on G 0 , assigning zero costs to the edges of the dual graph associated to the virtual edges. We obtain an orthogonal drawing of G 0 with minimum number of bends on the set ) be such a number of bends. Then, by Corollary 1 we compute the lower bound L fi at node fi, as Lower bounds (b i ) can be pre-computed with a suitable pre-processing by visiting T bottom-up. The pre-computation consists of two phases. We apply Tamassia's algorithm on the skeleton of each R- and P-node, with zero cost for virtual edges; in this way we associate a lower bound to each R- and P-node of T . Note that these pre-computed bounds do not depend on the choice of the reference edge, so they are computed only once, at the beginning of BB-ORTH. We visit T bottom-up summing for each Rand P-node the lower bounds of the children of to the lower bound of . Note that these pre-computed bounds depend on the choice of the reference edge, so they are re-computed at any choice of the reference edge. 3. We compute upper bounds. Namely, we consider the embedded partial graph G fi and complete it to a pertinent embedded graph G OE . The embedding of G OE is obtained by substituting the unknown values of X fi with embedding values in a random way. Then we apply Tamassia's algorithm to G OE so obtaining the upper bound. We also avoid multiple generations of the same embedded graph in completing the partial graph. We compute the optimal solution over all possible choices of the reference edge. To do so, our implementation of SPQR-trees supports efficient evert oper- ations. Also, to expedite the process we: reuse upper bounds computed during computations referring to different choices of the virtual edge; avoid to compute solutions referring to embeddings that have been already explored. If an edge e that has already been reference edge appears on the external face of some G fi , since we have already explored all the embeddings with e on the external face, we cut from the search tree all the descendants of fi. One of the consequences of the above discussion is summarized in the following theorem: Theorem 3 Let G be a biconnected planar graph. The enumeration schema adopted by BB-ORTH examines each planar embedding of G exactly once. 5 Implementation The system has been developed with the C++ language and has full Leda [16] compatibility. Namely, several classes of Leda have been refined into new classes: Embedded Graphs, preserve the embedding for each update operation. We have taken care of a new efficient implementation of several basic methods like DFS, BFS, st-numbering, topological sorting, etc; Directed Embedded Graphs, with several network flow facilities. In particular we implemented the minimum cost flow algorithm that searches for minimum cost augmenting paths with the Dijkstra algorithm, used in the original paper of Tamassia [19]. We implemented the corresponding priority queues with the Fibonacci-heaps of Leda; Planar Embedded Graphs, with faces and dual graph; Orthogonal planar embedded graphs, to describe orthogonal drawings; Drawn Graphs, with several graphical attributes; SPQR-trees, with several supporting classes, like split-components, skeletons, etc. As far as we know, this is the first implementation of the SPQR-trees. Fur- ther, to support experiments, we have developed a new Leda graph editor, based on the previous Leda editor. The graphical interface shows an animation of the algorithm in which the SPQR-tree is rooted at different reference edges. Also, it displays the relationships between the SPQR-tree and the search tree and the evolution of the search tree. Clicking at SPQR-tree nodes shows skeletons of nodes. Of course, to show the animation, the graphical interface exploits several graph drawing algorithms. Fig. 1 shows a 17-verticed graph drawn by the GIOTTO, and by BB-ORTH, respectively. Fig. 2 shows a 100-vertices graph drawn by the GIOTTO, and by BB-ORTH, respectively. Finally, we observe that a careful cases analysis of each P-node allows faster computations of preprocessing lower bounds; such analysis is based on the number of virtual edges of the skeleton of , on the values of the precomputed lower bounds of the children of and on length of the shortest paths in the pertinent graphs of the children of . 6 Experimental Setting and Computational Result We have tested our algorithm against two randomly generated test suites overall consisting of about 200 graphs. Such graphs are available on the web (www.inf.uniroma3.it/people/gdb/wp12/LOG.html) and have been generated as follows. It is well known that any embedded planar biconnected graph can be generated from the triangle graph by means of a sequence of insert-vertex and insert-edge operations. Insert-vertex subdivides an existing edge into two new edges separated by a new vertex. Insert-edge inserts a new edge between two existing vertices that are on the same face. Thus, we have implemented a generation mechanism that at each step randomly chooses which operation to apply and where to apply it. The two test suites differ in the probability distributions we have given for insert-vertex and insert-edge. Also, to stress the performance of the algorithm we have discarded graphs with "a small number" of triconnected components. Observe that we decided to define a new test suite instead of using the one of [5] because the biconnected components of those graphs do not have a sufficiently large number of triconnected components (and thus a sufficient number of different embeddings) to stress enough BB-ORTH. For each drawing obtained by the algorithm, we have measured the following quality parameters. bends: total number of edge bends. maxedgebends: max number of bends on an edge. unifbends: standard deviation of the number of edge bends. area: area of the smallest rectangle with horizontal and vertical sides covering the drawing. totaledgelen: total edge length. maxedgelen: maximum length of an edge. uniflen: standard deviation of the edge length. screenratio: deviation of the aspect ratio of the drawing (width/height or height/width) from the optimal one (4/3). Measured performance parameters are: cpu time, total number of search tree nodes, number of visited search tree nodes. Due to space limitations, we show the experimental results with respect to only. They are summarized in the graphics of Fig. 3 On the x-axis we either have the number of vertices or the total number (RP) of R- and P-nodes. All values in the graphics represent average values. Fig. 3.A, 3.C and 3.D compare respectively the number of bends, the area and the maxedgelen of the drawing of BB-ORTH with the ones of GIOTTO. Graphics with white points represent the behavior of BB-ORTH. Graphics with black points represent the behavior of GIOTTO. The improvement on the number of bends positively affects all the quality measures. In Fig. 3.B, 3.D and 3.F we show the percentages of the improvements. Concerning performance, Fig. 3.G shows the CPU-time of BB-ORTH on a RISC6000 IBM. The algorithm can take about one hour on graphs with vertices. Finally, Fig. 3.H compares the number of nodes of the search tree and the number of nodes that are actually visited. It shows that for high values of RP the percentage of visited nodes is dramatically low (below 10 %). 7 Conclusions and Open Problems Most graph drawing problems involve the solution of optimization problems that are computationally hard. Thus, for several years the resarch in graph drawing has focused on the development of efficient heuristics. However, the graphs to be drawn are frequenly of small size (it is unusual to draw an Entity-Relationship or Data Flow diagrams with more than 70-80 vertices) and the available work-stations allow faster and faster computation. Also, the requirements of the users in terms of aesthetic features of the interfaces is always growing. Hence, recently there has been an increasing attention towards the development of graph drawing tools that privilege the effectiveness of the drawing against the efficiency of the computation (see e.g., [15]). In this paper we presented an algorithm for computing an orthogonal drawing with the minimum number of bends of a graph. The computational results that we obtained are encouraging both in terms of the quality of the drawings and in terms of the time performance. Several problems are still open. For example: is it possible to apply variations of the techniques presented in this paper to solve the upward drawability testing problem? A directed graph is upward planar if it is planar and can be drawn with all the edges following the same direction. The problem to test a directed graph for upward drawability is NP-complete [10]; how difficult is it to find an enumeration schema and lower bounds for the problem of computing the orthogonal drawing in 3-space with the minimum number of bends? Acknowledgements We are grateful to Antonio Leonforte for implementing part of the system. We are also grateful to Sandra Follaro, Armando Parise, and Maurizio Patrignani for their support. --R A better heuristic for orthogonal graph drawings. Graph Drawing An experimental comparison of force-directed and randomized graph drawing algorithms Algorithms for drawing graphs: an annotated bibliography. An experimental comparison of three graph drawing algorithms. Spirality of orthogonal representations and optimal drawings of series-parallel graphs and 3-planar graphs Graph Algorithms. On the computational complexity of upward and rectilinear planarity testing. On the computational complexity of upward and rectilinear planarity testing. Comparing and evaluating layout algorithms within GraphEd. Dividing a graph into triconnected compo- nents A note on planar graph drawing algorithms. Exact and heuristic algorithms for 2-layer straightline crossing minimization LEDA: a platform for combinatorial and geometric computing. Planar graphs: Theory and algorithms. Improved algorithms and bounds for orthogonal drawings. On embedding a graph in the grid with the minimum number of bends. Automatic graph drawing and readability of diagrams. A unified approach to visibility representations of planar graphs. Efficient embedding of planar graphs in linear time. Graph Drawing --TR --CTR Markus Eiglsperger , Michael Kaufmann , Martin Siebenhaller, A topology-shape-metrics approach for the automatic layout of UML class diagrams, Proceedings of the ACM symposium on Software visualization, June 11-13, 2003, San Diego, California Giuseppe Di Battista , Walter Didimo , Maurizio Patrignani , Maurizio Pizzonia, Drawing database schemas, SoftwarePractice & Experience, v.32 n.11, p.1065-1098, September 2002 Markus Eiglsperger , Carsten Gutwenger , Michael Kaufmann , Joachim Kupke , Michael Jnger , Sebastian Leipert , Karsten Klein , Petra Mutzel , Martin Siebenhaller, Automatic layout of UML class diagrams in orthogonal style, Information Visualization, v.3 n.3, p.189-208, September 2004

planar embedding;orthogonal drawings;bends;branch and bound;planar graphs;graph drawing

354877

Xor-trees for efficient anonymous multicast and reception.

We examine the problem of efficient anonymous multicast and reception in general communication networks. We present algorithms that achieve anonymous communication, are protected against traffic analysis, and require O(1) amortized communication complexity on each link and low computational comlexity. The algorithms support sender anonymity, receiver(s) anonymity, or sender-receiver anonymity.

Introduction One of the primary objectives of an adversary is to locate and to destroy command-and-control centers - that is, sites that send commands and data to various stations/agents. Hence, one of the crucial ingredients in almost any network with command centers is to conceal and to confuse the adversary regarding which stations issue the commands. This paper shows how to use standard off-the-shelf cryptographic tools in a novel way in order to conceal the command-and-control centers, while still assuring easy communication between the centers and the recipients. Specifically, we show efficient solutions that hide who is the sender and the receiver (or both) of the message/directive in a variety of threat models. The proposed solutions are efficient in terms of communication overhead (i.e., how much additional information must be transmitted in order to confuse the adversary) and in terms of computation efficiency (i.e., how much computation must be performed for concealment). Moreover, we establish rigorous guarantees about the proposed solutions. 1.1 The problem considered Modern cryptographic techniques are extremely good in hiding all the contents of data, by means of encrypting the messages. However, hiding the contents of the message does not hide the fact that some message was sent from or received by a particular site. Thus, if some location (or network node) is sending and/or receiving a lot of messages, and if an adversary can monitor this fact, then even if an adversary does not understand what these messages are, just the fact that there are a lot of outgoing (or incoming) messages reveals that this site (or a network node) is sufficiently active to make it a likely target. The objective of this paper is to address this problem - that is, the problem of how to hide, in an efficient manner, which site (i.e. command-and-control center) transmits (or receives) a lot of data to (or from, respectively) other sites in the network. This question was addressed previously in the literature [Ch81, RS93] at the price of polynomial communication overhead for each bit of transmission per edge. We show an amortized solution which after a fixed pre-processing stage, can transmit an arbitrary polynomial-size message in an anonymous fashion using only bits over each link (of a spanning tree) for every data bit transmission across a link. 1.2 General setting and threat model We consider a network of processors/stations where each processor/station has a list of other stations with which it can communicate (we do not restrict here the means of communication, i.e. it could be computer networks, radio/satellite connections, etc.) Moreover, we do not restrict the topology of the network - our general methodology will work for an arbitrary network topology. One (or several) of the network nodes is a command-and-control center that wishes to send commands (i.e. messages) to other nodes in the network. To reiterate, the question we are addressing in this paper is how we can hide which site is broadcasting (or multicasting) data to (a subset of) other processors in the network. Before we explore this question further, we must specify what kind of attack we are defending against. A simple attack to defend against is of a restricted adversary (called outside adversary) who is allowed only to monitor communication channels, but is not allowed to infiltrate/monitor the internal contents of any processor of the network. (As a side remark, such weak attack is very easy to defend against: all processors simply transmit either noise or encrypted messages on each communication channel - if noise is indistinguishable from encrypted traffic this completely hides a communication pattern.) Of course, a more realistic adversary, (and the one that we are considering in this paper) is the (internal) adversary that can monitor all the communication between stations and which in addition is also trying to infiltrate the internal nodes of the network. That is, we consider the adversary that may mount a more sophisticated attack, where he manages to compromise the security of one or several internal nodes of the network, whereby he is now not only capable of monitoring the external traffic pattern but is also capable of examining every message and all the data which passes through (or stored at) this infiltrated node. Thus, we define an internal k-listening adversary, an adversary that can monitor all the communication lines between sites and also manages to monitor (the internal contents of) up to k sites of the network. (This, and similar definitions were considered before in the literature, see, for example [RS93, CKOR97] and references therein). We remark, though, that in this paper we restrict out attention only to listening adversary, that only monitors traffic, but does not try to sabotage it, similar to [FGY93, KMO94], but with different objectives. 1.3 Comparison with Previous Work One of the first works (if not the first one) to consider the problem of hiding the communication pattern in the network is the work of Chaum [Ch81] where he introduced the concept of a mix: A single processor in the network, called a mix, serves as a relay. A processor P that wants to send a message m to a processor Q encrypts m using Q's public key to obtain m 0 . Then P encrypts the pair (m using the public key of the mix. The double encrypted message is sent to the mix. The mix decrypts the message (to get the pair (m 0 ; q)) and forwards m 0 to q. Further work in this direction appear in [Pf85, PPW91, SGR97]. The single mix processor is not secure when this single processor is cooperating with the (outside) adversary; If the processor that serves as a mix is compromised, it can inform the adversary where the messages are forwarded to. Hence, as Chaum pointed out, a sequence of "mixes" must be employed at the price of additional communication and computation. Moreover, the single mix scheme operates under some statistic assumption on the pattern of communication. In case a single message is sent to the mix then an adversary that monitors the communication channels can observe the sender and the receiver of the particular message. An extension of the mix scheme is presented by Rackoff and Simon [RS93] who embedded an n-element sorting network of depth polynomial in log(n) that mixes incoming messages and requires only polynomially many (in log(n)) synchronous steps. In each such step every message is sent from one site of the network to another site of the network. Thus, the message delay may be proportional to log(n) times the diameter of the network. The statistic assumptions on the pattern of communication is somewhat relaxed in [RS93] by introducing dummy communication: Every processor sends a message simultaneously. However, the number of (real and dummy) messages arriving to each destination is available to the traffic analyzer. Rackoff and Simon also presented in [RS93] a scheme that copes with passive internal adversaries by the use of randomly chosen committees and multi-party computation (e.g., [GMW87, BGW88, CCD88, CFGN96, CKOR97].) More generally, secure multi-party computation can be used to hide the communication pattern in the network (see, for example, [GMW87, Ch88, WP90, BGW88, CCD88, CFGN96, CKOR97]) via secure function valuation. However, anonymous communication is a very restricted form of hiding participants' input and hence may benefit from less sophisticated and more efficient algorithms. In particular, Chaum suggested in [Ch88] to use the dc-net approach in order to achieve anonymous communication. Our approach is similar to the dining cryptographers solution in [Ch88], where a graph characterization of the random bits distribution is given. We present a specific choice (an efficient instance that satisfies Chaum's graph characterization) of selecting (a small number of) keys for each processor, and a procedure to securely distribute the keys and use O(1) amortized communication complexity on each link. Our algorithm is proven correct by a new argument proving that each bit communicated has an equal probability to be 0 and to be 1 for a particular adversary. In [Ch88] the case of anonymous sender is considered, in this work we suggest schemes also for the cases in which the receiver is (receivers are, respectively) anonymous, and in which both the sender and the receiver are anonymous to each other. In [Ch88] it is assumed that the underlying communication networks is a ring or that a back-off mechanism is repeatedly used to send data. In this work we consider the problem of anonymous communication on a spanning tree of a general graph communication network. We note that solutions for star and tree networks are briefly mentioned in [Pf85, PW87], with no details for the way communication starts and terminates for this specific networks. Our contributions is a detailed design for a (spanning) tree communication network. The details include: a new scheme for seeds selection that ensures anonymity in the presence of an outside adversary and k-listening internal dynamic adversary. Schemes for anonymous receiver, as well as anonymous sender and receiver. We specify the initialization (including seed distribution), the communication and the termination procedures that preserve anonymity for the case of a (spanning) tree communication network. In addition, we use an extra random sequence (that is produced by a pseudo random generator) shared by the sender (and the receiver(s)) to encrypt (decrypt, respectively) the message, avoiding the use of additional different scheme for encryption and decryption during the transmission of the (long) messages. This new approach fits transmission of a very long sequence of bits, such as video information to several recipient. Thus, it can be used for anonymous multicast such as multicast by cable TV. Our initialization scheme is designed to cope with the problem of the information revealed by the back-off mechanism (see [BB89]) by using a predefined ordered of transmission. We note that in this work we do not concern ourselves with active adversary that can corrupt the program or forge messages on the links as assumed in [Wa89]. The extensions suggested in [Wa89] is a design of a fail-stop broadcast instead of assuming reliable broadcast. In a network of n processors our algorithm (after a pre-processing stage) sends O(1) bits on each tree link in order to transmit a clear-text bit of data and each processor computes pseudo-random bits for the transmission of a clear-text bit. Multiple anonymous transmission is possible by executing in parallel several instances of our algorithm. Each instance uses part of the bandwidth of the communication links. Our algorithm is secure for both outside adversary and k-listening internal dynamic adversary. (We remark, though, that we are only considering eavesdropping "listening" adversary, similar to [FGY93, KMO94], and do not consider a Byzantine adversary which tries to actively disrupt the communication, as in [GMW87].) 1.4 A simple example In this subsection, we examine a very simple special case, in order to illustrate the issues being considered and a solution to this special case. We stress, though, that we develop a general framework that works for the general case (e.g. the case of general communication graph, unknown receiver, etc.) as well. Suppose we are dealing with a network having 9 nodes: P1 \Gamma! P2 \Gamma! P3 \Gamma! P4 \Gamma! P5 \Gamma! P6 \Gamma! P7 \Gamma! P8 \Gamma! R where R is the "receiver" node and one of the P i is the command-and-control center which must broadcast commands to R. The other P j 's for j 6= i are ``decoys'' which are used for transmission purposes from P i to R and also are used to "hide" which particular P i is the real command and control center. That is, in this simplified example, we only wish to hide from an adversary which of the P i is the real command and control center which sends messages to R. Before we explain our solution, we examine several inefficient, but natural to consider simple strategies and then explain what are their drawbacks. Communication-inefficient solution: One simple (but inefficient!) way to hide which P i is the command-and-control center is for every P i to broadcast an (encrypted) stream of messages to R. Thus, R receives 8 different streams of messages, ignores all the messages except those from the real command-and-control center, and decrypts that one. Every processor P i forwards messages of all the smaller-numbered processors and in addition sends its own message. Clearly, an adversary who is monitoring all the communication channels and which can also monitor the internal memory of one of the P i 's (which is not the actual command-and-control center) does not know which P j is broadcasting the actual message. Drawback: Notice that instead of one incoming message, R must receive 8 messages, thus the throughput of how much information the real command-and-control center can send to R is only 1of the total capacity! As the network becomes larger this solution becomes even more costly. Note that this solution enables the receiver to identify the sender. Computation-inefficient solution: In the previous example, the drawback was that the messages from decoy command-and-control nodes were taking up the bandwidth of the channel. In the following solution, we show how this difficulty can be avoided. In order to explain this solution, we shall use pseudo-random generators 1 [BM84, Ha90, ILL89]. We first pick 8 seeds for the pseudo-random generator, and give to processor P i seed s i . Processor P 1 stretches its seed s 1 into long pseudo-random sequence, and sends, at each time step the next bit of its sequence to processor P 2 . Processor P 2 takes the bit it got from processor P 1 and "xors" it with its own next bit from its pseudo-random sequence G(s 2 ) and sends it to P 3 and so forth. The processor P j which is the real command-and-control center additionally "xors" into each bit it sends out a bit of the actual message m i . Processor R is given all the 8 seeds so it can take the incoming message, (which is the message from command-and- control center "xored" with 8 different pseudo-random sequences.) Hence, R can compute all the 8 pseudo-random sequences, subtract (i.e. xor) the incoming message with all the 8 pseudo-random sequences and get the original command-and-control message m. The advantage of this solution is that any P j which is not a command-and-control center (and not R), clearly can not deduce which other processor is the real center. Moreover, the entire bandwidth of the channel between command-and-control processor and the receiver is used to send the messages from the center to the receiver. Drawback: The receiver must compute 8 different pseudo-random sequences in order to recover the actual message. As the network size grows, this becomes prohibitively expensive in terms of the computation that the receiver needs to perform in order to compute the actual message m. Our solution for this simple example: Here, we present a solution that is both computation- efficient and communication-efficient and is secure against an adversary that can monitor all the communication lines and additionally can learn internal memory contents of any one of the intermediate processors. The seed distribution (for a particular communication session) is as follows: Pick 9 random seeds for pseudo-random generator s ffl Give to the real command-and-control processor seed s 0 . ffl Additionally, give to processor P 1 seed fs processor P 3 two seeds fs 3 ; s 4 g, and so on. That is, we give to each processor P i for the seeds fs g. initial "seed" of truly random bits, and deterministically expands it into a long sequence of pseudo-random bits. There are many such commercially available pseudo-random generators, and any such "off-the-shelf " generator that is sufficiently secure and efficient will suffice. ffl give to receiver, R, one seed s 0 Suppose processor P 4 is the real command-and-control center. Then the distribution of seeds is as follows: Now, the transmission of the message is performed in the same fashion as in the previous solution - that is, each processor receives a bit-stream from its predecessor, "xors" a single bit from each pseudo-random sequence that it has, and sends it to the next processor. The command-and-control center "xors" bits of the message into each bit that it sends out. Notice, that adjacent processors "cancel" one of the pseudo-random sequences, by xoring it twice, but introduce a new sequence. For example, processor P 2 cancels s 2 , but "introduces" s 3 . Moreover, each processor must now only compute the output of at most three seeds. Yet, it can be easily verified that if the adversary monitors all the communication lines and in addition can learn seeds of any single processor P i which is not a command and control center, then it can not gain any information as to which other P i is the real command and control center, even after learning the two seeds that belong to processor P i . Of course, the simplified example that we presented works only provided that the adversary cannot monitor both the actual command-and-control center and can not monitor the memory contents of the receiver. (We note that these and other restrictions can be resolved - we address this further in the paper.) Moreover, it should be stressed that the restricted solution presented above does not work if the adversary is allowed to monitor more than one decoy processor. Note that our solution requires that the command-and-control and the receiver have a special common seed s 0 , one obvious extension is to ensure that every two processors have a distinct additional seed that is used for communication between themselves. We should point out that in the rest of the paper we show how the above scheme can be extended to one that is robust against adversaries that can monitor up-to k stations, where in our solution every processor is required to compute the number of different pseudo-random sequences proportional to k only (in particular, at most 2k 1). Moreover, we also show how to generalize the method to arbitrary-topology networks/infrastructures. Additionally, we show how initial distribution of seeds can be done without revealing the command-and-control center and how the actual location of the command-and-control center can be hidden from the recipients of the messages as well. At last, we show how communication from stations back to the command-and-control center could be achieved without the stations knowing at which node of the network the center is located and how totally anonymous communication can be achieved. solutions vs. Public-key solutions The above simple solution is a private-key solution, that is, we assume that before the protocol begins, a set of seeds for pseudo-random function must be distributed in a private and anonymous manner. Thus, we combine this solution with a preprocessing stage in which we distribute these seeds using a public-key solution, that is, a solution where we assume that all users/nodes only have corresponding public and private keys and do not share any information a-priori. Thus, our overall solution is a public-key solution, where before communication begins, we do not assume that users share any private data. As usual in many of such cryptographic setting, our overall efficiency comes from the fact that we switch from public-key to private key solution and then show how to (1) make an efficient private-key implement and (2) how to set up private keys in a pre-processing stage by using public keys in an anonymous and private manner. The rest of the paper is organized as follows. The problem statement appears in Section 2. The anonymous communication (our Xor-Tree Algorithm) which is the heart of our scheme appears in 3. Section 4 and 5 sketch the anonymous seeds transmission and the initialization and termination schemes, respectively. Extensions and concluding remarks appear in Section 6. Problem Statement A communication network is described by a communication graph E). The nodes, processors of the network. The edges of the graph represent bidirectional communication channels between the processors. Let us first define the assumptions and requirements used starting with the adversary models. The adversary is a passive listening adversary that does not intervene in the computation, in particular it does neither forge messages on the links, nor corrupt the program of the processors. ffl An outside adversary is an adversary that can monitor all the communication links but not the contents of the processors memory. ffl An internal dynamic k-listening adversary (inside adversary, in short) is an adversary that can choose to "bug" (i.e., listen to) the memory of up to k processors. The targeted processors are called corrupted, compromised, or colluding processors. Corrupted processors reveal all the information they know to the adversary, however they still behave according to the protocol. The adversary does not have to choose the k faulty processors in advance. While the adversary corrupts less than k processors the adversary can choose the next processor to be corrupted using the information the adversary gained so far from the processors that are already corrupted. The following assumptions are used in the first phase of our algorithm which is responsible for the seeds distribution. Each of the n processors has a public-key/private-key pair. The public key of a processor, P , is known to all the processors while the private key of P is known only to P . The anonymity of the communicating parties can be categorized into four cases: ffl Anonymous to the non participating processors: A processor P wishes to send a message to processor Q without revealing to the rest of the processors and to the inside and outside adversary the fact that P is communicating with Q. ffl Anonymous to the sender and the non participating processors: P wishes to receive a message from Q without revealing its identity to any processor including Q as well as to an inside and outside adversary. ffl Anonymous to the receiver(s) and the non participating processors: P wishes to send (or multicast) a message without revealing its identity to any processor as well as to an inside and an outside adversary. ffl Anonymous to the sender, to the receiver, and the non participating processors: A processor P wishes to communicate with some other processor, without knowing the identity of the processor, and without revealing its identity to any processor including the one it is communicating with, as well as to an inside and outside adversary. (This is similar to the "chat-room" world-wide-web applications, where two processors wish to communicate with one another totally anonymously, without revealing to each other or anybody else their identity.) The efficiency of a solution is measured by the communication overhead which is the number of bits sent over each link in order to send a bit of clear-text data. The efficiency is also measured by the computation overhead which is the maximal number of computation steps performed by each processor in order to transfer a bit of clear-text data. The algorithm is a combination of anonymous seeds transmission, initialization, communication and termination. In the anonymous seeds transmission phase, processors that would like to transmit, anonymously send seeds for a pseudo-random sequence generators to the rest of the processors. The anonymous seeds transmission phase also resolves conflicts of multiple requests for transmission by an anonymous back-off mechanism. Once the seeds are distributed the communication can be started. Careful communication initialization (and termination) procedure that hide the identity of the sender must be performed. We first describe the core of our algorithm which is the communication phase. During the communication phase seeds are used for the production of pseudo-random sequences. The anonymous seeds distribution is presented following the description of the anonymous communication phase. 3 Anonymous Communication 3.1 Computation-inefficient O(n) solution The communication algorithm is designed for a spanning tree T of a general communication graph, where the relation parent child is naturally defined by the election of a root. We start with a simple but inefficient algorithm which requires O(n) computation steps of a processor. (This algorithm is similar to the computation-inefficient solution presented in Section 1, but for the general-topology graph. We then show how to make it computation-efficient as well.) In this (computation-inefficient) solution the sender will chose a distinct seed for each processor. Then the sender can encrypt each bit of information using the seeds of all the processors including its own seeds. Each such seed is used for producing a pseudo-random sequence. The details of the algorithm appear in Figure 1. The symbol \Phi is used to denote the binary xor operation. Note that the i'th bit produced by the root is a result of xoring twice every of the i'th bits of the pseudo-random sequences except the senders' sequence: once by the sender and then during the communication upwards. Each encrypted bit of data will be xored by the receiver(s) using the senders' seed to reveal the clear-text. Note that the scheme is resilient to any number of colluding processors as long as the sender and the receiver(s) are non-faulty. This simple scheme requires a single node (the sender) to compute O(n) pseudo-random bits for each bit of data. (We remark that in contrast, our Xor-Tree Algorithm, requires the computation of only O(k) pseudo-random bits to cope with an outside adversary and an internal dynamic k- listening adversary.) The next Lemma state the communication and computation complexities of the algorithm presented in Figure 1. Lemma 3.1 The next two assertions hold for every bit of data to be transmitted over each edge of the spanning tree: ffl The communication overhead of the algorithm is O(1) per edge. ffl The computation overhead of our algorithm is O(n) pseudo-random bits to be computed by each processor per each bit of data. Proof: In each time unit two bits are sent in each link: one upwards and the other downwards. Since a bit of data is sent every time unit (possibly except the first and last h time units, where is the depth of the tree) the number of bits sent over a link to transmit a bit of data is O(1). The second assertion follows from the fact that the sender computes the greatest number of pseudo random bits in every time unit, namely O(n) pseudo-random bits in every time units. Seeds Distribution - ffl Assign (anonymously) a distinct seed s i to each processor P i . ffl Assign to the sender all the seeds s of all the processors and an additional seed s 0 . ffl Assign the receiver(s) with an additional seed, the seed s 0 . Upwards Communication : P is the sender - ffl Let d i be the i'th bit of data. l be the i'th bits received from the children (if any) of P . 0 be the i'th bit of the pseudo random sequence obtained from the additional seed s 0 of P j . n be the i'th bits of the pseudo random sequence obtained from the seeds s ffl The i'th bit P j sends to its parent (if any) is d i \Phi b 0 n . P is not the sender - l be the i'th bits received from the children (if any) of P j . j be the i'th bit of the pseudo random sequence obtained from the seed ffl The i'th bit that P communicants to its parent (if any) is . Downwards Communication - ffl The root processor calculates an output as if it has a parent and sends the result to every of its children. Every processor which is not the root, sends to its children every bit received from its parent. ffl The receiver(s) decrypts the downward communication by xoring the i'th bit that arrives from the parent with the i'th bit in the pseudo random sequence obtained from s 0 . Figure 1: O(n) Computation Steps Algorithm, for a processor P j . 3.2 Towards our O(k) solution: The choice of seeds For the realization of the communication phase of our O(k) solution we use n(k seeds where k is less than bn=2 \Gamma 1c. Each processor receives seeds. To describe the seeds distribution decisions of the sender we use k each consists of two layers of seeds. We order the processors by their (arbitrary assigned) indices we use the relation follows in a straight forward manner. The first level - Let L n be the seeds that the sender (randomly) chooses for the first level. The sender uses the sequence of seeds L 1 for the first layer of the first level and L 1 1 for the second layer. Note that L 1 2 is obtained by rotating seeds s 1 1 . The l'th level - Similarly, for the l'th level 1 - l - distinct seeds for this level L n to be the seeds of the l'th level and uses two sequences L l= L l and L l= s l l lis obtained by rotating L l l times. receives the seeds s l i+l and P receives the seeds s l Thus, at the end of this procedure every processor is assigned by 2k+2 distinct seeds. Figure 2: The choice of seeds. The seeds distribution procedure appears in Figure 2. An example for the choice of seeds for the processors appears in Figure 3. Seeds of 9 s 00 Figure 3: An example for the distribution of seeds, where 2. The choice of seeds made by the sender has the following properties: ffl Each seed is shared by exactly two processors. ffl For every processor P , P shares a (distinct) seed with every of the k+1 processors that immediately follow P , (if there are at least such k+1 processors), or with the rest of the processors including P n , otherwise. 3.3 The Xor-Tree Algorithm Here, we present out main algorithm, the Xor-Tree Algorithm. The Xor-Tree Algorithm appears in Figure 4. 3.4 An abstract game In this subsection we describe an abstract game that will serve us in analyzing and proving the correctness of the Xor-Tree Algorithm presented in the previous subsection. The adversary get to see the outputs of all the players. The adversary can pick k out of the players and see their seeds. We claim, and later prove, that when the adversary does not pick the sender then every one of the remaining (n \Gamma processors that are not picked by the adversary is equally likely to be the sender for any poly-bounded adversary 2 . We proceed by showing that the above assignment of seeds yields a special seed ds P for each processor P . We choose ds P out of the seeds assigned to each non-faulty processor P . We order the processors by their index in a cyclic fashion such that the processor that follows the i'th processor, i 6= n, is the processor with the index and the processor that follows the n'th processor is the first processor. Then we assign a new index for each processor such that the sender has the index one, the processor that follows the sender has the index two and so on and so forth. These new indices are used for the interpretation of next, follows, prior and last in the description of the choice of special seeds that appears in Figure 6. Recall that with overwhelming probability every two processors share at most one seed. Note that by our special seeds selection, described in Figure 6, the special seeds are not known to the k faulty processors. Theorem 3.2 In the abstract game any of the (n \Gamma non-faulty processors is equally likely to be the sender for any poly-bounded internal k-listening adversary. Proof: We prove that the i'th bit produced by any non-faulty processors is equally likely to be 0 or 1 (for any poly-bounded adversary). Let P be the first non-faulty processor that follows the sender (P is among the first k processors that follow the sender). Let ds P 1 be the special seed of the sender that is shared only with (the non-faulty processor) P . The i'th bit that the sender outputs is a result of a xor operation with the i'th bit of the pseudo-random 2 If the adversary can predict who is the sender then we can use this adversary to break a pseudo-random generator. Seeds Distribution - ffl Assign seeds to the processors as described in Figure 2. ffl Assign the sender with one additional seed, s 0 . ffl Assign the receiver(s) with an additional seed, the seed f the sender s 0 . Upwards Communication : is the sender - ffl Let d i be the i'th bit of data. l be the i'th bits received from the children (if any) of P j . 2k+2 be the i'th bits of the pseudo-random sequences obtained from the seeds of P j . be the i'th bit of the pseudo-random sequence obtained from the additional seed s 0 of P j . ffl The i'th bit P j sends to its parent (if any) is d 2k+3 . is not the sender - ffl The i'th bit that P j communicants to its parent (if any) is Downwards Communication - ffl The root processor calculates an output as if it has a parent and sends the result to every of its children. ffl Every non root processor send to its children every bit received from its parent. ffl The receiver(s) decrypts the downward communication by xoring the i'th bit that arrives from the parent with the i'th bit in the pseudo random sequence obtained from s 0 . Figure 4: The Xor-Tree Algorithm, for a processor P j . Seeds Assignment - Assign seeds to the processors as described in Figure 2. Assign the sender with one additional seed. Computation - Each processor, P , uses its seeds to compute pseudo-random sequences. At the i'th time unit the sender S computes the i'th bit of every of its pseudo-random sequences, xors these bits and the i'th bit of data and outputs the result. At the same time unit every other processor P computes the i'th bit of every of its pseudo-random sequences xors these bits and outputs the result. Figure 5: The Abstract Game. The sender P 1 - Each of the k+1 processors that immediately follows the sender shares exactly one seed with the sender. Since there are at most k colluding processors, one of these k+1 processors must be non-faulty. Pick, P , the first such non-faulty processor. Assign ds P 1 , the special seed of the sender, to be the seed that the sender shares with processors P that is not among the k last processors - If P is not among the last processors then P is assigned by 2k seeds of these seeds are from the first layers of the k+1 seed levels. These k+1 seeds are new - they do not appear in any processor prior to P . Since there are at most k colluding processors, one of the next k processors is non-faulty. Let Q be the first such non-faulty processor and assign ds P by the seed that P shares with Q. Repeat the procedure until you reach a non-faulty processor that is among the last k processors. A processors Q that is among the k last processors - Note that Q does not new seeds since some of its seeds are assigned to the first processors (at least the one in the k + 1'th level). Fortunately, Q shares a single new seed with every of the last processors. This fact allows us to continue the special seed selection procedure, by choosing the seed shared with the next non-faulty processor. Figure Choice of special seeds. sequence (among other pseudo-random sequences) obtained from ds P 1 . Since only P (that is a non-faulty processor) shares ds P 1 with the sender, it holds that the i'th bit output by the sender is equally likely to be 0 or 1 (for any poly-bounded internal k-listening adversary). A similar argument hold for the output of P , since there exists a special seed shared with the next non-faulty processor Q. In general it holds for the output of every non-faulty processor. The same argument holds if any of the non-faulty processors is the sender. Thus, for any polynomially-bounded k-internal and external adversary, the distribution of the output is indistinguishable of the identity of the sender. The fact that the adversary can be a dynamic adversary is implied by the Corollary 3.3. The proof of the corollary is similar to the proof of Theorem 3.2. Corollary 3.3 For any k 0 - k after the adversary chooses k 0 faulty processors any of the processors is equally likely to be the sender for any poly-bounded internal k'-listening adversary. 3.5 Reduction to the abstract game In this subsection we prove that if there is an algorithm that reveals information on the identity of the sender in the tree then there exists an algorithm that reveals information on the identity of the sender in the abstract game. The above reduction together with Theorem 3.2 yields the proof of correctness for the Xor-Tree algorithm. Assume that the adversary reveals information on the sender in a tree T of n processors. Then an abstract game of n nodes is mapped to the tree as follows: 1. Each processor of the abstract game is assigned to a node of the tree T . 2. The output of every processor to its parent is computed as follows: Let the hight of a processor P in T be the number of edges in the longest path P from P to a leaf such that P does not traverse the root. We start with the processors that are in hight i.e. the leaves. The output of the processors that were assigned to the leaves of the tree is not changed i.e. it is identical to their output in the abstract game. Once we computed the output of processors in hight h we use these computed outputs to compute the outputs of processors in hight h+ 1. Let Q be a processor in hight h+ 1, l be the i'th computed bits that are output by the children of Q, and let b Q be the original i'th output bit of Q in the abstract game. The computed output of Q is b 1 \Phi b 2 \Phi Figure 7: The Reduction. Theorem 3.4 In the Xor-Tree Algorithm any of the (n \Gamma non-faulty processors is equally likely to be the sender for any poly-bounded internal k-listening adversary. Proof: If there exists an adversary A that reveals information on the identity of the sender in a tree T then there exists an abstract game with the same number of processors and the same seeds distribution, such that the application of the reduction in Figure 7 yields the communication pattern on T and reveals information on the sender identity in the abstract game. This contradicts Theorem 3.2 and thus contradicts the existence of A. The next Lemma states the communication and computation overheads of the anonymous communication algorithm. Lemma 3.5 The next two assertions hold for every bit of data to be transmitted over each edge of the spanning tree: ffl The communication overhead of the algorithm is O(1) per edge. ffl The computation overhead of our algorithm is O(k) pseudo-random bits to be computed by each processor per each bit of data. Proof: In each time unit two bits are sent in each link: one upwards and the other downwards. Since a bit of data is sent every time unit (possibly except the first and last h time units, where is the depth of the tree) the number of bits send over each link to transmit a bit of data is O(1). The second assertion follows from the fact that in each time unit each processor generates at most 2k Anonymous Seeds Transmission We first outline the main ideas in the seeds transmission scheme and then give full details. Every processor has a public-key encryption, known to all other processors. A virtual ring defined by the Euler tour on the tree is used for the seeds transmission. Note that the indices of the processor used in this description are related to their location on the virtual ring. First all processors send messages to P 1 over the (virtual) ring. Those processors that wish to broadcast send a collection of seeds, and those processors that do not wish to broadcast, send dummy messages of equal length. To do so in an anonymous fashion (so that P 1 does not know which message is from which processor), k + 1 of Chaum's mixes [Ch81] are used, where k (real) processors just before P 1 in the Euler tour are used as mixes. Hence, P 1 can identify the number of non-dummy arriving messages but not their origin. In case more than one non-dummy message reaches P 1 , a standard back-off algorithm is initiated by P 1 . Once exactly one message (containing a collection of seeds) arrives to P 1 the seed distribution procedure described above (for sending a collection of seeds to P 1 ) is used to send the seeds to P 2 and so on. (At this point processors know that only one processor wishes to broadcast.) This procedure is repeated n times in order to allow the anonymous sender to transmit a collection of seeds to every processor. Notice that this process is quadratic in the size of the ring, the number of colluding processors k, and the length of the security parameter, (i.e., let g be a security parameter and k as before, then we send O((gkn) 2 ) bits per edge.) Thus, as long the message size p to be broadcasted is greater than O((gkn) 2 ) we achieve O(1) overall amortized cost per edge, and otherwise we get O((gkn) 2 =p) amortized cost. The details follow. The seeds transmission procedure uses a virtual ring R defined by an Euler tour of the tree T . Note that each edge of T appears exactly twice in R and therefore the number of edges and nodes in R is 2. The seeds transmission procedure starts with the transmission of seeds to the first processor P 1 . Let L be the list of processors in R in clockwise order starting with P 1 ; the indices 2 to 2n \Gamma 2 are implied by the Euler tour and not by the indices of the processors in T . Note that a single processor of T may appear more than once in L 1 . We use the term instance for each such appearance. Define the reduced list RL 1 to be a list of processors that is obtained from L 1 by removing all but the first instance of each processor. Thus, in RL 1 every processor of T appears exactly once. The communication of seeds uses the anti-clockwise direction. Define the last l real processors to be the first l processors in RL 1 . When transmitting seeds to P i , L i , RL i and the last l processors, are defined analogously. In the first stage every processor that wants to communicate with another processor sends an encrypted message with the seeds to be used by P 1 . Note that P 1 can be a faulty processor, thus a careful transmission must be carried on. Let L be the list of processors in R in anti-clockwise order i.e. L 1 in reversed order. Again L 1 includes more than one instance of each processor P of T . Define the active instance of a processor P of T in L 1 to be the last appearance of P in L 1 . Define an active message to be a message that arrives to an active instance of a processor. The details of the anonymous seeds transmission to P 1 appears in Figure 8. As we prove in the sequel no information concerning the identity of the requesting processors is revealed during the anonymous seeds transmission to P 1 except the information that can be concluded by the value of n t - the number of processors that would like to transmit. Once processors starts sending messages to P 2 in a fashion similar to the one used to send seeds to P 1 . Then processors sends seeds to P 3 and so on and so forth, till the processors send messages to P n . Note that when n there is exactly one sender for the next communication session and at the end of the seeds distribution procedure every processor holds the seeds distributed by the sender. Lemma 4.1 A coalition of k colluding processors cannot reveal the identity of the seeds distributors Proof: We prove the lemma for the transmission of the seeds from the sender to P 1 . Note that one of the last k+1 real processors must be non-faulty. If P 1 is non-faulty then no information starts - The first processor to send a message m n to P 1 is P n . If P n wants to transmit data then m n contains seeds to be used by P 1 , otherwise m n is an empty message i.e. a message that can be identified by P 1 as a null message. P n uses the public of the last k processors m n in a nested fashion; First encrypting m n with pu 1 then encrypting the resulting message with pu 2 and so on. P n sends the k 1-nested encrypted message m k+1 n to the processor that is next to the active instance of P n in L 1 . Non active message - When a processor P i receives a non active message it forwards the message to the next processor according to L 1 . Active message - We now proceed by describing the actions taken by a processor upon the arrival of an active message. ffl We first describe the action taken by a processor P i that is not among the last k+1 real processors. When an active message with fm k+1 arrives (to the active instance of) P its own k 1-encrypted message (again, containing seeds to be used by P 1 or null message) to the message received and sends the message to the next processor according to L 1 . ffl We now turn to consider a processor P i that is among the last k processors. When an active message with fm i arrives to using its private key to obtain fm g. its message to P 1 by the public keys of the last processors. P i randomly orders the set fm and sends the reordered set to the processor that is next according to L 1 (Note that following the first such reordering the j'th index of m j is not necessarily the index of the sender of m Arrival to receives an active message with fm 1 every message and finds out the number n t of the processors that would like to transmit. If n t 6= 1 then P 1 sends a message with the value of n t that traverses the virtual ring. Upon receiving such a message each processor that wants to transmit, randomly chooses a waiting time in the range of say, 1 to 2n t . The procedure of sending seeds to P 1 is repeated until n Figure 8: Anonymous Seeds Transmission to P 1 . concerning the identity of the seeds distributors is revealed to the adversary. Otherwise, when P 1 is faulty then let P i be the non faulty processor that is the last to reorder the set fm upon the arrival of fm i g. Since every arriving m i is encrypted with P i 's public key no set of k-faulty processors can decrypt m originated by a faulty processor). P i randomly order the set fm holds that a coalition of k processors cannot reveal the identity of the sender of any m i\Gamma1 in fm g. 5 Initialization and Termination When the seed distribution procedure is over, then the transmission of data may start. P n broadcasts a signal on the tree that notifies the leaves that they can start transmitting data. The leaves start sending data in a way that ensures that every non-leaf processor receives the i'th bit from its children simultaneously. Thus, the delay in starting transmission of a particular leaf l is proportional to the difference between the longest path from a leaf to the root and the distance of l from the root. Each non leaf processor waits for receiving the i'th bit from each of its children, uses these bits and its seeds to compute its own i'th bit and sends the output to its parent. Note that buffers can be used in case the processors are not completely synchronized. The sender can terminate the session by sending a termination message that is not encrypted by its additional seed. This message will be decrypted by the root that will broadcast it to the rest of the processors to notify the beginning of a new anonymous seeds transmission. 6 Extensions and Concluding Remarks Our treatment so far considered the anonymous sender case, which is also anonymous to the non participating processors. A simple modification of the algorithm can support the anonymous receiver case: The receiver plays a role of a sender of the previous solution in order to communicate in an anonymous fashion an additional seed to the sender. Then the sender uses the same scheme for the anonymous sender case with the seed the sender got from the receiver. To achieve anonymous communication in which both the sender and the receiver are anony- mous, do the following: The two participants, P and Q, that would like to communicate (each) send anonymously distinct seeds to P . It is possible that more than two participants will send anonymously distinct seeds to P 1 . In such a case, P 1 will broadcast the processors that more than two processors tried to anonymously chat and a back-off mechanism will be used until exactly two participants, P and Q, send seeds to P 1 . Then, P 1 will encrypt and broadcast the two seeds it got, each seed encrypted (using distinct intervals of the pseudo-random expansions of the two seeds) by the other seed. Hence, each of the two processors will use its seed to reveal the seed of the other processor. At this stage P and Q will continue and anonymously send seeds to P . The same procedure continues for Now P has a set of k seeds that are used for encryption of messages sent to Q and used for encryption messages sent to P . They both act as senders using the bit resulting from xoring the bits produced by the set of the k seeds as the bit of special seed known to the receiver in our anonymous sender scheme. The back-off mechanism ensures that one of P and Q starts the communication and then the other can replay (when the first allows him to, i.e., stops transmitting data). We remark that it is possible to have more than two participants by a similar scheme. The security of the above algorithm is derived from the fact that there must be a non-faulty processor among the processor therefore the adversary does not know at least one key used to encrypt and decrypt messages by the sender and the receiver. Acknowledgment We thank Oded Goldreich, Ron Rivest and the anonymous referees for helpful remarks. --R "Detection of disrupters in the DC protocol" "An efficient probabilistic public-key encryption scheme which hides all partial information" "How to Generate Cryptographically Strong Sequences of Pseudo-Random Bits" "Completeness Theorems for Non-Cryptographic Fault-Tolerant Distributed Computation" "Adaptively Secure Multi-Party Computation" "Randomness vs. Fault- Tolerance" "Untraceable Electronic Mail, Return Addresses, and Digital Pseudonyms" "Multiparty Unconditionally Secure Pro- tocols" "The Dining Cryptographers Problem: Unconditional Sender and Recipient Untraceability" "Achieving Electronic Privacy" "Eavesdropping Games: A Graph-Theoretic Approach to Privacy in Distributed Systems," "How To Play Any Mental Game" "Pseudo-Random Generators under Uniform Assumptions" "Pseudo-Random Generation from One-Way Functions," "Reducibility and Completeness in Multi-Party Private Computations" "How to Implement ISDNs Without User Observability - Some Re- marks" "Network without User Observability," "ISDN-MIXes - Untraceable Communication with Very Small Bandwidth Overhead," "Anonymous Connections and Onion Routing" "Unconditional Sender and Recipient Untraceability in spite of active attacks" "Cryptographic Defense Against Traffic Analysis" --TR How to generate cryptographically strong sequences of pseudo-random bits Networks without user observability How to play ANY mental game The dining cryptographers problem: <italic>unconditional sender and recipient untraceability</> Completeness theorems for non-cryptographic fault-tolerant distributed computation Multiparty unconditionally secure protocols Unconditional sender and recipient untraceability in spite of active attacks Detection of disrupters in the DC protocol The dining cryptographers in the disco Cryptographic defense against traffic analysis Adaptively secure multi-party computation Randomness vs. fault-tolerance The art of computer programming, volume 1 (3rd ed.) A Pseudorandom Generator from any One-way Function Untraceable electronic mail, return addresses, and digital pseudonyms ISDN-MIXes Anonymous Connections and Onion Routing --CTR Chin-Chen Chang , Chi-Yien Chung, An efficient protocol for anonymous multicast and reception, Information Processing Letters, v.85 n.2, p.99-103, 31 January Nicholas Hopper , Eugene Y. Vasserman, On the effectiveness of k;-anonymity against traffic analysis and surveillance, Proceedings of the 5th ACM workshop on Privacy in electronic society, October 30-30, 2006, Alexandria, Virginia, USA Steven S. Seiden , Peter P. Chen , R. F. Lax , J. Chen , Guoli Ding, New bounds for randomized busing, Theoretical Computer Science, v.332 n.1-3, p.63-81, 28 February 2005 Jiejun Kong , Dapeng Wu , Xiaoyan Hong , Mario Gerla, Mobile traffic sensor network versus motion-MIX: tracing and protecting mobile wireless nodes, Proceedings of the 3rd ACM workshop on Security of ad hoc and sensor networks, November 07-07, 2005, Alexandria, VA, USA

anonymous communication;anonymous multicast

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Configuring role-based access control to enforce mandatory and discretionary access control policies.

Access control models have traditionally included mandatory access control (or lattice-based access control) and discretionary access control. Subsequently, role-based access control has been introduced, along with claims that its mechanisms are general enough to simulate the traditional methods. In this paper we provide systematic constructions for various common forms of both of the traditional access control paradigms using the role-based access control (RBAC) models of Sandhu et al., commonly called RBAC96. We see that all of the features of the RBAC96 model are required, and that although for the manatory access control simulation, only one administrative role needs to be assumed, for the discretionary access control simulations, a complex set of administrative roles is required.

INTRODUCTION Role-based access control (RBAC) has recently received considerable attention as a promising alternative to traditional discretionary and mandatory access controls (see, for example, Proceedings of the ACM Workshop on Role-Based Access Con- trol, 1995-2000). In RBAC, permissions are associated with roles, and users are made members of appropriate roles thereby acquiring the roles' permissions. This greatly simplies management of permissions. Roles can be created for the various job functions in an organization and users then assigned roles based on their responsibilities and qualications. Users can be easily reassigned from one role to another. Roles can be granted new permissions as new applications and systems are incorporated, and permissions can be revoked from roles as needed. An important characteristic of RBAC is that by itself it is policy neutral. RBAC is a means for articulating policy rather than embodying a particular security policy (such as one-directional information ow in a lattice). The policy enforced in a particular system is the net result of the precise conguration and interactions of various RBAC components as directed by the system owner. Moreover, the access control policy can evolve incrementally over the system life cycle, and in large systems it is almost certain to do so. The ability to modify policy to meet the changing needs of an organization is an important benet of RBAC. Traditional access control models include mandatory access control (MAC), which we shall call lattice-based access control (LBAC) here [Denning 1976; Sandhu 1993], and discretionary access control (DAC) [Lampson 1971; Sandhu and Samarati 1994; Sandhu and Samarati 1997]. Since the introduction of RBAC, several authors have discussed the relationship between RBAC and these traditional models [Sandhu 1996; Sandhu and Munawer 1998; Munawer 2000; Nyanchama and Osborn 1996; Nyanchama and Osborn 1994]. The claim that RBAC is more general than all of these traditional models has often been made. The purpose of this paper is to show how RBAC can be congured to enforce these traditional models. Classic LBAC models are specically constructed to incorporate the policy of one-directional information ow in a lattice. This one-directional information ow can be applied for condentiality, integrity, condentiality and integrity together, or for aggregation policies such as Chinese Walls [Sandhu 1993]. There is nonetheless strong similarity between the concept of a security label and a role. In particular, the same user cleared to, for example, Secret can on dierent occasions login to a system at Secret and Unclassied levels. In a sense the user determines what role (Secret or Unclassied) should be activated in a particular session. This leads us naturally to ask whether or not LBAC can be simulated using RBAC. If RBAC is policy neutral and has adequate generality it should indeed be able to do so, particularly since the notion of a role and the level of a login session are so similar. This question is theoretically signicant because a positive answer would establish that LBAC is just one instance of RBAC, thereby relating two distinct access control models that have been developed with dierent motivations. A positive answer is also practically signicant, because it implies that the same Trusted Computing Base can be congured to enforce RBAC in general and LBAC in particular. This addresses the long held desire of multi-level security advocates that technology which meets needs of the larger commercial marketplace be appli- Conguring RBAC to Enforce MAC and DAC 3 cable to LBAC. The classical approach to fullling this desire has been to argue that LBAC has applications in the commercial sector. So far this argument has not been terribly productive. RBAC, on the other hand, is specically motivated by needs of the commercial sector. Its customization to LBAC might be a more productive approach to dual-use technology. In this paper we answer this question positively by demonstrating that several variations of LBAC can be easily accommodated in RBAC by conguring a few components. 1 We use the family of RBAC models recently developed by Sandhu et al [Sandhu et al. 1996; Sandhu et al. 1999] for this purpose. This family is commonly called the RBAC96 model. Our constructions show that the concepts of role hierarchies and constraints are critical to achieving this result. Changes in the role hierarchy and constraints lead to dierent variations of LBAC. A simulation of LBAC in RBAC was rst given by Nyanchama and Osborn [Nyan- chama and Osborn 1994]; however, they do not exploit role hierarchies and constraints and cannot handle variations so easily as the constructions of this paper. Discretionary access control (DAC) has been used extensively in commercial ap- plications, particularly in operating systems and relational database systems. The central idea of DAC is that the owner of an object, who is usually its creator, has discretionary authority over who else can access that object. DAC, in other words, involves owner-based administration of access rights. Whereas for LBAC, we do not need to discuss a complex administration of access rights, we will see that for DAC, the administrative roles developed in Sandhu, Bhamidipati, and Munawer [1999] are crucial. Because each object could potentially be owned by a unique owner, the number of administrative roles can be quite large. However, we will show that the role administration facilities in the RBAC96 model are adequate to build a simulation of these sometimes administratively complex systems. The rest of this paper is organized as follows. We review the family of RBAC96 models due to Sandhu, Coyne, Feinstein, and Youman [1996] in section 2. This is followed by a quick review of LBAC in section 3. The simulation of several LBAC variations in RBAC96 is described in section 4. This is followed by a brief discussion in Section 5 of other RBAC96 congurations which also satisfy LBAC properties. Section 6 introduces several major variations of DAC. In Section 7 we show how each of these variations can be simulated in RBAC96. Section 8 summarizes the results. Preliminary versions of some of these results have appeared in Sandhu [1996], Sandhu and Munawer [1998], Nyanchama and Osborn [1996] and Osborn [1997]. 2. RBAC MODELS A general RBAC model including administrative roles was dened by Sandhu et al [Sandhu et al. 1996]. It is summarized in Figure 1. The model is based on three sets of entities called users (U ), roles (R), and permissions (P ). Intuitively, a It should be noted that RBAC will only prevent overt ows of information. This is true of any access control model, including LBAC. Information ow contrary to the one-directional requirement in a lattice by means of so-called covert channels is outside the purview of access control per se. Neither LBAC nor RBAC addresses the covert channel issue directly. Techniques used to deal with covert channels in LBAC can be used for the same purpose in RBAC. 4 Osborn, Sandhu and Munawer user is a human being or an autonomous agent, a role is a job function or job title within the organization with some associated semantics regarding the authority and responsibility conferred on a member of the role, and a permission is an approval of a particular mode of access to one or more objects in the system. ADMINIS- ROLES AR user roles HIERARCHY ROLE RH ROLE HIERARCHY ARH IONS PERMISSION ASSIGNMENT PA ROLES R PERMISSION APA ASSIGNMENT ADMIN. IONS CONSTRAINTS U USERS USER ASSIGNMENT UA USER ASSIGNMENT AUA Fig. 1. The RBAC96 Model The user assignment (UA) and permission assignment (PA) relations of Figure 1 are both many-to-many relationships (indicated by the double-headed arrows). A user can be a member of many roles, and a role can have many users. Similarly, a role can have many permissions, and the same permission can be assigned to many roles. There is a partially ordered role hierarchy RH, also written as , where x y signies that role x inherits the permissions assigned to role y. In the work of Nyanchama and Osborn [Nyanchama and Osborn 1994; Nyanchama and Osborn 1999; Nyanchama and Osborn 1996], the role hierarchy is presented as an acyclic directed graph, and direct relationships in the role hierarchy are referred to as edges. Inheritance along the role hierarchy is transitive; multiple inheritance is allowed in partial orders. Figure 1 shows a set of sessions S. Each session relates one user to possibly many roles. Intuitively, a user establishes a session during which the user activates Conguring RBAC to Enforce MAC and DAC 5 some subset of roles that he or she is a member of (directly or indirectly by means of the role hierarchy). The double-headed arrow from a session to R indicates that multiple roles can be simultaneously activated. The permissions available to the user are the union of permissions from all roles activated in that session. Each session is associated with a single user, as indicated by the single-headed arrow from the session to U . This association remains constant for the life of a session. A user may have multiple sessions open at the same time, each in a dierent window on the workstation screen for instance. Each session may have a dierent combination of active roles. The concept of a session equates to the traditional notion of a subject in access control. A subject (or session) is a unit of access control, and a user may have multiple subjects (or sessions) with dierent permissions active at the same time. The bottom half of Figure 1 shows administrative roles and permissions. RBAC96 distinguishes roles and permissions from administrative roles and permissions re- spectively, where the latter are used to manage the former. Administration of administrative roles and permissions is under control of the chief security o-cer or delegated in part to administrative roles. The administrative aspects RBAC96 elaborated in [Sandhu et al. 1999] are relevant for the DAC discussion in Section 6. For the purposes of the LBAC discussion, we assume a single security o-cer is the only one who can congure various components of RBAC96. Finally, Figure 1 shows a collection of constraints. Constraints can apply to any of the preceding components. An example of constraints is mutually disjoint roles, such as purchasing manager and accounts payable manager, where the same user is not permitted to be a member of both roles. The following denition formalizes the above discussion. Definition 1. The RBAC96 model has the following components: |U , a set of users R and AR, disjoint sets of (regular) roles and administrative roles P and AP , disjoint sets of (regular) permissions and administrative permissions S, a set of sessions many-to-many permission to role assignment relation APA AP AR, a many-to-many permission to administrative role assignment relation |UA U R, a many-to-many user to role assignment relation AUA U AR, a many-to-many user to administrative role assignment relation |RH R R, a partially ordered role hierarchy ARH AR AR, partially ordered administrative role hierarchy (both hierarchies are written as in inx notation) mapping each session s i to the single user user(s i ) (constant for the session's lifetime), roles maps each session s i to a set of roles and administrative roles (which can change with time) session s i has the permissions [ 6 Osborn, Sandhu and Munawer |there is a collection of constraints stipulating which values of the various components enumerated above are allowed or forbidden. 3. LBAC (OR MAC) MODELS Lattice based access control is concerned with enforcing one directional information ow in a lattice of security labels. It is typically applied in addition to classical discretionary access controls, but in this section we will focus only on the MAC component. A simulation of DAC in RBAC96 is found in Section 7. Depending upon the nature of the lattice, the one-directional information ow enforced by LBAC can be applied for condentiality, integrity, condentiality and integrity together, or for aggregation policies such as Chinese Walls [Sandhu 1993]. There are also variations of LBAC where the one-directional information ow is partly relaxed to achieve selective downgrading of information or for integrity applications [Bell 1987; Lee 1988; Schockley 1988]. The mandatory access control policy is expressed in terms of security labels attached to subjects and objects. A label on an object is called a security classi- cation, while a label on a user is called a security clearance. It is important to understand that a Secret user may run the same program, such as a text editor, as a Secret subject or as an Unclassied subject. Even though both subjects run the same program on behalf of the same user, they obtain dierent privileges due to their security labels. It is usually assumed that the security labels on subjects and objects, once assigned, cannot be changed (except by the security o-cer). This last assumption, that security labels do not change, is known as tranquillity. (Non- tranquil LBAC can also be simulated in RBAC96 but is outside the scope of this paper.) The security labels form a lattice structure as dened below. Definition 2. (Security Lattice) There is a nite lattice of security labels SC with a partially ordered dominance relation and a least upper bound operator. 2 An example of a security lattice is shown in Figure 2. Information is only permitted to ow upward in the lattice. In this example, H and L respectively denote high and low, and M1 and M2 are two incomparable labels intermediate to H and L. This is a typical condentiality lattice where information can ow from low to high but not vice versa. The specic mandatory access rules usually specied for a lattice are as follows, where signies the security label of the indicated subject or object. Definition 3. (Simple Security Property) Subject s can read object if Definition 4. (Liberal ?-property) Subject s can write object (o). 2 The ?-property is pronounced as the star-property. For integrity reasons sometimes a stricter form of the ?-property is stipulated. The liberal ?-property allows a low subject to write a high object. This means that high data may be maliciously or accidently destroyed or damaged by low subjects. To avoid this possibility we can employ the strict ?-property given below. Definition 5. (Strict ?-property) Subject s can write object (o). 2 Conguring RBAC to Enforce MAC and DAC 7 Fig. 2. A Partially Ordered Lattice The liberal ?-property is also referred to as write-up and the strict ?-property as non-write-up or write-equal. In variations of LBAC the simple-security property is usually left unchanged as we will do in all our examples. Variations of the ?-property in LBAC whereby the one-directional information ow is partly relaxed to achieve selective downgrading of information or for integrity applications [Bell 1987; Lee 1988; Schockley 1988] will be considered later. 4. CONFIGURING RBAC FOR LBAC We now show how dierent variations of LBAC can be simulated in RBAC96. It turns out that we can achieve this by systematically changing the role hierarchy and dening appropriate constraints. This suggests that role hierarchies and constraints are central to dening policy in RBAC96. 4.1 A Basic Lattice We begin by considering the example lattice of Figure 2 with the liberal ?-property. Subjects with labels higher up in the lattice have more power with respect to read operations but have less power with respect to write operations. Thus this lattice has a dual character. In role hierarchies subjects (sessions) with roles higher in the hierarchy always have more power than those with roles lower in the hierarchy. To accommodate the dual character of a lattice for LBAC we will use two dual hierarchies in RBAC96, one for read and one for write. These two role hierarchies for the lattice of Figure 2 are shown in Figure 3(a). Each lattice label x is modeled as two roles xR and xW for read and write at label x respectively. The relationship among the four read roles and the four write roles is respectively shown on the left and right hand sides of Figure 3(a). The duality between the left and right lattices is obvious from the diagrams. To complete the construction we need to enforce appropriate constraints to re ect the labels on subjects in LBAC. Each user in LBAC has a unique security clearance. This is enforced by requiring that each user in RBAC96 is assigned to exactly two roles xR and LW. An LBAC user can login at any label dominated by the user's clearance. This requirement is captured in RBAC96 by requiring that each session has exactly two matching roles yR and yW. The condition that x y, that is the 8 Osborn, Sandhu and Munawer HR LR M1R M2R M1W M2W (a) Liberal ?-Property HR LR M1R M2R HW LW M2W (b) Strict ?-Property Fig. 3. Role Hierarchies for the Lattice of Figure 2 user's clearance dominates the label of any login session established by the user, is not explicitly required because it is directly imposed by the RBAC96 construction. Note that, by virtue of membership in LW, each user can activate any write role. However, the write role activated in a particular session must match the session's read role. Thus, both the role hierarchy and constraints of RBAC96 are exploited in this construction. LBAC is enforced in terms of read and write operations. In RBAC96 this means our permissions are read and writes on individual objects written as (o,r) and (o,w) respectively. An LBAC object has a single sensitivity label associated with it. This is expressed in RBAC96 by requiring that each pair of permissions (o,r) and (o,w) be assigned to exactly one matching pair of xR and xW roles respectively. By assigning permissions (o,r) and (o,w) to roles xR and xW respectively, we are implicitly setting the sensitivity label of object o to x. 4.2 The General Construction Based on the above discussion we have the following construction for arbitrary lattices (actually the construction works for partial orders with a lower-most security class). Given SC with security labels and partial order LBAC , an equivalent RBAC96 system is given by: Construction 1. (Liberal ?-Property) |RH which consists of two disjoint role hierarchies. The rst role hierarchy consists of the \read" roles fL and has the same partial order as LBAC ; the second partial consists of the \write" roles fL and has a partial order which is the inverse of LBAC . Conguring RBAC to Enforce MAC and DAC 9 is an object in the systemg |Constraint on UA: Each user is assigned to exactly two roles xR and LW where x is the label assigned to the user and LW is the write role corresponding to the lowermost security level according to LBAC |Constraint on sessions: Each session has exactly two roles yR and yW |Constraints on PA: |(o,r) is assigned to xR i (o,w) is assigned to xW |(o,r) is assigned to exactly one role xR such that x is the label of Theorem 1. An RBAC96 system dened by Construction 1 satises the Simple Security Property and the Liberal ?-Property. Proof: (a) Simple Security Property: Subjects in the LBAC terminology correspond to RBAC96 sessions. For subject s to read o, (o,r) must be in the permissions assigned to a role, either directly or indirectly, which is among the roles available to session s, which corresponds to exactly one user u. For u to be involved in this session, this role must be in the UA for u (either directly or indirectly). Let z and y. By the constraints on PA given in Construction 1, (o,r) is assigned directly to exactly one role xR, where and by the construction of RH, is inherited by roles yR such that y LBAC x. For s to be able to read o, it must have one of these yR in its session. By the denition of roles in a session from Denition 1, any role junior to zR can be in a session for u, i.e. z LBAC y. In other words, a session for u can involve one reading role yR such that z LBAC y. Therefore, the RBAC96 system dened above allows subject s to read object and which is precisely the Simple Security Property. (b) Liberal ?-Property: Each user, u, is assigned by UA to xR, where x is the clearance of the user. According to LBAC, the user can read data classied at level x or at levels dominated by x. It also means that the user can start a session at a level dominated by x. So, if a user cleared to say level x, wishes to run a session at level y, such that x LBAC y, the constraints in Construction 1 allow the session to have the two active roles yR and yW. Because every user is assigned to LW, it is possible for every user to have a session with yW as one of its roles. The structure of the two role hierarchies means that if the yW role is available to a user in a session, the user can write objects for which the permission (o,w) is in yW. By construction of the role hierarchy, the session can write to level y or levels dominated by y. In LBAC terms, the subject, s, corresponds to the session, and within a session a write can be performed if (o,w) is in the permissions of a role, which by the construction is only if (o) LBAC (s). This is precisely the Liberal ?-Property.4.3 LBAC Variations Variations in LBAC can be accommodated by modifying this basic construction in dierent ways. In particular, the strict ?-property retains the hierarchy on read roles but treats write roles as incomparable to each other as shown in Figure 3(b) for the example of our basic lattice. Construction 2. (Strict ?-Property) Identical to construction 1 except RH has a partial order among the read roles identical to the LBAC partial order, and no relationships among the write roles. 2 Theorem 2. An RBAC96 system dened by Construction 2 satises the Simple Security Property and the Strict ?-Property. The proof of this and subsequent similar results is omitted. Next we consider a version of LBAC in which subjects are given more power than allowed by the simple security and ?-properties [Bell 1987]. The basic idea is to allow subjects to violate the ?-property in a controlled manner. This is achieved by associating a pair of security labels r and w with each subject (objects still have a single security label). The simple security property is applied with respect to r and the liberal ?-property with respect to w . In the LBAC model of [Bell 1987] it is required that r should dominate w . With this constraint the subject can read and write in the range of labels between r and w which is called the trusted range. If r and w are equal the model reduces to the usual LBAC model with the trusted range being a single label. The preceding discussion is remarkably close to our RBAC constructions. The two labels r and w correspond directly to the two roles xR and yW we have introduced earlier. The dominance required between r and w is trivially recast as a dominance constraint between x and y. This leads to the following construction: Construction 3. (Liberal ?-Property with Trusted Range) Identical to construction |Constraint on UA: Each user is assigned to exactly two roles xR and yW such that x y in the original lattice |Constraint on sessions: Each session has exactly two roles xR and yW such that x y in the original lattice 2 Lee [1988] and Schockley [1988] have argued that the Clark-Wilson integrity model [Clark and Wilson 1987] can be supported using LBAC. Their models are similar to the above except that no dominance relation is required between x and y. Thus the write range may be completely disjoint with the read range of a subject. This is easily expressed in RBAC96 as follows. Construction 4. (Liberal ?-Property with Independent Write Range) Identical to construction 3 except x y is not required in the constraint on UA and the constraint on sessions. 2 A variation of the above is to use the strict ?-property as follows. Construction 5. (Strict ?-Property with Designated Write) Identical to construction |Constraint on UA: Each user is assigned to exactly two roles xR and yW |Constraint on sessions: Each session has exactly two roles xR and yW 2 Construction 5 can also be directly obtained from construction 4 by requiring the strict ?-property instead of the liberal ?-property. Construction 5 can accommodate Clark-Wilson transformation procedures as outlined by Lee [1988] and Schockley Conguring RBAC to Enforce MAC and DAC 11 [1988]. (Lee and Schockley actually use the liberal ?-property in their constructions, but their lattices are such that the constructions are more directly expressed in terms of the strict ?-property.) 5. EXTENDING THE POSSIBLE RBAC CONFIGURATIONS In the previous section, we looked at specic mappings of dierent kinds of LBAC to an RBAC system with the same properties. In this section we examine whether or not more arbitrary RBAC systems which do not necessarily follow the constructions in Section 4 still satisfy LBAC properties. In order to do this, we assume that all users and objects have security labels, and that permissions involve only reads and writes. In the previous discussion, all constructions created role hierarchies with disjoint read and write roles. This is not strictly necessary; the role hierarchy in Figure 5 could be the construction for the strict ?-property with the following modications: |Constraint on UA: Each user is assigned to all roles, xRW such that the clearance the user dominates the security label x |Constraint on sessions: Each session has exactly one role: yRW |Constraints on PA: |(o,r) is assigned to xR i (o,w) is assigned to xRW |(o,r) is assigned to exactly one role xR 2 HR LR M1R M2R Fig. 4. Alternate role hierarchy for Strict ?-property Nevertheless, the structure of role hierarchies which do map to valid LBAC cong- urations is greatly restricted, as the examples in Osborn [1997] show. For example, a role with permissions to both read and write a high data object and a low data object cannot be assigned to a high user as this would allow write down, and cannot be assigned to a low user, as this would allow read up. If a role had only read permissions for some objects classied at M1, and other objects classied at M2 (cf Figure 2), a subject cleared at H could be assigned to this role. As far as the read operation is concerned, a subject can have a role r in its session if the label of the subject dominates the level of all o such that (o,r) is in the role. Since the least upper bound is dened for the security lattice, this can always be determined. Similarly, for write operations, if a greatest lower bound is dened for the security levels, then the Liberal ?-property is satised in a session if the security level of the subject dominates the greatest lower bound of all o such that (o,w) is in the role. If such a greatest lower bound does not exist, such a role should not be in any user's UA. (If it could be determined that (s) (o) for all o such that (o,w) is in the role, then this (s) would be a lower bound, and then a greatest lower bound would exist.) We introduce the following two denitions to capture the maximum read level of objects in a role, and the minimum write level if one exists. Definition 6. The r-level of a role r (denoted r-level(r)) is the least upper bound (lub) of the security levels of the objects for which (o,r) is in the permissions of r. Definition 7. The w-level of a role r (denoted w-level(r)) is the greatest lower bound (glb) of the security levels of the objects o for which (o,w) is in the permissions of r, if such a glb exists. If the glb does not exist, the w-level is undened. The following theorem follows from these denitions. Theorem 3. An RBAC96 conguration satises the simple security property and the Liberal ?-Property if all of the following hold: |Constraint on Users: ((8u 2 U) [(u) is given]) |Constraints on Permissions: is an object in the systemg |Constraint on UA: |Constraint on Sessions: An example showing a possible role hierarchy is given in Figure 5, where the underlying security lattice contains labels funclassied, secret, top secretg and roles are indicated by, for example, (ru,rs) meaning the permissions in the role include read of some unclassied and some secret object(s) (each role may have permissions inherited because of the role hierarchy). The roles labeled ru1 and ru3 at the bottom have read access to distinct objects labeled unclassied; ru2 inherits the permissions of ru1 and has additional read access to objects at the unclassied level. The role labeled (ru,ws) contains permission to read some unclassied objects and some secret objects. This role could be assigned in UA to either unclassied users or to secret users. Notice the role at the top of the role hierarchy, labeled (ru,rs,rts,ws,wts). This role cannot be assigned to any user without violating either Conguring RBAC to Enforce MAC and DAC 13 the Simple Security Property or the Liberal-? Property. Note that if this role is deleted from the role hierarchy, we have an example of a role hierarchy which satises the Simple Security Property and the Liberal-? Property, and which does not conform to any of the constructions of Section 4. Not valid in any User Assignment ru,rs In UA for unclassified users ru,ws ws,wts ws ru,rs ws ru,rs ws,wts In UA for Top- Users ru,rs,rts ws,wts ru,rs,rts users In UA for Fig. 5. A Role Hierarchy and its User Assignments An RBAC96 conguration satises the strict ?-property if all of the above conditions hold, changing the Constraint on Sessions to: |Constraint on Sessions: (8s 2 sessions) 6. DAC MODELS In this section we discuss the DAC policies that will be considered in this paper. The central idea of DAC is that the owner of an object, who is usually its creator, has discretionary authority over who else can access that object. In other words the core DAC policy is owner-based administration of access rights. There are many variations of DAC policy, particularly concerning how the owner's discretionary power can be delegated to other users and how access is revoked. This has been recognized since the earliest formulations of DAC [Lampson 1971; Graham and Denning 1972]. 14 Osborn, Sandhu and Munawer Our approach here is to identify major variations of DAC and demonstrate their construction in RBAC96. The constructions are such that it will be obvious how they can be extended to handle other related DAC variations. This is an intuitive, but well-founded, justication for the claim that DAC can be simulated in RBAC. 2 The DAC policies we consider all share the following characteristics. |The creator of an object becomes its owner. |There is only one owner of an object. In some cases ownership remains xed with the original creator, whereas in other cases it can be transferred to another user. (This assumption is not critical to our constructions. It will be obvious how multiple owners could be handled.) |Destruction of an object can only be done by its owner. With this in mind we now dene the following variations of DAC with respect to granting of access. (1) Strict DAC requires that the owner is the only one who has discretionary authority to grant access to an object and that ownership cannot be transferred. For example, suppose Alice has created an object (Alice is owner of the object) and grants read access to Bob. Strict DAC requires that Bob cannot propagate access to the object to another user. (Of course, Bob can copy the contents of Alice's object into an object that he owns, and then propagate access to the copy. This is why DAC is unable to enforce information ow controls, particularly with respect to Trojan Horses.) (2) Liberal DAC allows the owner to delegate discretionary authority for granting access to an object to other users. We dene the following variations of liberal DAC. (a) One Level Grant: The owner can delegate grant authority to other users but they cannot further delegate this power. So Alice being the owner of object O can grant access to Bob who can grant access to Charles. But Bob cannot grant Charles the power to further grant access to Dorothy. (b) Two Level Grant: In addition to a one-level grant the owner can allow some users to further delegate grant authority to other users. Thus, Alice can now authorize Bob for two-level grants, so Bob can grant access to Charles, with the power to further grant access to Dorothy. However, Bob cannot grant the two-level grant authority to Charles. (We could consider n-level grant but it will be obvious how to do this from the two level construction.) (c) Multilevel Grant: In this case the power to delegate the power to grant implies that this authority can itself be delegated. Thus Alice can authorize Bob, who can further authorize Charles, who can further authorize Dorothy, and so on indenitely. (3) DAC with Change of Ownership: This variation allows a user to transfer ownership of an object to another user. It can be combined with strict or liberal DAC in all the above variations. 2 A formal proof would require a formal denition of DAC encompassing all its variations, and a construction to handle all of these in RBAC96. This approach is pursued in [Munawer 2000]. Conguring RBAC to Enforce MAC and DAC 15 For revocation we consider two cases as follows. (1) Grant-Independent Revocation: Revocation is independent of the granter. Thus Bob may be granted access by Alice but have it revoked by Charles. (2) Grant-Dependent Revocation: Revocation is strongly tied to the granter. Thus if Bob receives access from Alice, access can only be revoked by Alice. In our constructions we will initially assume grant-independent revocation and then consider how to simulate grant-dependent revocation. In general, we will also assume that anyone with authority to grant also has authority to revoke. This coupling often occurs in practice. Where appropriate, we can decouple these in our simulations because, as we will see, they are represented by dierent permissions. These DAC policies certainly do not exhaust all possibilities. Rather these are representative policies whose simulation will indicate how other variations can also be handled. 7. CONFIGURING RBAC FOR DAC To specify the above variations in RBAC96 it su-ces to consider DAC with one operation, which we choose to be the read operation. Similar constructions for other operations such as write, execute and append, are easily possible. 3 Before considering specic DAC variations, we rst describe common aspects of our constructions 7.1 Common Aspects The basic idea in our constructions is to simulate the owner-centric policies of DAC using roles that are associated with each object. 7.1.1 Create an Object. For every object O that is created in the system we require the simultaneous creation of three administrative roles and one regular role as follows. |Three administrative roles in AR: OWN O, PARENT O and PARENTwith- GRANT O |One regular role in R: READ O Role OWN O has privileges to add and remove users from the role PAREN- TwithGRANT O which in turn has privileges to add and remove users from the role PARENT O The relationship between these roles is shown in Figure 6. In Figure administrative roles are shown with darker circles than regular roles. In Figure 6(a), the dashed right arrows indicate that the role on the left contains the administrative permissions governing the role on the right. Figure 6(b) shows the administrative role hierarchy, with the senior role above its immediate junior, connected by an edge. For instance role OWN O has administrative authority over roles PARENTwithGRANT O as indicated in gure 6(a). In addition due to the 3 More complex operations such as copy can be viewed as a read of the original object and a write (and possibly creation) of the copy. It can be useful to associate some default permissions with the copy. For example, the copy may start with access related to that of the original object or it may start with some other default. Specic policies here could be simulated by extending our constructions. inheritance via the role hierarchy of gure 6(a) OWN O also has administrative authority over PARENT O and READ O. OWN_O READ_O (a) OWN_O PARENTwithGRANT_O PARENT_O (b) PARENT_O PARENTwithGRANT_O Fig. 6. (a)Administration of roles associated with an object (b) Administrative role hierarchy In addition we require simultaneous creation of the following eight permissions along with creation of each object O. |canRead O: authorizes the read operation on object O. It is assigned to the role READ O. |destroyObject O: authorizes deletion of the object. It is assigned to the role OWN O. |addReadUser O, deleteReadUser O: respectively authorize the operations to add users to the role READ O and remove them from this role. They are assigned to the role PARENT O. |addParent O, deleteParent O: respectively authorize the operations to add users to the role PARENT O and remove them from this role. They are assigned to the role PARENTwithGRANT O. |addParentWithGrant O, deleteParentWithGrant O: respectively authorize the operations to add users to the role PARENT O and remove them from this role. They are assigned to the role OWN O. These permissions are assigned to the indicated roles when the object is created and thereafter they cannot be removed from these roles or assigned to other roles. 7.1.2 Destroy an Object. Destroying an object O requires deletion of the four roles namely OWN O, PARENT O, PARENTwithGRANT O and READ O and the eight permissions (in addition to destroying the object itself). This can be done only by the owner, by virtue of exercising the destroyObject O permission. Conguring RBAC to Enforce MAC and DAC 17 7.2 Strict DAC In strict DAC only the owner can grant/revoke read access to/from other users. The creator is the owner of the object. By virtue of membership (via seniority) in PARENT O and PARENTwithGRANT O, the owner can change assignments of the role READ O. Membership of the three administrative roles cannot change, so only the owner will have this power. This policy can be enforced by imposing a cardinality constraint of 1 on OWN O and of 0 on PARENT O and PARENTwith- GRANT O. This policy could be simulated using just two roles OWN O and READ O, and giving the addReadUser O and deleteReadUser O permissions directly to OWN O at creation of O. For consistency with subsequent variations we have introduced all required roles from the start. 7.3 Liberal DAC The three variations of liberal DAC described in Section 6 are now considered in turn. 7.3.1 One-Level Grant. The one-level grant DAC policy can be simulated by removing the cardinality constraint of strict DAC on membership in PARENT O. The owner can assign users to the PARENT O role who in turn can assign users to the READ O role. But the cardinality constraint of 0 on PARENTwithGRANT O remains. 7.3.2 Two-Level Grant. In the two level grant DAC policy the cardinality constraint on PARENTwithGRANT O is also removed. Now the owner can assign users to PARENTwithGRANT O who can further assign users to PARENT O. Note that members of PARENTwithGRANT O can also assign users directly to READ O, so they have discretion in this regard. Similarly the owner can assign users to PARENTwithGRANT O, PARENT O or READ O as deemed appropri- ate. (N-level grants can be similarly simulated by having N roles, PARENTwith- GRANT O N 1 , PARENTwithGRANT O N 2 , . , PARENTwithGRANT O, PARENT O.) 7.3.3 Multilevel Grant. To grant access beyond two levels we authorize the role PARENTwithGRANT O to assign users to PARENTwithGRANT O. We achieve this by assigning the addParentWithGrant O permission to the role PARENTwith- GRANT O when object O is created. As per our general policy of coupling grant and revoke authority, we also assign the deleteParentWithGrant O permission to the role PARENTwithGRANT O when O is created. This coupling policy is arguably unreasonable in the context of grant-independent revoke, so the deletePar- entWithGrant O permission could be retained only with the OWN O role if so desired. For grant-dependent revoke the coupling is more reasonable. 7.4 DAC with Change of Ownership Change of ownership can be easily accomplished by suitable redenition of the administrative authority of a member of OWN O. Recall that change of ownership in this context means transfer of ownership from one user to another. Thus the OWN O role needs a permission that enables this transfer to occur and this per- mission can only be assigned to this role. A member of OWN O can assign another user to OWN O but at the cost of loosing their own membership. 7.5 Multiple Ownership Multiple ownership can also be accommodated by removing the cardinality constraint on membership in the OWN O role. Since all members of OWN O have identical power, including the ability to revoke other owners, it would be appropriate with grant-independent revoke to distinguish the original owner. Alternately, we can have grant-dependent revoke of ownership. 7.6 Grant-Dependent Revoke So far we have considered grant-independent revocation where revocation is independent of granter. Now nally we consider how to simulate grant-dependent revoke in RBAC96. In this case only the user who has granted access to another user can revoke the access (with possible exception of the owner who is allowed to revoke everything). U1_PARENT_O U2_PARENT_O Un_PARENT_O U1_READ_O U2_READ_O Un_READ_O Fig. 7. Read O Roles associated with members of PARENT O Specically, let us consider the one level grant DAC policy simulated earlier by allowing members of PARENT O role to assign users to READ O role. To simulate grant-dependent revocation with this one level grant policy we need a dierent administrative role U PARENT O and a dierent regular role U READ O for each user U authorized to do a one-level grant by the owner. These roles are automatically created when the owner authorizes user U. We also need two new administrative permissions created at the same time as follows. |addU ReadUser O, deleteU ReadUser O: respectively authorize the operations to add users to the role U READ O and remove them from this role. They are assigned to the role U PARENT O. Conguring RBAC to Enforce MAC and DAC 19 Ui PARENT O manages the membership assignments of Ui READ O role as indicated in Figure 7 for user Ui. U PARENT O has a membership cardinality constraint of one. Moreover, its membership cannot be changed. Thus user U will be the only one granting and revoking users from U READ O. The U READ O role itself is assigned the permission canRead O at the moment of creation. As before all of this enforced by RBAC96 constraints. We can allow the owner to revoke users from the U READ O role by making U PARENT O junior to OWN O in the administrative role hierarchy. Simulation of grant-dependent revocation can be similarly simulated with respect to the PARENT O and PARENTwithGRANT O roles. Extension to multiple ownership is also possible. 8. CONCLUSIONS We have shown that the common forms of LBAC and DAC models can be simulated and enforced in RBAC96 with systematic constructions. All of the components of the RBAC96 model shown in Figure 1 were required to carry out these simulations. Users and permissions are essential to express any access control model. The Role Hierarchy is important in the LBAC simulation. The Administrative Role Hierarchy is essential in the enforcement of DAC policies, as is the administrative user to role assignment relation. We observe however that the permissions that have been granted to users in a DAC system can give an arbitrarily rich role hierarchy, as was noted in a conversion of relational database permissions to role graphs by Osborn, Reid, and Wesson [1996]. Constraints play a role in all of the constructions. It is important to note that the LBAC simulation assumes a single administrative role, whereas the DAC simulation requires a large number of administrative roles, which are dynamically created and destroyed. RBAC Models One Administrative Role Complex Administrative Roles of Section 7 of Section 4 of Section 5 a. LBAC construction b. other LBAC configurations c. DAC configurations Fig. 8. Containment of Models We can represent some of our ndings using the Venn diagram in Figure 8. The 20 Osborn, Sandhu and Munawer area on the left of the gure indicates that in this subset of RBAC96 congurations, there is no need for administrative roles except for an assumed single administrative role. On the right, the administrative part of the RBAC96 model is fully utilized. Part (a) represents the subset of possible RBAC96 congurations which are built by constructions 1 and 2. Area (b) shows that there are other congurations not built by these two constructions which still satisfy LBAC properties. Part (c) of the diagram represents in general the RBAC96 congurations built by the various constructions in Section 7. Note that these latter constructions all fall in the region where the administrative roles of the RBAC96 model are being fully utilized. Future work should now focus on what happens in the rest of the RBAC96 Models not included in the areas constructed in this paper. Models for decentralized role administration which fall in between these extremes have been proposed by Sandhu, Bhamidipati, and Munawer [1999]. These models allow for large numbers of administrative roles but this number is expected to be much smaller than the number of objects in the system. In conclusion, then, we have shown with various systematic constructions how to simulate and enforce traditional LBAC and DAC access control models in RBAC96. --R Secure computer systems: A network interpretation. A comparison of commercial and military computer security policies. A lattice model of secure information ow. 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Implementing the Clark/Wilson integrity policy using current tech- nology --TR Role-Based Access Control Models Modeling mandatory access control in role-based security systems Mandatory access control and role-based access control revisited On the interaction between role-based access control and relational databases How to do discretionary access control using roles The role graph model and conflict of interest The ARBAC97 model for role-based administration of roles A lattice model of secure information flow Lattice-Based Access Control Models Access Rights Administration in Role-Based Security Systems Role Hierarchies and Constraints for Lattice-Based Access Controls Administrative models for role-based access control --CTR Sylvia L. Osborn, Information flow analysis of an RBAC system, Proceedings of the seventh ACM symposium on Access control models and technologies, June 03-04, 2002, Monterey, California, USA Cungang Yang , Chang N. 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Mancini, Security and privacy issues of handheld and wearable wireless devices, Communications of the ACM, v.46 n.9, p.74-79, September Engineering authority and trust in cyberspace: the OM-AM and RBAC way, Proceedings of the fifth ACM workshop on Role-based access control, p.111-119, July 26-28, 2000, Berlin, Germany Manuel Koch , Luigi V. Mancini , Francesco Parisi-Presicce, A graph-based formalism for RBAC, ACM Transactions on Information and System Security (TISSEC), v.5 n.3, p.332-365, August 2002 Wolfgang Essmayr , Stefan Probst , Edgar Weippl, Role-Based Access Controls: Status, Dissemination, and Prospects for Generic Security Mechanisms, Electronic Commerce Research, v.4 n.1-2, p.127-156, January-April 2004 David F. Ferraiolo , Serban Gavrila , Vincent Hu , D. Richard Kuhn, Composing and combining policies under the policy machine, Proceedings of the tenth ACM symposium on Access control models and technologies, June 01-03, 2005, Stockholm, Sweden Jason Crampton, On permissions, inheritance and role hierarchies, Proceedings of the 10th ACM conference on Computer and communications security, October 27-30, 2003, Washington D.C., USA Yang , Raimund K. Ege , Huiqun Yu, Mediation security specification and enforcement for heterogeneous databases, Proceedings of the 2005 ACM symposium on Applied computing, March 13-17, 2005, Santa Fe, New Mexico James B. D. Joshi , Elisa Bertino , Usman Latif , Arif Ghafoor, A Generalized Temporal Role-Based Access Control Model, IEEE Transactions on Knowledge and Data Engineering, v.17 n.1, p.4-23, January 2005 Patrick C. K. Hung , Dickson K. W. Chiu , W. W. Fung , William K. Cheung , Raymond Wong , Samuel P. M. Choi , Eleanna Kafeza , James Kwok , Jousha C. C. Pun , Vivying S. Y. Cheng, Towards end-to-end privacy control in the outsourcing of marketing activities: a web service integration solution, Proceedings of the 7th international conference on Electronic commerce, August 15-17, 2005, Xi'an, China Siqing Du , James B. D. Joshi, Supporting authorization query and inter-domain role mapping in presence of hybrid role hierarchy, Proceedings of the eleventh ACM symposium on Access control models and technologies, June 07-09, 2006, Lake Tahoe, California, USA W. T. Tsai , X. Liu , Y. Chen , R. Paul, Simulation Verification and Validation by Dynamic Policy Enforcement, Proceedings of the 38th annual Symposium on Simulation, p.91-98, April 04-06, 2005 James B. D. Joshi , Basit Shafiq , Arif Ghafoor , Elisa Bertino, Dependencies and separation of duty constraints in GTRBAC, Proceedings of the eighth ACM symposium on Access control models and technologies, June 02-03, 2003, Como, Italy Mahesh V. Tripunitara , Ninghui Li, Comparing the expressive power of access control models, Proceedings of the 11th ACM conference on Computer and communications security, October 25-29, 2004, Washington DC, USA Jingzhu Wang , Sylvia L. Osborn, A role-based approach to access control for XML databases, Proceedings of the ninth ACM symposium on Access control models and technologies, June 02-04, 2004, Yorktown Heights, New York, USA James B. D. Joshi , Elisa Bertino, Fine-grained role-based delegation in presence of the hybrid role hierarchy, Proceedings of the eleventh ACM symposium on Access control models and technologies, June 07-09, 2006, Lake Tahoe, California, USA Jason Crampton , George Loizou, Administrative scope: A foundation for role-based administrative models, ACM Transactions on Information and System Security (TISSEC), v.6 n.2, p.201-231, May David F. Ferraiolo, An argument for the role-based access control model, Proceedings of the sixth ACM symposium on Access control models and technologies, p.142-143, May 2001, Chantilly, Virginia, United States Timothy Fraser , David Ferraiolo , Mikel L. Matthews , Casey Schaufler , Stephen Smalley , Robert Watson, Panel: which access control technique will provide the greatest overall benefit, Proceedings of the sixth ACM symposium on Access control models and technologies, p.141-149, May 2001, Chantilly, Virginia, United States Basit Shafiq , James B. D. Joshi , Elisa Bertino , Arif Ghafoor, Secure Interoperation in a Multidomain Environment Employing RBAC Policies, IEEE Transactions on Knowledge and Data Engineering, v.17 n.11, p.1557-1577, November 2005 Thuong Doan , Steven Demurjian , T. C. 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Sreedhar, Data-centric security: role analysis and role typestates, Proceedings of the eleventh ACM symposium on Access control models and technologies, June 07-09, 2006, Lake Tahoe, California, USA James B. D. Joshi , Elisa Bertino , Arif Ghafoor, An Analysis of Expressiveness and Design Issues for the Generalized Temporal Role-Based Access Control Model, IEEE Transactions on Dependable and Secure Computing, v.2 n.2, p.157-175, April 2005 Yanjun Zuo , Brajendra Panda, Component based trust management in the context of a virtual organization, Proceedings of the 2005 ACM symposium on Applied computing, March 13-17, 2005, Santa Fe, New Mexico Ting Yu , Divesh Srivastava , Laks V. S. Lakshmanan , H. V. Jagadish, A compressed accessibility map for XML, ACM Transactions on Database Systems (TODS), v.29 n.2, p.363-402, June 2004 Shih-Chien Chou, Providing flexible access control to an information flow control model, Journal of Systems and Software, v.73 n.3, p.425-439, November-December 2004 Lawrence A. Gordon , Martin P. Loeb, The economics of information security investment, ACM Transactions on Information and System Security (TISSEC), v.5 n.4, p.438-457, November 2002 Shih-Chien Chou , Chin-Yi Chang, An information flow control model for C applications based on access control lists, Journal of Systems and Software, v.78 n.1, p.84-100, October 2005 Rafae Bhatti , Arif Ghafoor , Elisa Bertino , James B. D. Joshi, X-GTRBAC: an XML-based policy specification framework and architecture for enterprise-wide access control, ACM Transactions on Information and System Security (TISSEC), v.8 n.2, p.187-227, May 2005 Charles E. Phillips, Jr. , T.C. Ting , Steven A. Demurjian, Information sharing and security in dynamic coalitions, Proceedings of the seventh ACM symposium on Access control models and technologies, June 03-04, 2002, Monterey, California, USA Shih-Chien Chou, Embedding role-based access control model in object-oriented systems to protect privacy, Journal of Systems and Software, v.71 n.1-2, p.143-161, April 2004 David F. Ferraiolo , Ravi Sandhu , Serban Gavrila , D. Richard Kuhn , Ramaswamy Chandramouli, Proposed NIST standard for role-based access control, ACM Transactions on Information and System Security (TISSEC), v.4 n.3, p.224-274, August 2001 Joon S. Park , Ravi Sandhu , Gail-Joon Ahn, Role-based access control on the web, ACM Transactions on Information and System Security (TISSEC), v.4 n.1, p.37-71, Feb. 2001 W. T. 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mandatory access control;lattice-based access control;role-based access control;discretionary access control

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Automated systematic testing for constraint-based interactive services.

Constraint-based languages can express in a concise way the complex logic of a new generation of interactive services for applications such as banking or stock trading, that must support multiple types of interfaces for accessing the same data. These include automatic speech-recognition interfaces where inputs may be provided in any order by users of the service. We study in this paper how to systematically test event-driven applications developed using such languages. We show how such applications can be tested automatically, without the need for any manually-written test cases, and efficiently, by taking advantage of their capability of taking unordered sets of events as inputs.

INTRODUCTION Today, it is becoming more and more commonplace for modern interactive services, such as those for personal banking or stock trading, to have more than one interface for accessing the same data. For example, many banks allow customers to access personal banking services from an automated teller machine, bank-by- phone interface, or web-based interface. Furthermore, telephone-based services are starting to support automatic speech recognition and natural language understanding, adding further flexibility in interaction with the user. When multiple interfaces are provided to the same service, duplication can be a serious problem in that there may be a different service logic (i.e., the code that defines the essence of the service) for every different user interface to the service. Moreover, to support natural language interfaces, services must allow users considerable flexibility in the way they input their service requests. Thus, it is desirable for the programming idioms and methods for such services to provide the following capabilities: allow requests to be phrased in various ways (e.g., needed information can be provided in any order), prompting for missing information, correction of erroneous information, lookahead (to allow the user to speak several commands at once), and backtracking to earlier points in the service logic. Constraint-based languages (for a foundational overview, see [16, 19]) provide a suitable paradigm to support the required flexibility in user inputs. Examples of such languages in dialogue management include methods based on frames [1], forms [9, 15], approaches based on AND/OR trees [22] and Sisl (Several Interfaces, Single Logic) [4]. The powerful expressiveness of constraint-based languages allows the construction of succinct programs with complex reactive behavior. This motivates the need for reliable yet cost-effective testing techniques and tools suitable to check the correctness of business-critical applications developed using such languages. We study in this paper how to automatically and efficiently check temporal properties of programs written in constraint-based languages for interactive services. We first present a nondeterministic algorithm for systematically testing the logic of a program in a constraint-based language. The possible use of arbitrary host language code and potential elaborate access to external data (such as database lookups) in a full program makes the use of classical constraint satisfaction and model checking techniques problematic. Consequently, this algorithm assumes, as is usually done in testing, that the user specifies a fixed set of possible data values for each user input. The algorithm then dynamically detects at run-time the set of input events that an application is currently ready to accept, and uses that information to drive the execution of the application by sending it inputs selected nondeter- ministically. Used in conjunction with the systematic state-space exploration tool VeriSoft [11], which supports a special system call (called VS toss) simulating nondeterminism, this nondeterministic test driver can systematically generate all possible behaviors of any such program. These behaviors can then be monitored and checked against user-specified safety properties. This algorithm thus eliminates the need for manually-written test cases, and is automatically applicable to any program. This testing technique is made possible by the structured interface that an event-driven constraint-based program provides to its environment. Unfortunately, the expressiveness of constraint-based languages makes this algorithm quite inefficient. For instance, a service that awaits 10 inputs from the user (in no particular order) has 10! different paths to collect the inputs. However, the flip side of this ability to accept unordered sets of inputs that can be provided in any order, is the inability of constraint languages to (observation- ally) distinguish permutations of the unordered sets of inputs. Thus, generating all these sequences is not always necessary to check the temporal properties that we are considering. Our second algorithm makes systematic testing significantly more efficient by exploiting this observation and taking advantage of a form of symmetry induced by constraints and their ability to specify sets of input events rather than single events. While the basic problems and ideas that we discuss in this work are common to any deterministic constraint-based language in our domain of interest, our concrete presentation is in the context of the Sisl language mentioned earlier. Sisl includes a deterministic constraint-based domain-specific language for developing event-driven reactive services. It is implemented as a library in Java. Sisl supports the development of services that are shared by multiple user interfaces including automatic speech recognition based or text-based natural language understanding, telephone voice access, web, and graphics based interfaces. Sisl is currently being used to prototype a new generation of call processing services for a Lucent Technologies switching product. This paper is organized as follows. In the next section, we describe the Sisl language and formalize its semantics. Then, we present algorithms for automatically and efficiently checking safety properties of Sisl programs. These algorithms have been implemented and implementation issues are discussed in Section 4. A simple example of Sisl application is presented in Section 5. Results of experiments with this example are then discussed. Finally, we present concluding remarks and compare our results with other related work. 2. SISL: SEVERAL INTERFACES, SINGLE 2.1 Overview of Sisl Sisl applications consist of a single service logic, together with multiple user interfaces. The service logic and the interfaces communicate via events: the user interfaces collect information from the user and send it to the service logic in the form of events. The service logic reacts to the events, and reports to the user interfaces its set of enabled events; i.e. the set of events it is ready to accept in its current state. The user interfaces utilize this information to appropriately prompt the user. Any user interface that performs these functions can be used in conjunction with a Sisl service logic. For example, in the context of an interactive natural language service that browses and displays organizational data from a corporate database, Sisl may inform the natural-language interface that it can accept an organization e.g., if the user has said "Show me the size of the organization." The interface may then prompt the user, e.g. by saying "What organization do you want to see?" If the user responds with something that the natural-language interface recognizes as an organization, the interface will then generate a Sisl event indicating the organization was detected. This event will then be processed by the Sisl service logic. In Sisl, the service logic of an application is specified as a reactive constraint graph, which is a directed graph with an enriched structure on the nodes. The traversal of reactive constraint graphs is driven by the reception of events from the environment: these events have an associated label (the event name) and may carry associated data. In response, the graph traverses its nodes and executes actions; the reaction of the graph ends when it needs to wait for the next event to be sent by the environment. Predicates on events play an important role in reactive constraint graphs. In par- ticular, nodes may contain an ordered set of predicates, which indicate a conjunction of information to be received from the user. Upon receipt of the appropriate events, predicates in the current node are evaluated in order. Satisfaction of all predicates at a node triggers a node change in the graph, as may violation of any predicate Reactive constraint graphs are implemented as a Java library. An important aspect of reactive constraint graphs is that nodes may have associated actions, consisting of arbitrary Java code, which is executed upon entry into/exit from the node. These actions can hence have side effects on local and global variables of the Java program, or on external databases. We note that individual predicates do not have actions associated with them. 2.2 The Sisl process algebra In this section, we describe a process algebra to succinctly describe Sisl programs. Events. We begin with a description of the events used in the process algebra. A set of (input) labels I . A set of values V , that are carried on labels from I . A label with an associated value is called an event, and we write e to range over [I V]. A signature is a subset of I . We use for a signature, and jj for its size. A predicate over a signature is a boolean valued function from V jj ! Bool; that is, it maps every set f(a; v)ja to ftrue; falseg. The signature of a predicate is sometimes written (). Syntax. Terms of the process algebra have the following abstract (where and viol denote processes). (Sequential composition) viol (Constraint) We refer to P i viol as the target node of predicate i . Informal dynamic semantics. The process combinators have the following intuitive meaning. Sequential composition is standard: means that P1 is executed first and P2 is executed after terminates. Choice, functions like prefixing in process alge- bras: the process waits for an event with label in the a i 's and with data v i . Such a term corresponds to a "choice" node in the reactive constraint graph, and represents a disjunction of events to be received from the user. For every event in this set, the Sisl service logic automatically sends out a corresponding prompt to the user. The choice node then waits for the user to send an event in the specified set. When such an event arrives, the corresponding transition is taken, and control transfers to the child node. To ensure determinism of the Sisl program, all events (a must be distinct. next viol n), the constraint combinator, has the most interesting dynamic behavior. Such a term corresponds to a "constraint" node in the reactive constraint graph, and has an associated ordered set of predicates i on events. Intuitively, a constraint node is awaiting all the events in [ Thus, this node represents a conjunction of information to be received from the user. These events can be sent to the constraint node in any order. When the control reaches the constraint node, the Sisl service logic automatically sends out a prompt event for every event that is still needed in order to evaluate some predicate. In addition, it automatically sends out a optional prompt for all other events mentioned in the predicates - these correspond to information that can be corrected by the user. In every round of interaction, the constraint node waits for the user to send any event that is mentioned in its associated predicates. Each predicate associated with a constraint node is evaluated in order as soon as all of its events have arrived. If an event is resent by the user interfaces (i.e, information is corrected), predicates with that event in their signature are re-evaluated in order. There are two ways to exit a constraint node. All of its predicates have been evaluated and are satisfied. In this case, control transfers to P next Some predicate i with non-null P i viol evaluates to false, and all predicates j with j < i evaluate to true. That is, predicate i is the first predicate in order that is false, and its viol is non-null. In this case, control transfers to P i viol As mentioned earlier, node changes may cause side-effecting actions to be executed; however, evaluation, satisfaction, or violation of individual predicates do not cause any side effects to occur, other than the node changes described above. 2.3 Sisl: The state machine semantics We will describe a associated space of state machines associated with Sisl, and describe the semantics of Sisl processes as state machines A state machine in the Sisl semantics is a tuple (I; where I is the set of input labels, S is a set of states, s0 is the unique start state, F S is the set of final states, and the transition relation T S [I V ] S is deterministic, i.e., given a state s and any event e, there is at most one s 0 such that (s; e; s 0 ) is in the transition relation. In this case, we write s e Final states do not have any outgoing transitions. Constructions We will describe the denotations of Sisl combinators as constructions on the state machines. In this section we will use P (perhaps with super/subscripts) to range over the members of the given class of state machines. The state machine corresponding to null has a single state that is both the start state and the final state. It has no transitions. Sequential composition This is described in a standard fashion by assumption on final states. All arcs into the final states of P1 are redirected to the unique start state of P2 in P1 ; P2 . Formally, the sequential composition of state machines and is the state machine Choice: This is described by building the state machine for adding a new start state with transitions labeled to the start state of the state machine for P i . Formally, given state machines P the resulting state machine P is (I; new state, Constraint: viol The set S of intermediate states the Sisl program goes through while collecting events in K are determined by partial maps f Intuitively, the domain of a partial function f , written dom(f), indicates the labels on which data has been received. With this intuition, it is clear that the information still required to satisfy the requirements of this constraint node is given by the transitions that are enabled at this state, and have labels undefinedg. Furthermore, the start state is given by the partial map with empty domain, since no information has been received. We are identical on all other labels. f is consistent if f is such that for all predicates i , if dom(f) includes all a f is inconsistent otherwise, i.e., there is some predicate i such that dom(f) includes f is complete, if dom(f) = K, incomplete otherwise. All four combinations of the above two parameters are possible. Formally, the set S of states is all partial maps f is either inconsistent or incomplete. Let transition relation T on states in S be defined as follows: (a; v) be a consistent and complete map. There is a transition labeled (a; v) from state with label f to P next . That is, P next is the target of transitions that make the information contained in a state complete in a consistent fashion. (a; v) be an inconsistent map, with i being the first i that is falsified in g. If P viol i is not null, then there is an arc labeled (a; v) from state with label f to the start state of P i viol . That is, causing a predicate violation with a non-null target node causes control to move to that node. There is a transition labeled (a; v) from a state with label f to a state with label - g is consistent and incomplete, or - g is inconsistent, and P viol i is null, where i is the first predicate in order that is falsified in g. That is, if no predicates are violated, or if a predicate with a null target node is violated, then control remains in the constraint node. A transition labeled (a; v) adds or changes information on label a at a state. If it changes the information, it is called an overwrite event: (a; v) is an overwrite event if it causes a transition from some state f in which a 2 dom(f). The start state is given by the partial map with empty domain. The final states are given by the union of the final states of P next and the non-null P i viol 's. Formally, given state machines P next and the non-null P i the resulting state machine is is the partial map with empty domain, In the following we write target( i ) to refer to the target node of predicate i . 3. AUTOMATIC TESTING OF SISL PROGRAM In this section, we show how Sisl programs can be automatically and systematically tested for violations of safety properties using nondeterministic testing algorithms. Used in conjunction with the systematic state-space exploration tool VeriSoft [11], which supports a special system call "VS toss" simulating nondeterminism, these nondeterministic testing algorithms can systematically drive the execution of the Sisl application being tested in order to exhibit all its possible behaviors. Safety properties can be represented by prefix-closed finite automata on finite words [3]. We assume such a representation AP , and define a safety property L(P ) as the set of finite words accepted by the finite automaton AP . Let O be the alphabet of AP . By construction, O is contained in the set of input events I of the state machine M representing the Sisl program to be tested. We call input events in O observable events. Let w jO denote the projection of a word w 2 I over a set O I . DEFINITION 3.1. Let be the state machine defining the semantics of a Sisl program as defined in the previous section but with (which is therefore omitted). Let s w denote an execution of M that goes from state s to state s 0 after receiving a finite sequence w of events which does not include any overwrite events. Let O I denote a set of observable events. We define the set of observable behaviors of M as the language LO (M) on finite words on O such that LO jO g. In other words, the set of observable behaviors of a Sisl program with respect to a set of observable events O is defined as the set of finite sequences of observable events that the Sisl program can take as inputs excluding overwrite events. For technical convenience and efficiency reasons, we deliberately ignore overwrite events in this definition since their occurrence is an artifact of the user-interface of the system that does not affect transitions from nodes to nodes, and hence the logic of the Sisl program. The problem we address in this work is thus how to check automatically and efficiently that LO (M) L(P ). A naive solution to this problem consists of driving the execution of the Sisl program by a test driver that nondeterministically sends any enabled input event and associated valid data value to whenever M is ready to take a new input, the execution of the nondeterministic test driver being itself under the control of VeriSoft. Checking LO (M) L(P ) can then be done by monitoring all possible executions of M in conjunction with this test driver, and checking that all its observable behaviors are accepted by AP . However, this naive approach would generate a state space typically so large that it would render any analysis intractable: for instance, any input event that takes a 32-bit integer as argument would immediately generate a branching point with 2 branches. Clearly, data values are the cause of this unacceptable state explosion In the case of constraint nodes for instance, one could think that an analysis of the predicates associated to such nodes using constraint satisfaction techniques might be used to generate automatically data values. However, this approach is problematic in our context since predicates in Sisl programs are implemented by arbitrary Java code, can be quite elaborate and may involve accesses to external data (for instance, the evaluation of a predicate may involve database lookups to fetch and test provisioning data for the subscriber of the service). Also, the evaluation of a predicate on a same set of input data values may change over time (when the evaluation of a predicate may depend on data previously modified during the current execution). How to close automatically any open reactive (Java) program with its most general environment is an interesting but hard problem [5] that is beyond the scope of the present work. Therefore, we will simply assume here, as is usually done in testing, that the user specifies a fixed set values(a) of possible data values for each input event a in I . Whenever event a is provided as input to the Sisl program during testing, a data value v in values(a) is chosen nondeterministically by the test driver and then passed as argument of event a to the Sisl service. For a given set V of sets values(a) of data values, we define the restriction of M by V as follows. DEFINITION 3.2. Let be the state machine defining the semantics of a Sisl program as defined in the previous section. Let V be a (complete) valuation function that associates with each input event a a finite nonempty set values(a) of data values: 8a We call the restriction of by V the state machine (a)g. The restriction of M by V is thus the set of states the Sisl program represented by M can reach when data values associated to input events are taken from V exclusively. Note that, since Sisl programs are deterministic, the successor state s 0 reached after receiving an input event a(v) in a state s is always unique. In the remainder of this section, we discuss a restricted version of our original problem, namely how to check automatically and efficiently that LO (MV ) L(P ), instead of LO (M) L(P ). A simple algorithm for checking whether LO (MV ) L(P ) is presented in Figure 1. At any reached state, this algorithm non-deterministically selects an enabled input event a (Step 2.b) and a data value v in values(a) (Step 2.c), and then sends a(v) to the Sisl program being tested (Step 2.d). Nondeterminism is simulated by the special operation "VS toss" supported by VeriSoft. This operation takes as argument a positive integer n, and returns an integer in [0; n]. The operation is nondeterministic: the execution of a transition starting with VS toss(n) may yield up to n different successor states, corresponding to different values returned by VS toss. The execution of this nondeterministic and nonterminating test- driver algorithm is controled by VeriSoft. VeriSoft provides the value to be returned for each call to VS toss in order to systematically explore all the possibilities. It also forces the termination of every execution when a certain depth is reached. This maximum depth is specified by the user via one of the several parameters that the user can set to control the state-space search performed by VeriSoft, and is measured here by the number of calls to VS toss executed so far in the current run of the algorithm. Checking LO (MV ) L(P ) can be done as follows. Whenever an event a(v) is sent to the Sisl service SSV to be tested, the automaton AP representing the property P is evaluated on a(v). If a 2 O, the automaton may move and reach a new state. By con- struction, if AP reaches a non-accepting state, this means that the property P is violated. An error is then reported, and the search stops. The last scenario executed is saved in memory, and can be replayed later by the user with the VeriSoft simulator. The algorithm of Figure 1 generates all possible sequences of input events (and associated data values) that can be taken as input by the Sisl program. In the rest of this section, we show that generating all these sequences is actually not necessary to check the type of safety properties we consider here. Precisely, we show that the Initialize the Sisl service SSV . 2. Loop forever: (a) Let E be the set of non-overwrite events in I that are enabled in the current state s. If (b) Let i =VS toss(jEj 1); let a be the i th element of E. (c) Let j =VS toss(jvalues(a)j 1); let v be the j th element of values(a). (d) Send a(v) to SSV . Figure 1: Simple Algorithm algorithm of Figure 1 can be optimized so that it does not generate all possible sequences of input events enabled in constraint nodes provided that these events are not observable. EXAMPLE 3.3. Let P be a sisl program, and let 1 any predicates, with signatures given by Let O, the set of observable events be empty. All interleavings of unobservable input events that drive the system to a same successor node of a constraint node have the same void effect on the property being checked; so, there is no need to generate all of these. Conse- quently, the testing of this constraint node requires the generation of one sequence of input events rather than the ([ i i )! sequences generated by the naive algorithm. On the other hand, if O is ([ i i ), the testing of this constraint node requires the generation of all the ([ i i )! sequences generated by the naive algorithm. This observation is exploited in the algorithm of Figure 2. This algorithm behaves as the previous one except in the case of constraint nodes. In that case (Step 2.c), the algorithm starts (Step 2.c.ii) by nondeterministically choosing a data value va to be associated with each input event a enabled in the constraint node. Then (Step 2.c.iii), it evaluates successively all the predicates i of the constraint node. Predicates i that are violated by the selected set of data values va for each a 2 E are added to a set Marked of violated predicates unless there exists another predicate j previously added in Marked whose signature ( j ) is contained in the signature whose signature contains the same set of observable events as (Step 2.c.iii.A). If the selected set of data values does not satisfy all the predicates of the node (Step 2.c.iv), one violated predicate i in the set Marked such that target( i ) 6= null is nondeterministically chosen, and the input events in its signature are selected in the set S of events to be provided to the Sisl program (Step 2.c.iv.E); otherwise, all predicates are satisfied, and set S is the set of all enabled input events that are enabled in the node (Step 2.c.v). Then, all unobservable input events in set S (if any) are sent to the Sisl program (Step 2.c.vi). Finally, all remaining (hence observable) input events in S are sent to the Sisl program one by one in some random order picked nondeterministically among the set of all possible interleavings of these events (Step 2.c.vii). The correctness of the algorithm of Figure 2 can be proved by showing that there is a weak bisimulation between the nodes reached during the execution of the algorithm and the nodes of MV , the restriction of M by V . This in turn guarantees that all observable behaviors of MV can be observed during the nondeterministic executions of the algorithm of Figure 2. Let node(s) be the current node the Sisl program when it is in state s. We write n w denote that there exists a sequence of non-overwrite input events w 0 such that s w 0 jO and no node other than n and n 0 are traversed during the transition from s to s 0 . THEOREM 3.4. Let be the restriction of the state machine M defining the semantics of a Sisl program by a valuation function V . Let M be the state machine defined with the set T 0 representing the set of state transitions performed by the Sisl program when it is being tested by the algorithm of Figure 2. Then, for any reachable state s in MV , we have for if n w if n w in MV . PROOF. (Sketch) The proof is immediate if n is not a constraint node. Consider the case where n is a constraint node. Let s be a reachable state in MV such that To simplify the presentation, assume s is the first state reached when entering node n during that visit of n. (Other cases can be treated in a similar way.) Let us show that every node transition n w in MV can be matched by a node transition n w (the converse is immediate). there exists a sequence of non-overwrite input events w 0 such that s w 0 jO no node other than n and n 0 are traversed during the transition from s to s 0 . For simplicity, assume that s 0 denote the first state reached when entering node n 0 when executing w 0 from s. (Again, other cases can be treated in a similar way.) If n 0 is the node Pnext reached when all predicates in n are sat- isfied, then the set fva ja 2 w 0 g of data values associated to input events provided during the execution of w 0 from state s to s 0 satisfy all the predicates in n. Thus, there exists an execution of the algorithm of Figure 2 that can select this set of data values in Step 2.c.ii. In that case, none of the predicates of node n will be marked, the algorithm will select all the input events in w to be sent to the Sisl program (Step 2.c.v), and there exists one execution of the algorithm that will send all the observable events in w 0 in the same order as in w (Step 2.c.vii). Otherwise, n 0 is the (non-null) target node target( i ) reached after a predicate i of n is violated. This means that the set fva ja 2 of data values associated to input events provided during the execution of w 0 from state s to s 0 violates predicate i . Since is reachable, this also means that no other predicate and such that violated (other- wise, target( j ) would be reached instead of target( i ) when executing w 0 from s). There exists an execution of the algorithm of Figure 2 that can select in Step 2.c.ii the set of data values fva ja 2 Initialize the Sisl service SSV . 2. Loop forever: (a) Let E be the set of non-overwrite events in I that are enabled in the current state s. If (b) If the current node of SSV is NOT a constraint node: let a be the i th element of E. ii. Let j =VS toss(jvalues(a)j 1); let v be the j th element of values(a). iii. Send a(v) to SSV . (c) If the current node of SSV is a constraint node: be the sequence of predicates associated with the constraint node. ii. Loop through all events a in E (in any order): A. Let j =VS toss(jvalues(a)j 1); let va be the j th element of values(a). iii. Loop through all i from 1 to m (in A. If i is violated by the data values fva ja 2 Eg AND Marked implies then add i to set Marked. iv. If jMarkedj > 0: A. Remove from Marked all i such that target( i B. If C. Let i =VS toss(jMarkedj 1). D. Let i be the i th element of Marked. v. Else: A. Let vi. For all a 2 (S n O), send a(va) to SSV . vii. Loop until (S \ A. Let i =VS toss(jS \ Oj 1); let a be the i th element of S \ O. B. Send a(va) to SSV . C. Remove event a from set S. Figure 2: Optimized Algorithm g. This set of data values violates predicate i , which will then be added to the set Marked in Step 2.c.iii of the algorithm, unless there exists another violated predicate j with j < i, previously added in Marked and such that target( i in that case, violating this other predicate also leads to node n 0 via a sequence w 00 of input events such that w 00 jO In any case, a predicate whose violation leads to n 0 via a sequence w 000 of events such that w 000 jO w is added to Marked. If this predicate is selected in Step 2.c.iv of the algo- rithm, all the events in its signature (which includes all events in w) are then sent to the Sisl program, and there exists one execution of the algorithm that will send all the observable events in w 000 in the same order as in w (Step 2.c.vii). An immediate corollary of the above theorem is that all observable behaviors of MV are generated by the algorithm of Figure 1, which can thus be used to check whether LO (MV ) L(P ). We can also prove that the optimization for constraint nodes performed by Step 2.c of the algorithm of Figure 2 is optimal in the following sense. THEOREM 3.5. Let be defined as in Theorem 3.4. Let s be any reachable state s in MV such that is a constraint node. For any given set fva ja 2 Eg of data values associated to input events enabled when the node is entered, if n w exists exactly one transition n w PROOF. (Sketch) By contradiction. Assume that there exists two transitions n w This implies that there exist two predicates i and j of n that are both violated by the set fva ja 2 Eg of data values, whose signatures are not included in each other, and such that target( i transitions are labeled by w, we have wg. Therefore, by the condition of Step 2.c.iii.A of the algorithm of Figure 2, i and j cannot be both be added to the set Marked, and hence the algorithm cannot visit node n 0 twice by executing from s two different sequences of events w 0 and w 00 such that w 0 jO jO 4. IMPLEMENTATION ISSUES To automatically and systematically test Sisl programs for violation of safety properties using the algorithms presented in the previous section, we have integrated VeriSoft and Sisl. VeriSoft is a tool for systematically exploring the state spaces of systems composed of several concurrent processes executing arbitrary code written in any language. The state space of a system is a directed graph that represents the combined behavior of all the components of the system. Paths in this graph correspond to sequences of operations (scenarios) that can be observed during executions of the system. VeriSoft systematically explores the state space of a system by controlling and observing the execution of all the components, and by reinitializing their executions. VeriSoft drives the execution of the whole system by intercepting, suspending and resuming the execution of specific operations (system calls) executed by the implementation being tested. Examples of operations intercepted by VeriSoft are operations on communication objects (e.g., sending or receiving a message), and the VS toss(n) operation mentioned earlier, which simulates nondeterminism and introduces a branching point with n+ 1 branches in the state space whenever it is executed. VeriSoft can always guarantee a complete coverage of the state space up to some depth; in other words, all possible executions of the system up to that depth are guaranteed to be covered. Since VeriSoft can typically generate, execute and evaluate thousands of tests per minute, it can quickly reveal behaviors that are virtually impossible to detect using conventional testing techniques. More details about the state-space exploration techniques used by VeriSoft are given in [11]. VeriSoft has been applied successfully for analyzing several software products developed in Lucent Technologies, such as telephone-switching applications and implementations of network protocols (e.g., see [12]). In order to use VeriSoft for controlling the execution of the non-deterministic algorithms from Section 3, we have built a "VeriSoft interface" to Sisl. This interface provides the necessary information requested by the algorithms of the previous section (such as the current set of enabled input events, etc. These algorithms were implemented in a straightforward manner in Java. Calls to the external operation VS toss were performed using the Java Native Interface. VeriSoft can then control the execution of the resulting single process formed by the combination of the Sisl application being tested and its nondeterministic test driver, by intercepting calls to VS toss and providing the value returned by these calls, and by creating and destroying the Java Virtual Machine to reinitialize the program. For testing of safety properties, we used the specification-based testing package of Triveni [6], a framework for event-driven concurrent programming in Java. This implementation uses a standard algorithm [20] to translate a given safety formula in propositional linear-time temporal logic formulas into a finite-state automaton whose language is the set of finite event sequences that violate the formula. 5. EXAMPLE AND EXPERIMENTS Consider Table 1 which describes an interactive banking service called the Teller. To motivate the Sisl implementation of this service, we describe the transfer of funds in more detail. The transfer capability establishes a number of constraints among the three input events (source account, target account, and transfer amount) required to make a the specified source and target accounts both must be valid accounts for the previously given (login,PIN) the dollar amount must be greater than zero and less than or equal to the balance of the source account; it must be possible to transfer money between the source and target accounts. These constraints capture the minimum requirements on the input for a transfer transaction to proceed. Perhaps more important is what these constraints do not specify: for example, they do not specify an ordering on the three inputs, or what to do if a user has repeatedly entered incorrect information. Figure 3 depicts the Sisl service logic for the Teller, specified as a reactive constraint graph. It uses constraint nodes to describe the requirements on the transfer capability, deposit capability, and withdrawal capability. In particular, there is a constraint node for each transaction type. In order to enter the service, the user must first provide a startService event (e.g. dialing into the service or going to the web page); this is not depicted in Figure 3. The user then must successfully log in with a valid login and pin combi- nation. Since the login and pin may be provided in either order, a constraint node expresses this requirement. For expository simplic- ity, we assume that the login and pin must be identical for the login to be successful. After the user has successfully logged in, the service provides a choice among the different transaction types. If a startTransfer event is provided, for example, control flows to the transfer constraint node. The user is then prompted for a source ac- count, target account and amount, in any order. If the user provides a source account and an amount which is greater than the balance of the source account, for example, the constraint amt <= Bal- ance(src) will be violated. Since no explicit failure target nodes have been specified, control flow will remain in the current node. If the user provides consistent information about both accounts and the amount, the transfer will be performed and control reverts back to the choice node on transaction types. The self-loop annotated with "!has Quit" on the choice node indicates that the the subgraph from this node will be repeatedly executed until the precondition becomes false (i.e. the user has quit the service). Some temporal properties of interest for the Teller include: The service can accept a source account only if the current transaction type is either a withdrawal or transfer. The service can accept a target account only if the current transaction type is either a deposit or transfer. The service can begin a deposit transaction only if the user has given a login and pin in the past and has not quit the service. Each property described above in English has an equivalent formula in propositional linear-time temporal logic. In our terminol- ogy, the set of observable events when analyzing each property is the set of events that occur in the corresponding formula. For ex- ample, the set of observable events when analying the first property is fsrc,startWithdrawal, startDeposit, startTransferg; in particular, the reference to the "last" transaction type necessitates the startDe- posit event to occur in the corresponding formula. In our implementation, the two valid accounts are checking and savings, and transfers are permitted only between accounts of different i.e. between checking and savings, and vice-versa. Money market accounts are not considered to be valid accounts in this example. Initially, the balance on all accounts is zero, and the hasQuit variable is set to false. Our implementation of this portion of the Teller consists of approximately 200 lines of text in a mark-up language, which is automatically translated by the Sisl toolset into approximately 500 lines of Java code. It currently has applet, HTML, automatic speech recognition, and VoiceXML [21] interfaces, each about 300 lines, all sharing the same service logic. 5.1 Results of Experiments To evaluate our approach and compare the efficiency of the algorithms presented in Section 3, we performed systematic state-space explorations on the Teller service. The Teller is an interactive banking service. The service is login protected; the customer must authenticate themselves by entering an identifier (login) and PIN (password) to access the functions. As customers may have many money accounts, most functions require the customer to select the account(s) involved. Once authenticated, the customer may: Make a deposit. Make a withdrawal. The service makes sure the customer has enough money in the account, then withdraws the specified amount. Transfer funds between accounts. The service prompts the customer to select a source and target account, and a transfer amount, and performs the transfer if (1) the customer has enough money in the source account, and (2) transfers are permitted between the two accounts. Quit the service. Table 1: A high-level description of the Teller. login==pin isValid(src) amt <= Balance(src) Do the withdrawal and notify Do the deposit and notify isValid(src) amt <= Balance(src) Do the transfer and notify startWithdrawal startDeposit startTransfer Choice Node hasQuit:=false userQuit Figure 3: The Reactive Constraint Graph for the Teller We first selected the following data to be associated to each event: the name and pin events each were assigned two names from the same set fJohn, Maryg, the src, tgt, amt events each were assigned three types from the set fchecking, savings, moneymarketg, and the amt event was assigned values from the set f0; 100g. For the analysis, we first tested the following temporal prop- erty: the service can accept a target account only if the current transaction type is a deposit. The set of observable events of this property is ftgt, startDeposit, startWithdrawal, startTransferg. As expected, both algorithms reported a violation trace in which the current transaction type is a transfer and a target account was accepted We then ran a set of experiments in which the set of observable events was empty, in order to measure the efficiency of the algo- rithms. This actually tests that there were no uncaught run-time exceptions in the Sisl (Java) program along all paths up to a certain depth, as measured in the number of events sent to the service logic. We ran these tests in succession, each time increasing the maximum depth of the paths to be explored. Our results are summarized in the Figure 4. The plot on the left depicts the number of paths explored by the algorithms against the maximum depth of paths to be explored, while the plot on the right depicts the running time of the algorithms in seconds against the maximum depth of paths to be explored. Some interesting observations can be made about the experimental data. First, as expected, the running times of both algorithms are proportional to the number of paths explored. Second, consider the rate at which the number of paths and running times increase with respect to the maximum path depth; this rate is significantly less for the optimized algorithm. An important observation about the experimental data is that for maximum depths of 5 and 6, the optimized algorithm explores the same number of paths and hence has the same running time! This phenomenon occurs because the optimized algorithm performs the bulk of its work upon entry into a constraint node. This is especially true in the case of an empty set of observable events: in this case, all the work is done by the optimized algorithm upon entry into the constraint node. For example, in the Teller, paths of depth 4 consist of a startService event followed by a consistent name and pin event in either order, followed by a transaction type. At depth 5, control enters one of the transaction constraint nodes. The optimized algorithm performs all its work in choosing the event data, computing the marked predicates, and choosing a marked predicate upon entry into the node. It then merely sends the corresponding events in a fixed order. Hence the set of paths is explored is identical at depths 5 and 6, and the running time is the same (except for the extremely minor activity of sending an additional event). The similar phenomenon occurs at depths 8, 9, and 10. The results show that even for small examples such as the Teller, the optimized algorithm can provide a significant improvement in efficiency: for example, at depth 11, the simple algorithm takes over one and a half hours to complete, while the optimized algorithm takes only eleven minutes. Hence, the latter can be used to efficiently and systematically test much more complicated services simple simple optimized Figure 4: Experimental Results for Number of Paths Explored (left) and Running Time in Seconds (right) vs. Path Depth 6. CONCLUDING REMARKS 6.1 Sisl Applications We are currently using Sisl in several projects involving multi-modal interactive services. For example, Sisl is being used to prototype a new generation of call processing services for a Lucent Technologies switching product. As part of this research/development collaboration, we are developing some call processing features that may form the basis of new product offerings. The service logics for these features are expected to be quite complex, and need to be tested thoroughly. We are planning to use the techniques and tools presented in this paper to test these applications. We are also planning to test two other Sisl applications being developed at Lucent Technologies: an interactive service based on a system for visual exploration and analysis of data, and some collaborative applications in which users may interact with the system through a rich collection of devices. 6.2 Related Work We conclude with a comparison of our approach with other related work. Combining an open reactive system with its most general environment is related to the idea of "hiding" a set of visible actions of a process in a process calculus [13, 17]. Closing automatically open reactive (event-driven) programs for systematic testing (model-checking) purposes has been studied in [5, 8]. For sequential (data-driven) programs, numerous algorithms have also been proposed to automatically generate a set of input data that is sufficient to exercise and test all the possible paths in the control-flow graph of a program, for instance. This previous work makes extensive use of static analysis techniques (e.g., [7, 18, 2]), which automatically extract information about the dynamic behavior of a sequential program by examining its text. In contrast, the algorithms presented here dynamically detects at run-time the set of input events that the application under test is currently ready to accept, and uses that information to drive its execution, without using any static analysis techniques. This makes our algorithms directly applicable to any host language (Java, Perl scripts, etc.) and environment (including external databases, etc. The observation exploited by our second algorithm, namely that interleavings of input events at a constrained node have sometimes the same effect on the overall behavior of the system, is somewhat similar to the intuition behind partial-order reduction algorithms used in model-checking to prune the state spaces of concurrent systems (e.g., see [10]). A major difference is that these algorithms exploit a notion of "independence" (commutativity) on actions executed by interacting concurrent processes. In contrast, constraint-based programs are purely sequential. The reduction we obtain here is derived directly from the structure of the program and takes advantage of a form of symmetry induced by constraint nodes and their ability to specify sets of input events rather than single events. Another example of programming language construct inducing symmetry that can be exploited during verification (sys- tematic testing) is the "scalarset" [14]. 7. --R Development principles for dialog-based interfaces Compilers: Principles Recognizing safety and liveness. Sisl: Several interfaces Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints. Model Checking for Programming Languages using VeriSoft. Model Checking Without a Model: An Analysis of the Heart-Beat Monitor of a Telephone Switch using VeriSoft Communicating Sequential Processes. Better verification through symmetry. A speech interface for forms on WWW. Constraint logic programming. Communication and Concurrency. Program Flow Analysis: Theory and Applications. Concurrent Constraint Programming. An automata-theoretic approach to automatic program verification An event driven model for dialogue systems. --TR Communicating sequential processes Compilers: principles, techniques, and tools Constraint logic programming Communication and concurrency Concurrent constraint programming Model checking for programming languages using VeriSoft Model checking without a model Automatically closing open reactive programs Filter-based model checking of partial systems Abstract interpretation Partial-Order Methods for the Verification of Concurrent Systems Program Flow Analysis Better Verification Through Symmetry Design and Implementation of Triveni --CTR Jean Berstel , Stefano Crespi Reghizzi , Gilles Roussel , Pierluigi San Pietro, A scalable formal method for design and automatic checking of user interfaces, Proceedings of the 23rd International Conference on Software Engineering, p.453-462, May 12-19, 2001, Toronto, Ontario, Canada Jean Berstel , Stefano Crespi Reghizzi , Gilles Roussel , Pierluigi San Pietro, A scalable formal method for design and automatic checking of user interfaces, ACM Transactions on Software Engineering and Methodology (TOSEM), v.14 n.2, p.124-167, April 2005 Patrice Godefroid, Software Model Checking: The VeriSoft Approach, Formal Methods in System Design, v.26 n.2, p.77-101, March 2005

model checking;interactive services;testing;state-space reduction;constraint-based languages;state explosion;verification

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Interpreting Stale Load Information.

AbstractIn this paper, we examine the problem of balancing load in a large-scale distributed system when information about server loads may be stale. It is well-known that sending each request to the machine with the apparent lowest load can behave badly in such systems, yet this technique is common in practice. Other systems use round-robin or random selection algorithms that entirely ignore load information or that only use a small subset of the load information. Rather than risk extremely bad performance on one hand or ignore the chance to use load information to improve performance on the other, we develop strategies that interpret load information based on its age. Through simulation, we examine several simple algorithms that use such load interpretation strategies under a range of workloads. Our experiments suggest that by properly interpreting load information, systems can: 1) match the performance of the most aggressive algorithms when load information is fresh relative to the job arrival rate, 2) outperform the best of the other algorithms we examine by as much as percent when information is moderately old, significantly outperform random load distribution when information is older still, and 4) avoid pathological behavior even when information is extremely old.

Introduction When balancing load in a distributed system, it is well known that the strategy of sending each request to the least-loaded machine can behave badly if load information is old [11, 18, 21]. In such systems a "herd effect" often develops, and machines that appear to be underutilized quickly become overloaded because everyone sends their requests to those machines until new load information is propagated. To combat this problem, some systems This work was supported in part by an NSF CISE grant (CDA-9624082) and grants Novell and Sun. Dahlin was also supported by an NSF CAREER grant (9733842). adopt randomized strategies that ignore load information or that only use a small subset of load information, but these systems may give up the opportunity to avoid heavily loaded machines. Load balancing with stale information is becoming an increasingly important problem for distributed operating systems. Many recent experimental operating systems have included process migration facilities [2, 6, 9, 16, 17, 23, 24, 25, 26, 30] and it is now common for workstation clusters to include production load sharing programs such as LSF [31] or DQS [10]. In addition, many network DNS servers, routers, and switches include the ability to multiplex incoming requests among equivalent servers [1, 5, 8], and several run-time systems for distributed parallel computing on clusters or metacomputers include modules to balance requests among nodes [12, 14]. Server load may also be combined with locality information for wide area network (WAN) information systems such as selecting an HTTP server or cache [13, 22, 28]. As such systems include larger numbers of nodes or the distance between nodes increases, it becomes more expensive to distribute up-to-date load information. Thus, it is important for such systems to make the best use of old information. This paper attempts to systematically develop algorithms for using old information. The core idea is to use not only each server's last reported load information (L i ), but also to use the age of that information (T ) and an estimate of the rate at which new jobs arrive to change that information (-). For example, under a periodic update model of load information [21] that updates server load information every T seconds, clients using our algorithm calculate the fraction of requests they should send to each server in order to equalize the load across servers by the end of the epoch. Then, for each new request during an epoch, clients randomly choose a server according to these probabilities. In this paper, we devise load interpretation (LI) algorithms by analyzing the relevant queuing systems. We then evaluate these algorithms via simulation under a range of load information models and workloads. For our LI algorithms, if load information is fresh (e.g., T or - or both are small), then the algorithms tend to send requests to machines that recently reported low load, and the algorithms match the performance of aggressive algorithms while exceeding the performance of algorithms that use random subsets of load information or pure random algorithms that use no load information at all. Conversely, if load information is stale, the LI algorithms tend to distribute jobs uniformly across servers and thus perform as well as randomized algorithms and dramatically better than algorithms that naively use load information. Finally, for load information of modest age, the LI algorithms outperform current alternatives by as much 60%. Other algorithms that attempt to cope with stale load information, such as those proposed by Mitzenmacher [21], have the added benefit that by restricting the amount of load information that clients may consider when dispatching jobs, they may reduce the amount of load information that must be sent across the network. We examine variations of the LI algorithms that base their decisions on similarly reduced information. We conclude that even with severely restricted information, the algorithms that use LI can outperform those that do not. Furthermore, modest amounts of load information allow the LI algorithms to achieve nearly their full performance. Thus, LI decouples the question of how much load information should be used from the question of how to interpret that information. The primary disadvantage of our approach is that it requires clients to estimate or be told the job arrival rate (-) and the age of load information (T ). If this information is not avail- able, or if clients misestimate these values, our algorithms can have poor performance. We note, however, that although other algorithms that make use of stale load information do not explicitly track these factors, those algorithm do implicitly assume that these parameters fall within the range of values for which load information can be considered "fresh;" if the parameters fall outside of this range, those algorithms can perform quite badly. Con- versely, because our algorithms explicitly include these parameters, they gracefully degrade as information becomes relatively more stale. The rest of this paper proceeds as follows. Section 2 describes related work with a particular emphasis on Mitzenmacher's recent study [21], on which we base much of our methodology and several of our system models. Section 3 introduces our models of old information and Section 4 describes the load interpretation algorithms we use. Section 5 contains our experimental evaluation of the algorithms, and Section 6 summarizes our conclusions. Related work Awerbuch et. al [3] examined load balancing with very limited information. However, their model differs considerably from ours. In particular, they focus on the task of selecting a good server for a job when other jobs are placed by an adversary. In our model, jobs are placed by entities that act in their own best interest but that do not seek to interfere with one another. This difference allows us to more aggressively use past information to predict the future. A number of theoretical studies [4, 7, 15, 20, 27] have suggested that load balancing algorithms can often be quite effective even if the amount of information examined is severely restricted. We explore how to combine this idea with LI in Section 5.6. Several studies have examined the behavior of load balancing with old or limited information in queuing studies. Eager et. al [11] found that simple strategies using limited information worked well. Mirchandaney et. al [18, 19] found that as delay increases, random assignment performs as well as strategies that use load information. Several system have used the heuristic of weighing recent information more heavily than old information. For example, the Smart Clients prototype [29] distributed network requests across a group of servers using such a heuristic. Additionally, a common technique in process migration facilities is to use an exponentially decaying average for to estimate load on a machine (e.g., Load new = Load old k current 1). Unfortunately, the algorithms used by these systems are somewhat ad hoc and it is not clear under what circumstances to use these algorithms or how to set their constants. A goal of our study is to construct a systematic framework for using old load information. Our study most closely resembles Mitzenmacher's work [21]. Mitzenmacher examined a system in which arriving jobs are sent to one of several servers based on stale information about the servers' loads. The goal in such a system is to minimize average response time. He examined a family of algorithms that make each server choice from small random subsets of the servers to avoid the "herd effect" that can cause systems to exhibit poor behavior when clients chase the apparently least loaded server. Under Mitzenmacher's algorithm, if there are n servers, instead of sending a request to the least loaded of the n servers, a client randomly selects a subset of size k of the servers, and sends its request to the least loaded server from that subset. Note that when this algorithm is equivalent to uniform random selection without load information and that when it is equivalent to sending each request to the apparently least loaded server. In addition to the formulating these k-subset algorithms as a solution to this problem, Mitzenmacher uses a fluid limit approach to develop analytic models for these systems for the case when (n ! 1); however, the primary results in the study come from simulating the queuing systems, and we follow a similar simulation methodology here. Mitzenmacher concludes that the version of the algorithm is a good choice in most situations. He finds that it seldom performs significantly worse and generally performs significantly better than the more aggressive algorithms (e.g., or even the modestly aggressive outperforms the uniform random algorithm for a wide range of update frequencies. We believe, however, that this approach still has drawbacks. In particular, we note that as -the update frequency of load information-changes, the optimal value of k also changes. For example, under Mitzenmacher's periodic update model and one sample workload he examines, quickly becomes much better than larger values of T . Similarly, although outperforms for such a workload, the reverse is true for larger values of T . For example, when algorithm is a factor of 2 better than the variation. We also note that under Mitzenmacher's algorithms, the resulting arrival rate at a server depends only on the server's rank in the sorted list of server loads, not on the magnitude of difference in the queue lengths between servers. Furthermore, the least loaded servers receive no requests at all during a phase. More generally, if servers are ordered by load, with lowest load and s n\Gamma1 the highest, a given request will be sent to server s i if and only if (1) servers s 0 through s i\Gamma1 are not chosen as part of the random subset of k servers and (2) server s i is chosen as part of that subset. Because the probability that any server s j is chosen as part of the k-server subset is k , the probability that conditions (1) and (2) hold Fraction of Requests Server Rank Figure 1: Distribution of requests to servers under the k-subset algorithm. (The numerator is the number of ways to choose place them in the slots from slot the denominator is the number of ways to choose k elements from n elements assuming that element s i is always chosen.) Figure 1 illustrates the resulting distributions for a range of k's. These distributions have something of right flavor-more heavily loaded nodes get fewer requests than more lightly loaded nodes. However, it is not obvious that the slope of any one of the lines is, in general, right. The figure also illustrates why large values of k are inappropriate when T is large: a large fraction of requests are concentrated on a small number of servers for a long period of time. 3 Models of old information There are several reasonable ways to model a delay from when load information is sampled to when a decision is made to when the job under consideration arrives at its server. Different models will be appropriate for different practical systems, and Mitzenmacher found significant differences in system behavior among models [21]. We therefore examine performance under three models so that we can understand our results under a wide range of situations and so that we can compare our results to directly to those in the literature. We take the first two models, periodic update and continuous update, from Mitzenmacher's study. 1 Our third model, update-on-access, abstracts some additional systems of practical interest. We describe these models in more detail below. 3.1 Periodic and continuous update Mitzenmacher's periodic update and continuous update models can be visualized as variations of a bulletin board model. Under the periodic update model, we imagine that every T seconds a bulletin board that is visible to all arriving jobs is updated to reflect the current load of all servers. The period between bulletin board updates is termed a phase. Load information will thus be accurate at the beginning of a phase and may grow increasingly inaccurate as the phase progresses. Under the continuous update model, the bulletin board is constantly updated with load information, but on average the board state is T seconds behind the true system state. Each request thus bases its decisions on the state of the system on average T seconds earlier. Mitzenmacher finds that the probability distribution of T had a significant impact on the effectiveness of different algorithms. For a given average delay T , distributions with high variance in which some requests see newer information and others see older information outperform distributions with less variance where all jobs see data that are about T seconds old. Note that the real systems abstracted by these models would typically not include a centralized bulletin board. The periodic model could represent, for instance, a system that periodically gathers load information from all servers and then multicasts it to clients. The continuous update model could represent a system where an arriving job probes the servers for load information and then chooses a server but where there is a delay T due to network latency and transfer time from when the servers send their load information to when the client's job arrives at its destination server. 3.2 Update-on-access The final model we examine was not examined by Mitzenmacher. In our update-on-access model, we explicitly model separate clients sending requests to the servers, and different clients may have different views of the system load. In particular, when a client sends a request to a server, we assume that the server replies with a message that contains the system's current load and that snapshot of system load may be used by the client's next request. In such a system, the average load update delay, T , is equal to a client's inter-request time. Thus, the update-on-access model assumes that jobs sent by active clients will have a fresher picture of load than jobs sent by inactive clients. Mitzenmacher found a third model, individual updates to have similar behavior to the periodic update model, so we omit analysis of this model for compactness. We consider this model because it may be applicable for problems such as the server selection problem on the Internet [13, 22] where it may be prohibitively expensive to maintain load information at clients that are not actively using a service, but where it may be possible for clients to maintain good pictures of server load while they are actively using a service. Furthermore, we hypothesize that if a system exhibits bursty access patterns, it may be able to perform good load balancing even though average node's load information is, on average, quite stale. A disadvantage to using the update-on-access model is that it is more complex than the previous models. For example, under this model, results depend not only on the aggregate request rate but also on the number of clients generating a given number of requests. If there are many clients generating a certain number of requests, their load information will be on average older than if there is one client generating the same number of requests. 4 Algorithms for interpreting old information In this section we describe our algorithms for load balancing, which work by interpreting load information in the context of its age. We first describe the basic algorithm under the periodic update model and then describe a more aggressive algorithm under the same model. Finally, we describe minor variations of the algorithms to adapt them for the continuous update and update-on-access models. In general, our algorithms for interpreting load information follow two principles that distinguish them from previous algorithms. First, we consider the magnitude of imbalance between nodes, not just the nodes' ranks. Second, we modify our interpretation of information based on its age and the arrival rate of requests in the system to account for expected changes to system state over time. In the descriptions below, we use the following notation: Average age of the load information (The specific meaning of T depends on the update model.) n Number of servers - Average per-server arrival rate Reported load (queue length) on server i arrive T The number of requests expected to arrive during a phase The probability that an arriving request will be sent to server i 4.1 Algorithms for periodic update model The during a phase of length T , arrive arrive in the system. The goal of the Basic Load Interpretation (Basic LI) algorithm is to determine what fraction of those requests should be sent to each server in order to balance load (represented as server queue length) so that the sum of the jobs at the servers at the start of the phase plus the jobs that arrive during the phase are equal across all servers. 2 So, if we begin with L tot jobs at the servers and L i jobs at server i, the probability P i that we should send an arriving job to server i is arrive T if 8i( L tot +arrive T see below otherwise (1) The first term in the numerator is the number of jobs that should end up at each server to evenly divide the incoming jobs plus the current jobs. The second term in the numerator is the jobs already at server i. So the numerator is the number of jobs that should be sent to server i during this phase. The denominator is the total number of jobs that are expected to arrive during the phase. Thus the Basic LI algorithm is to send each arriving request to server i with probability P i as calculated above for the current phase. Note that if L tot +arrive T i, then the phase is too short to completely equalize the load. In that case, we want to place the arrive T requests in the least loaded buckets to even things out as well as we can. We use the following simple procedure in that case: at the start of the phase, pretend to place arrive T requests greedily and sequentially in the least loaded buckets and keep track of the number of requests placed in each bucket (tmp i ). During the phase, send each arriving request to server i with probability arrive T 4.1.1 More aggressive algorithm The above algorithm seems sub-optimal in the following sense: it tries to equalize the load across servers by the end of a phase. Thus, if the phase is long, the system may spend a long time with significantly unbalanced server load. A more aggressive algorithm might attempt to subdivide the phase and use the first part of the phase to bring all machines to an even state and then distribute requests uniformly across servers for the rest of the phase. Our aggressive algorithm works as follows: without loss of generality assume that the servers have been sorted by L i (with so that machine i is the ith least loaded server and set L sentinal value. Then, subdivide the phase into n intervals. During evenly distribute arrivals across machines to bring their 2 Notice that we make the simplifying assumption that the departure rate is the same at all servers so that we can ignore the effect of departing jobs on the relative server queue lengths. This assumption will be correct if all servers are always busy, but it will be incorrect if some servers are idle at any time during the phase. This assumption can be justified because we are primarily concerned that our algorithms work well when the system is heavily loaded, and in that case, queues will seldom be empty. The impact of this simplification is that for lightly-loaded systems, we will overestimate the queue length at lightly-loaded machines and send too few requests to them. In such a case, our probability distribution will be somewhat more uniform across servers than would be ideal, and our algorithms will not be as aggressive as they could be. Our experiments suggest that this simplification has little performance impact. loads up to L j+1 . Thus, subinterval j is of length -n , and during subinterval j, the probability that an arriving request should be sent to machine i, is: (2) 4.2 Algorithms for other update models Adapting the Basic LI algorithm to the continuous update or update-on-access model is simple. We use Equation 1 to calculate the probabilities P i for sending incoming requests to each server. The only difference is that for the periodic update model this calculation is based on the L i estimates that hold during the entire phase, but under the new models the may change with each request. P i can now be thought of as a current estimate of the instantaneous rate at which requests should be sent to each server. Adapting the Aggressive LI algorithm is more problematic. We use Equation 2 to calculate the values based on the current L i array. However, under the continuous update model, we are effectively always "at the end of a phase" in that the information is T seconds old. And, although the aggressive algorithm is more aggressive than the basic algorithm during the early subintervals of a phase (e.g., j near 0), it is less aggressive during later subintervals (e.g., j near n). Thus, the "aggressive" algorithm may actually be less aggressive than the basic algorithm under these update models when T is large. 5 Evaluation In this section we evaluate the algorithms under a range of update models and workloads. Our primary methodology is to simulate the queuing systems. We model task arrivals as a Poisson stream of rate -n for a collection of n servers. When a task arrives, we send it to one of the server queues based on the algorithm under study. Server queues follow a first-in-first-out discipline. We select default system parameters to match those used in Mitzenmacher's study [21] to facilitate direct comparison of the algorithms. In particular, unless otherwise noted, and we assume that each server has a service rate of 1 and the service time for each task is exponentially distributed with a mean time of 1. We initially examine the Basic LI and Aggressive LI algorithms under the periodic update, continuous update, and update-on-access models and compare their performance to the k-subset algorithms examined by Mitzenmacher. We then explore three key questions for the LI algorithms: (1) What is the impact of bursty arrival patterns? (2) What is the impact of misestimating the system arrival rate? (3) What is the impact of limiting the amount of load information available to the algorithms? Basic LI Aggressive LI Figure 2: Service time v. update delay for periodic update model 5.1 Periodic update model Figure shows system performance under the periodic update model for the default param- eters. The performance of the LI algorithms is good over a wide range of update intervals. When T is large, the LI algorithms do not suffer the poor performance that the k-subset algorithms encounter for any k ? 1. In fact, the LI algorithms maintain a measurable advantage over the oblivious random algorithm even for large values of T . For example, when outperforms the oblivious algorithm by 9% and Aggressive LI outperforms the oblivious algorithm by 22%. For more modest values of T , the advantages are larger. For example, at Aggressive LI is 60% faster than any of the k-subset algorithms and Basic LI is 41% faster than any of the k-subset algorithms. Figure 3 details the performance of the algorithms for small values of T . Aggressive LI outperforms all other algorithms by at least a few percent down to the smallest value of T we examined Basic LI is generally better than and always at least as good as any k-subset algorithm over this range of T . Figure 4 shows the performance of the system under a workload with a lighter load than our default. When load is lighter, the need for load balancing is less pronounced and the gains by any algorithm over random are more modest. When information is fresh, the algorithms can perform up to a factor of two better than the oblivious algorithm. When information is stale, the performance of the k-subset algorithms is not nearly as bad as it was for the heavier load, although they do exhibit poor behavior compared to the oblivious algorithm for large T . Over the entire range of staleness examined (0:1 - T - 200), the Basic LI and Aggressive LI algorithms perform as well as or better than the best k-subset or oblivious algorithm. Basic LI Aggressive LI Figure 3: Detail: service time v. update delay for periodic update model Average Response Time Update Interval (T) Basic LI Aggressive LI Figure 4: Service time v. update delay for periodic update model Basic LI Aggressive LI Figure 5: Service time v. update delay for periodic update model Figure 5 shows the performance of the system with servers rather than the standard 100. The results are qualitatively similar to the results for the standard 5.2 Continuous update model Figure 6 shows the performance of the algorithms under the continuous update model. Because system behavior depends on the distribution of the delay parameter, we show results for four distributions of delay, all with average value T . In order of increasing variation, they are: constant(T ), uniform( Tto 3T), uniform(0 to 2T ), and exponential(T ). As the earlier discussion suggests, the Aggressive LI algorithm is actually less aggressive than the Basic LI algorithm, and Basic generally outperforms aggressive for this model. We will therefore focus on the Basic LI algorithm. Mitzenmacher notes that for a given T , the k-subset algorithms' performance improves for distributions that contain a mix of more recent and older information. This relationship seems present but less pronounced for the LI algorithms. As a result, as the distribution's variability increases, the advantage of LI over the k-subset algorithms shrinks. Thus, Basic LI seems a clear choice for the constant and uniform T distributions: for any value of T , its performance is as good as any of the k-subset algorithms and for any given k-subset algorithm there is some range of T where Basic LI's performance is significantly better. For the exponential distribution of T , however, the k-subset algorithms enjoy an advantage of up to 16% over Basic LI. Figure 7 tests the hypothesis that the relatively poor performance of Basic LI in this situation is because the algorithm calculates P i using the expected value of T whereas each individual request may see significantly different values of T . In this figure, still varies according to the specified distribution, but rather than knowing the average Average Response Time Update Interval (T) Basic LI Aggressive LI (a) Constant T (b) Uniform Tto 3T01020 Average Response Time Update Interval (T) Basic LI Aggressive LI26100 50 100 150 200 Average Response Time Update Interval (T) Basic LI Aggressive LI (c) Uniform 0 to 2T (d) Exponential with mean T Figure update delay for continuous update model when clients only know T , the average delay. (a) - (d) show result for different distributions of delay around the (a) Uniform Tto 3T01020 Average Response Time Update Interval (T) Basic LI Aggressive LI26100 50 100 150 200 Average Response Time Update Interval (T) Basic LI Aggressive LI (b) Uniform 0 to 2T (c) Exponential with mean T Figure 7: Service time v. update delay for continuous update model when clients know the age of information actually encountered for each request. (a) - (c) show result for different Figure 8: Service time v. update delay for update-on-access model. value of T , each request knows the value of T that holds for that request, and the algorithm calculates its P i vector using this more certain information. Compared to the performance in Figure 6, this extra information improves performance for each distribution of T , and the improvement becomes more pronounced for distributions with more variation. From this we conclude that good estimates of the delay between when load information is gathered and when a request will arrive at a server are important for getting the best performance from the LI algorithms. 5.3 Update-on-access model Figure 8 shows performance for the update-on-access model. In this model, we simulate some number of clients, and each client uses the load information gathered after sending one request to decide where to send the next request. Thus, T equals the per-client inter-request time. To vary T for a fixed total arrival rate, we simply vary the number of clients from which the requests are issued. For this model, all of the algorithms perform reasonably well. It appears that the per-client updates desynchronize the clients enough to reduce the herd effect. The Basic LI algorithm outperforms all of the others and provides a modest speedup over a wide range of update intervals. Figure 9: Service time v. update delay for update-on-access model under bursty workload 5.4 Bursty arrivals Figure 9 shows performance under a bursty-arrival version of the update-on-access model. As with the standard update-on-access model, each client uses the server loads discovered during one request to route the next one. To generate our bursty-arrivals workload, rather than assume that each client produces exponentially-distributed arrivals, we assume that a client whose average inter-request time is T produces a burst of b requests separated by seconds, with the bursts separated by exponential(T b) seconds. For Figure 9, The bursty workload significantly increases the performance of all of the algorithms that use server load compared to the oblivious algorithm. Although over time, a client's picture of server load is on average T seconds old, an average request sees a much fresher picture of the L i vector. This suggests that it may often be possible to significantly outperform the oblivious strategy even for challenging workloads such as internet server selection [13, 22] where information will likely be old on average, but where a client's requests to a service are bursty. Once again, the Basic LI algorithm is the best or tied for the best over the entire range of T examined (0:1 - T - 200). 5.5 Impact of imprecise information The primary drawback to the LI algorithms is that they require good estimates of T and -. Subsection 5.2 examined the impact of uncertainty about T . In this subsection, we examine what happens when the estimate of - is incorrect. We believe that servers supporting the Average Response Time Update Interval (T) Basic LI (8*Load) Basic LI (4*Load) Basic LI (2*Load) Basic LI (1*Load) Basic LI (0.5*Load) Basic LI (0.25*Load) Basic LI (0.125*Load) Figure 10: Service time v. update delay for periodic update model when clients mis-estimate the arrival rate. LI algorithms would be equipped to inform clients both of their current load and of the arrival rate of requests they anticipate. For example, a server might report the arrival rate it had seen over some recent period of time, or it might report the maximum request rate it anticipates encountering. However, it may be difficult for some systems to accurately predict future request patterns based on history. Figure shows performance under the periodic update model when the LI algorithm uses an incorrect estimate of -. Each line shows performance when the - used for calculating P i is multiplied by an error factor e between 1and 8. If we overestimate the load, the algorithm is more conservative than it should be and performance suffers a bit. If we underestimate the load, the algorithm sends too many requests to the apparently-least-loaded servers, and performance is very poor. From this, we conclude that systems should err on the side of caution when estimating -. From Figure note that if - estimated = 2- actual , performance is only marginally worse than actual . Also note that for these experiments, - actual = 0:9 and the the system would be unstable if - actual - 1:0. In other words, to overestimate - by a factor of two, one would have to predict a service rate 1.8 times larger than could ever be sustained by the system. We suggest the following strategy for estimating -: if the system's maximum achievable throughput is known, use that throughput as an estimate of - for purposes of the LI algo- rithms. When the system is heavily loaded, that estimate will be only a little bit higher than the actual arrival rate; when it is lightly loaded, the estimate will be far too high. But, as we have seen, the algorithm is relatively insensitive to overestimates of arrival rate. Further- more, overestimating the arrival rate does little harm when the system is lightly loaded. In Average Response Time Arrival Rate (lambda) Basic LI (actual lambda) Basic LI (assume lambda = 1.0) Figure 11: Service time v. arrival rate (-) for periodic update model with 20. The graph compares the standard algorithms as well as a variation of the Basic LI algorithm that overestimates - as the maximum achievable system throughput that case, the conservative estimate of - tends to make the LI algorithm distribute requests uniformly across the servers, which is an acceptable strategy when load is low. Figure 11 illustrates the effect of assuming - estimated = 1:0 as we vary - actual for a system with The two Basic LI lines-one with exact and the other with conservative estimates of -are almost indistinguishable. For all points, the difference between the two results is less than 4.5%; when - 0:7, the difference is always less than 1.5%. 5.6 Impact of reduced information The k-subset algorithms have an additional purpose beyond attempting to cope with stale load information: by restricting the amount of load information that clients may consider when dispatching jobs, they may reduce the amount of load information sent across the network. A number of theoretical [4, 7, 15, 20, 27] and empirical [11, 18] studies have suggested that load balancing algorithms can often be quite effective even if the amount of information they have is severely restricted. The Basic LI algorithm can also be adapted to use a subset of server load information rather than requiring a vector of all servers' loads. In the k-subset version of the Basic LI algorithm (Basic LI-k), we select a random subset of k servers and use the algorithm to determine how to bias requests among those k nodes. In particular, we modify Equation 1 to use P 0 arrays of size k rather than n, to compute L 0 tot from the smaller L 0 i array, to replace n with k, and to calculate arrive 0 k. Note that, as for the standard k-subset algorithms, we select a different subset for each incoming request. Average Response Time Update Interval (T) Basic LI (k=2) Basic LI (k=3) Basic LI (k=100) (a)26100 50 100 150 200 Average Response Time Update Interval (T) Basic LI (k=2) Basic LI (k=3) Basic LI (k=10) Basic LI (k=100) (b) Figure 12: Service time v. update delay for k-subset version of Basic LI algorithm for (a) update-on-access model and (b) continuous update with fixed delay model. Figure 12 examines the impact of restricting the information available to the Basic LI al- gorithm. This experiment suggests that the Basic LI-k algorithm can achieve good perfor- mance. Under the update-on-access model, the original k-subset algorithms perform well, and the LI-2 algorithm's performance is similar to that of the standard algorithms. Unlike the standard k-subset algorithms, as the LI-k algorithm is given more information, its performance becomes better. The LI-3 algorithm outperforms the all of the standard k-subset algorithms by a noticeable amount, and the full Basic LI algorithm widens the margin. Under the continuous update with fixed delay model (Figure 12-b), the performance of the LI-k algorithms is also good. In this case, the original k-subset algorithms behave badly, but the LI-k versions behave nearly identically with the Basic LI system. In this experiment, the versions of the LI algorithm are slightly better than the version, with smaller k consistently giving slightly better performance. We do not have an explanation for the improving behavior with reduced information in this experiment. From these experiments, we conclude that LI can be an effective technique in environments where we wish to restrict how much load information is distributed through the system. Modest amounts of load information allow the LI algorithms to achieve nearly their full performance. Thus, LI decouples the question of how much load information should be used from the question of how to interpret that information. 6 Conclusions The primary contribution of this paper is to present a simple strategy for interpreting stale load information. This approach resolves the paradox that under some algorithms, using additional information often results in worse performance than using less information or none at all. The Load Interpretation (LI) strategy we propose interprets load information based on its age so that a system is essentially always better off when it has and uses more information. When information is fresh, the algorithm aggressively addresses imbalances; when the information is stale, the algorithm is more conservative. We believe that this approach may open the door to safely using load information to attempt to outperform random request distribution in environments where it is difficult to maintain fresh information or where the system designer does not know the age of the information a priori. Our experiments suggest that by interpreting load information, systems can (1) match the performance of the most aggressive algorithms when load information is fresh, (2) outperform current algorithms by as much as 60% when information is moderately old, (3) significantly outperform random load distribution when information is older still, and (4) avoid pathological behavior even when information is extremely old. --R The Next Step in Server Load Balancing. Designing a Process Migration Facility: The Charlotte Experience. Making Commitments in the Face of Uncertainty: How to Pick a Winner Almost Every Time. Balanced Allocations. DNS Support for Load Balancing. Network Tasking in the Locus Distributed Unix System. Towards Developing Universal Dynamic Mapping Algorithms. Cisco Distributed Director. Transparent Process Migration: Design Alternatives and the Sprite Implementation. Research Toward a Heterogeneous Networked Computing Cluster: The Distributed Queuing System Version 3.0. Adaptive Load Sharing in Homogeneous Distributed Systems. Locating Nearby Copies of Replicated Internet Servers. Load distribution: Implementation for the Mach Microkernel. Analysis of the Effects of Delays on Load Sharing. Adaptive Load Sharing in Heterogeneous Distributed Systems. The Power of Two Choices in Randomized Load Balancing. How Useful is Old Information. Performance Characteristics of Mirror Servers on the Internet. Using Idle Workstations in a Shared Computing Environment. Process Migration in DEMOS/MP. Experiences with the Amoeba Distributed Operating System. Queuing Systems with Selection of the Shortest of Two Queues: an Asymptotic Approach. Squid Internet Object Cache. Using Smart Clients to Build Scalable Services. Attacking the Process Migration Bottleneck. A Load Sharing Facility for Large --TR --CTR Michael Rabinovich , Zhen Xiao , Amit Aggarwal, Computing on the edge: a platform for replicating internet applications, Web content caching and distribution: proceedings of the 8th international workshop, Kluwer Academic Publishers, Norwell, MA, 2004 Mauro Andreolini , Michele Colajanni , Riccardo Lancellotti , Francesca Mazzoni, Fine grain performance evaluation of e-commerce sites, ACM SIGMETRICS Performance Evaluation Review, v.32 n.3, December 2004 Simon Fischer , Berthold Vcking, Adaptive routing with stale information, Proceedings of the twenty-fourth annual ACM symposium on Principles of distributed computing, July 17-20, 2005, Las Vegas, NV, USA Request Redirection Algorithms for Distributed Web Systems, IEEE Transactions on Parallel and Distributed Systems, v.14 n.4, p.355-368, April Giovanni Aloisio , Massimo Cafaro , Euro Blasi , Italo Epicoco, The Grid Resource Broker, a ubiquitous grid computing framework, Scientific Programming, v.10 n.2, p.113-119, April 2002 Mauro Andreolini , Michele Colajanni , Ruggero Morselli, Performance study of dispatching algorithms in multi-tier web architectures, ACM SIGMETRICS Performance Evaluation Review, v.30 n.2, September 2002 Suman Nath , Phillip B. Gibbons , Srinivasan Seshan, Adaptive data placement for wide-area sensing services, Proceedings of the 4th conference on USENIX Conference on File and Storage Technologies, p.4-4, December 13-16, 2005, San Francisco, CA Bogumil Zieba , Marten van Sinderen , Maarten Wegdam, Quality-constrained routing in publish/subscribe systems, Proceedings of the 3rd international workshop on Middleware for pervasive and ad-hoc computing, p.1-8, November 28-December 02, 2005, Grenoble, France Mauro Andreolini , Sara Casolari, Load prediction models in web-based systems, Proceedings of the 1st international conference on Performance evaluation methodolgies and tools, October 11-13, 2006, Pisa, Italy Yu, The state of the art in locally distributed Web-server systems, ACM Computing Surveys (CSUR), v.34 n.2, p.263-311, June 2002

server selection;load balancing;distributed systems;queuing theory;stale information

355190

A Protocol-Centric Approach to on-the-Fly Race Detection.

AbstractWe present the design and evaluation of a new data-race-detection technique. Our technique executes at runtime rather than post-mortem, and handles unmodified shared-memory applications that run on top of CVM, a software distributed shared memory system. We do not assume explicit associations between synchronization and shared data, and require neither compiler support nor program source. Instead, we use a binary code re-writer to instrument instructions that may access shared memory. The most novel aspect of our system is that we are able to use information from the underlying memory system implementation in order to reduce the number of comparisons made at runtime. We present an experimental evaluation of our techniques by using our system to look for data races in five common shared-memory programs. We quantify the effect of several optimizations to the basic technique: data flow analysis, instrumentation batching, runtime code modification, and instrumentation inlining. Our system correctly found races in three of the five programs, including two from a standard benchmark suite. The slowdown of this debugging technique averages less than 2.5 for our applications.

Introduction Despite the potential savings in time and effort, data-race detection techniques are not yet an accepted tool of builders of parallel and distributed systems. Part of the problem is surely the restricted domain in which most such mechanisms operate, i.e., parallelizing compilers. Compiler support is usually deemed necessary because race-detection is generally NP-complete [19]. This paper presents the design and evaluation of an on-the-fly race-detection technique for explicitly parallel shared-memory applications. This technique is applicable to shared memory programs written for the lazy-release-consistent (LRC) [11] (see Section 3.1) memory model. Our work differs from previous work [3, 4, 7, 9, 18, 17] in that data-race detection is performed both on-the-fly and without compiler support. In common with other dynamic systems, we address only the problem of detecting data races that occur in a given execution, not the more general problem of detecting all races allowed by program semantics [19, 25]. Earlier work [21] introduced this approach by demonstrating its use on a less complex single-writer protocol. This paper extends this earlier work through the use of a more advanced multi-writer protocol, and through a series of optimizations to the basic technique. We find data races by running applications on a modified version of the Coherent Virtual Memory (CVM) [13, 14] software distributed shared memory (DSM) system. DSMs support the abstraction of shared memory for parallel applications running on CPUs connected by general-purpose interconnects, such as networks of workstations or distributed memory machines like the IBM SP-2. The key intuition of this work is the following: LRC implementations already maintain enough ordering information to make a constant-time determination of whether any two accesses are concurrent. In addition to the LRC information, we track individual shared accesses through binary instrumentation, and run a simple race-detection algorithm at existing global synchronization points. This last task is made much easier precisely because of the synchronization ordering information maintained by LRC. The system can automatically generate global synchronization points in long-running applications if there are none originally. We used this technique to check for data races in implementations of five common parallel applications. Our system correctly found races in three. Water-Nsquared and Spatial, from the Splash2 [27] benchmark suite, had data races that constituted real bugs. These bugs have been reported to the Splash authors and fixed in their current version. While the races could affect correctness, they were unlikely to occur on the platforms for which SPLASH was originally intended. Barnes, on the other hand, had been modified locally in order to eliminate unnecessary synchronizations. The races introduced by these modifications did not affect the correctness of the application. Since overhead is still potentially exponential, we describe a variety of techniques that greatly reduce the number of comparisons that need to be made. Those portions of the race-detection procedure that have the largest theoretical complexity turn out to be only the third or fourth-most expensive component of the overall overhead. Specifically, we show that i) we can statically eliminate over 99% of all load and store instructions as potential race participants, ii) we eliminate over 80% potential comparisons at runtime through use of LRC ordering information, and iii) the average slowdown from use of our techniques is currently less than 2.8 on our applications, and could be reduced even further with support for inlining of instrumentation code. While this overhead is still too high for the system to be used all of the time, it is low enough for use when traditional debugging techniques are insufficient, or even to be a part of standard debugging toolbox for parallel programs. Problem Definition We paraphrase Adve's [1] terminology to define the data races detected by our system. We define the happened-before-1 partial order, denoted hb1 !, over shared accesses and synchronization acquires and releases as follows: 1. If a and b are ordinary shared memory accesses, releases, or acquires on the same processor, and a occurs before b in program order, then a b. 2. If a is a release on processor p 1 , and b is the corresponding acquire on processor p 2 , then a b. For a lock, an acquire corresponds to a release if there are no intervening acquires or releases of that lock. For a barrier, an acquire corresponds to a release if the acquire is a departure from and the release is an arrival to the same instance of the barrier. 3. If a hb1 c, then a hb1 Given Definition 1, we define data races as follows: Shared accesses a and b constitute a data race if and only if: 1. a and b both access the same word of shared data, and at least one is a write, and 2. Neither a hb1 ! a is true. approximates the notion of actual data races defined by Netzer [20]. In common with most other implemented systems, both with and without compiler support, we make no claim to detect all data races allowed by the semantics of the program (the feasible races discussed by Netzer [20]). As such, a program running to completion on our system without data races is not a guarantee that subsequent executions will be free of data races as well. However, we do detect all races that occur in any given execution. Figure 1 shows two possible executions of code in which processes access shared variable x and synchronize through synchronization variable L. The access pair w 1 in the execution on the left does not constitute a data race because semantics do not enforce an ordering on lock acquisitions, the execution might instead have happened as shown in 1(b). In this case, r 1 (x) is not ordered with respect to w 1 (x), and the two therefore constitute a race. Note that not all data races cause incorrect results to be generated. "Correct" results will be generated even for the execution on the right if r 1 completes before w 1 is issued. In order for the system to distinguish between the accesses in Figure 1, the system must be able to detect and understand the semantics of all synchronization used by the programs. In practice, this requirement means that programs must use only system-provided synchronization. Any synchronization implemented on top of the shared memory abstraction is invisible to the system, and could result in spurious race warnings. However, the above requirement is no stricter than that of the underlying DSM system. Programs must use system-visible synchronization in order to run on any release-consistent system. Our data-race detection system imposes no additional consistency or synchronization constraints. 3 Lazy Release Consistency and Data Races (a) (b) Figure 1. If the ordering of accesses to lock L is non-deterministic, either (a) or (b) is a possible ordering of events. The ordering given in (a) is not a data race because there is a release-acquire sequence between each pair of conflicting accesses. (b) has a race between r 1 (x) and w 1 (x). 3.1 Lazy Release Consistency Lazy release consistency [11] is a variant of eager release consistency (ERC) [8], a relaxed memory consistency that allows the effects of shared memory accesses to be delayed until selected synchronization accesses occur. Simplifying matters somewhat, shared memory accesses are labeled either as ordinary or as synchronization accesses, with the latter category further divided into acquire and release accesses. Acquires and releases may be thought of as conventional synchronization operations on a lock, but other synchronization mechanisms can be mapped on to this model as well. Essentially, ERC requires ordinary shared memory accesses to be performed before the next release by the same processor. ERC implementations can delay the effects of shared memory accesses as long as they meet this constraint. Under LRC protocols, processors further delay performing modifications remotely until subsequent acquires by other processors, and the modifications are only performed at the other processor that performed the acquire. The central intuition of LRC is that competing accesses to shared locations in correct programs will be separated by synchronization. By deferring coherence operations until synchronization is acquired, consistency information can be piggy-backed on existing synchronization messages. To do so, LRC divides the execution of each process into intervals, each identified by an interval index. Figure 2 shows an execution of two processors, each of which has two intervals. The second interval of P 1 , for example, is denoted oe 2 1 . Each time a process executes a release or an acquire, a new interval begins and the current interval index is incremented. We can relate intervals of different processes through a happens-before-1 partial ordering similar to that defined above for shared accesses: 1. intervals on a single processor are totally ordered by program order, 2. interval oe i if oe j begins with the acquire corresponding to the release that concluded interval oe i , and w(y 2 reply sync request Figure 2. Shared data x 1 and x 2 are assumed to be on the same page, and y 1 and y 2 are co-located on another page. A page-based comparison of concurrent intervals would flag concurrent interval pairs oe 1 1 and oe 2 containing possibly conflicting references to common pages. Comparison of the associated bitmaps would reveal that the former has only false sharing, while the latter has a true race in r(x 1 ) from oe 2 1 and w(x 1 ) from oe 2 . 3. the transitive closure of the above. LRC protocols append consistency information to all synchronization messages. This information consists of structures describing intervals seen by the releaser, together with enough information to reconstruct the hb1 ! ordering on all visible intervals. For example, the message granting the lock to P 2 in Figure 2 contains information about all intervals seen by P 1 at the time of the release that had not yet been seen by P 2 , i.e., oe 1 1 . The system also records the fact that oe 1hb1 . While we discuss only locks and barriers in this paper, the notion of synchronization acquires and releases can be easily mapped to other synchronization models as well. 3.2 Data-Race Detection in an LRC System Intuitively, a data race is a pair of accesses that do not have intervening synchronization, such that at least one of the accesses is a write. In Figure 2, the read of x 1 by P 1 and the write of x 1 by P 2 constitute a data race, because intervals oe 2 1 and oe 2 are concurrent (not ordered). Detecting data races generally requires comparing each shared access against every other shared access. With an LRC system, as with any other system based on hb1 partial ordering, we can limit comparisons only to accesses in pairs of concurrent intervals. For example, interval pair oe 1 2 in Figure 2 is not concurrent, and so we do not have to check further in order to determine if there is a data race formed by accesses in those intervals. We only perform word-level comparisons if we have first verified that the pages accessed by the two intervals overlap. For example, assume that y 1 and y 2 of Figure 2 reside on the same page. A comparison of pages accessed by concurrent 1 and oe 1 would reveal that they access overlapping pages, i.e. the page containing y 1 and y 2 . We would therefore need to perform a bitmap comparison in order to determine if the accesses constitute false sharing or true sharing (i.e., a data race). In this case, the answer would be false sharing because the accesses are to distinct locations. However, if P 2 's first write were to z, a variable on a completely different page, our comparison of pages accessed by the two intervals would reveal no overlap. No bitmap comparison would be performed, even though the intervals are concurrent. Implementation 4.1 System and its Changes We implemented our race-detection system on top of CVM [13, 14], a software DSM that supports multiple protocols and consistency models. Like commercially available systems such as TreadMarks [12], CVM is written entirely as a user-level library and runs on most UNIX-like systems. Unlike TreadMarks, CVM was created specifically as a platform for protocol experimentation. The system is written in C++, and opaque interfaces are strictly enforced between different functional units of the system whenever possible. The base system provides a set of classes that implement a generic protocol, lightweight threads, and network communication. The latter functionality consists of efficient, end-to-end protocols built on top of UDP. New shared memory protocols are created by deriving classes from the base Page and Protocol classes. Only those methods that differ from the base class's methods need to be defined in the derived class. The underlying system calls protocol hooks before and after page faults, synchronization, and I/O events take place. Since many of the methods are inlined, the resulting system is able to perform within a few percent of a severely optimized system, TreadMarks, running a similar protocol. CVM was also designed to take advantage of generalized synchronization interfaces, as well as to use multi-threading for latency toleration. Our detection mechanism is based on CVM's multi-writerLRC protocol. This protocol propagates modifications in the form of diffs, which are run-length encodings of modifications to a single page [12]. Diffs are created through word-by-word comparisons of the current contents of a page with a copy of the page saved before any modifications were made. We made only three modifications to the basic CVM implementation: (i) we added instrumentation to collect read and access information, (ii) we added lists of pages read (read notices) to message types that already carry analogous information about pages written, and (iii) we potentially add an extra message round at barriers in order to retrieve word-level access information, if necessary. 4.2 Instrumentation We use the ATOM [26] code-rewriter to instrument shared accesses with calls to analysis routines. ATOM allows executable binaries to be analyzed and modified. We use ATOM to identify and instrument all loads and stores that may access shared memory. Although ATOM is currently available only for DEC Alpha systems, a port is currently underway to Intel's x86 architecture. Moreover, tools that provide similar support for other architectures are becoming more common. Examples are EEL [16] (SPARC and MIPS), Shade [5] (SPARC), and Etch [22] (x86). The base instrumentation consists of a procedure call to an analysis routine that checks if the instruction accesses shared memory. If so, the routine sets bits corresponding to the access's page and position in the page in order to indicate that the page and word have been accessed. The analysis routine consists of including 10 instructions for saving registers to and restoring registers from the stack. The actual call to the analysis routine, together with instructions that save some registers outside the call, consume 7 or 8 more instructions. A small number of additional instructions are needed for the batching and runtime code modification optimizations. Information about which pages were accessed, together with the bitmaps themselves, is placed in known locations for bitmap comparison barrier arrivals barrier releases interval comparison bitmap requests bitmaps Figure 3. The barrier algorithm has the following steps: 1) read and write notices are sent to the barrier master, 2) the barrier master identifies concurrent intervals with overlapping page access lists, 3) bitmaps are requested for the overlapping pages from step 2, and comparisons are used to identify data races. Dotted lines indicate events that occur only if sharing is detected in step 2. CVM for use during the execution of the application. All data structures, including bitmaps, are statically allocated in order to reduce runtime cost. Shared memory and bitmaps are allocated at fixed memory locations in order to decrease the cost of the instrumentation code. 4.3 Algorithm The overall procedure for detecting data races, illustrated in Figure 3, is the following: 1. We use ATOM to instrument all shared loads and stores in the application binary. The problem here lies in determining which references are shared and which are not. In the base case, we simply instrument any reference that does not use the stack pointer, the global variables pointer 1 , or any register that has been loaded with one of these values in the current basic block. We also skip library references, as we know from inspection that our applications make no library calls that modify shared memory. In the absence of guarantees to the contrary, we can easily instrument non-CVM libraries as well. Such instrumentation would not have affect our slowdown because our applications spend time in libraries only during initialization. Section 4.5 describes several extensions to this basic technique that either eliminate more memory accesses as candidates for instrumentation, or decrease the cost of the resulting instrumentation. 2. Most synchronization messages in the base CVM protocol carry consistency information in the form of interval structures. Each interval structure contains one or more write notices that enumerate pages written during that interval. In CVM, we augmented these interval structures to also carry read notices, or lists of pages read during that interval. assumes that all shared data is allocated dynamically. Interval structures also contain version vectors that identify the logical time associated with the interval, and permit checks for concurrency. 3. Worker processes in any LRC system append interval structures (together with other consistency information) to barrier arrival messages. At each barrier, therefore, the barrier master has complete and current information on all intervals in the entire system. This information is sufficient for the barrier master to locally determine the set of all pairs of concurrent intervals. Although the algorithm must potentially compare the version vector of each interval of a given processor with the version vector of each interval of every other processor, exploiting synchronization and program order allows many of the comparisons to be omitted. 4. For each pair of concurrent intervals, the read and write notices are checked for overlap. A data race might exist on any page that is either written in two concurrent intervals, or read in one interval and written in the other. Such interval pairs, together with a list of overlapping pages, are placed on the check list. Steps 5 and 6 are performed only if this check list is non-empty, i.e., there is a data-race or there is false sharing (see Figure 3). 5. Barrier release messages are augmented to carry requests for bitmaps corresponding to accesses covered by the check list. Each read or write notice has a corresponding bitmap that describes precisely which words of the page were accessed during that interval. Hence, each pair of concurrent intervals has up to four bitmaps (read and write for each interval) that might be needed in order to detect races. These bitmaps are returned to the barrier master for each interval pair on the check list. 6. The barrier master compares bitmaps from overlapping pages in concurrent intervals. A single bitmap comparison is a constant time process, dependent only on page size. In the case of a read-write or write-write overlap, the algorithm has determined that a data race exists, and prints the address of the offending race. We currently use a very simple interval comparison algorithm to find pairs of concurrent intervals, primarily because the major system overhead is elsewhere. The upper bound on the number of intervals per processor pair that the comparison algorithm must compare is O(i 2 ), where i is the maximum number of intervals of a single processor since the last barrier. However, the algorithm needs only to examine intervals created during the last barrier epoch, where a barrier epoch is the interval of time between two successive barriers. By definition, these intervals are separated from intervals in previous epochs by synchronization, and are therefore ordered with respect to them. Since each interval potentially needs to be compared against every other interval of another process in the current epoch, the total comparison time per barrier is bounded by is the number of processes and i is the maximum number of intervals of any process in the current epoch. In practice, however, the number of such comparisons is usually quite small. Applications that use only barriers have one interval per process per barrier epoch. More than one interval per barrier is only created through additional peer-to-peer synchronization, such as exclusive locks. However, peer-to-peer synchronization also imposes ordering on intervals of the synchronizing processes. For example, a lock release and subsequent acquire order intervals prior to the release with respect to those subsequent to the acquire. Since an ordered pair of intervals is by definition not concurrent, the same act that creates intervals also removes many interval pairs from consideration for data races. Hence, programs with many intervals between barriers usually also have ordering constraints that reduce the number of concurrent intervals. Note that this technique does not require frequent barriers. We can accommodate long-running, barrier-free applications by forcing global synchronization to occur when system buffers are filled. 4.4 Example We illustrate the use of this technique through an example based on Figure 2. Figure 2 shows a portion of the execution of two processes, together with their synchronization and memory accesses. Only memory accesses that were not statically identified as non-shared are shown. Further, data items x 1 and x 2 are located on the same page, and y 1 and y 2 are on another. If we assume that barriers occur immediately before and after the accesses in this figure, then the events of the figure correspond to a single barrier epoch. Barrier arrival messages from P 1 and P 2 will therefore contain information about four intervals: oe 1 2 . Interval structures oe 1 2 each contain a single write notice, while oe 2 contains two read notices. The reads of x 1 and x 2 are represented by a single read notice because they are located on the same page. Upon the arrival of all processes at the second barrier, there are six possible interval pairs. We can eliminate oe 1 1 and 2 because of program order, and oe 1 2 because of synchronization order. Finally, oe 1 2 can be eliminated because these intervals access no pages in common. This leaves oe 2 2 and oe 2 2 as possible causes of races or false sharing. In the following, we use the notation oe j r(x) to refer to the read bitmap of page x during interval oe j and similar notation for writes. Barrier release messages will include requests for bitmaps oe 2 1 r(y) and oe 1 2 w(y) in order to judge the first pair, and oe 2 2 w(x) for the second pair. The comparison of oe 2 1 r(y) and oe 1 2 w(y) will only reveal false sharing the intervals access different data items that just happen to be located on the same page. By contrast, the comparison of oe 2 show that a data race exists because x 1 is accessed in both intervals, and one of the accesses is a write. 4.5 Optimizations This section describes three enhancements to our basic technique. 4.5.1 Dataflow analysis We use a limited form of iterative interprocedural register dataflow analysis in order to identify additional non-shared memory accesses. Our technique consists of creating a data-flow graph and associating incoming and outgoing sets of registers with each basic block. The registers in each set define registers known not to be pointers into shared memory. During each iteration of the analysis, the outgoing set is defined as the incoming set minus registers that are loaded in that block. Incoming sets are then redefined as the intersection of the incoming set for the previous iteration and the outgoing sets of all preceding blocks. The procedure continues until all incoming register sets stabilize. Any values left in registers of the incoming register sets are known not to be pointers into shared space. Memory accesses using such registers do not need to be instrumented. We made two main assumptions here. First, we simplify interprocedural analysis by exploiting the fact that function arguments are usually passed through registers. Tracking parameters that are not passed in this manner entails tracking the order of stack accesses in the caller and callee blocks. We conservatively assume that parameters passed by any other method might name pointers to shared data. Second, we assume that there are no function calls through register pointers. Such calls complicate data flow analysis because the destination of the calls can not be identified statically. The system could easily be modified to disable data-flow optimization when such calls are detected. 4.5.2 Batching Calls to instrumentation routines can be batched by combining the accesses checks for multiple instructions into a single procedure call. We implemented three different types of batching for accesses within a single basic blocks: ffl batching of accesses to the same memory location with the same reference type (either both load or both store) ffl batching of accesses to the same memory location with different reference types ffl batching of accesses to consecutive memory location with the same reference type The largest performance improvement is provided by the first method, i.e, batching of accesses to the same memory location with the same reference type. Instrumentation for all but the first such access can be eliminated because we care only that the data is accessed, we do not care how many times it is accessed. Duplicated loads or stores to the same memory location might occur within a basic block because of register pressure or aliasing. An example of the latter case is a pair of loads through one register, sandwiched around a store through another register. The compiler's static analysis generally has no way of determining whether the loads and the store access the same, or distinct locations in memory. Hence, the second load is left in the basic block. The other batching methods are less useful because no instrumentation is eliminated. However, the instrumentation that remains is less costly than without batching. Both the second and third methods avoid procedure calls by consolidating the instrumentation for multiple accesses into a single routine. The resulting instrumentation also needs to check whether the are shared only once. This is valid even for accesses to consecutive memory locations because we assume that shared and non-shared regions are not located contiguously in the address space. Instrumentation generated by the second method has the additional advantage of being able to use the same bitmap offset calculation for all accesses. 4.5.3 Runtime code modification We use self-modifying code to remove instrumentation from instructions that turn out to reference private data. Memory reference instrumentation consists of a check to distinguish private and shared references, and code to record the access if it references shared memory. With runtime code modification, we overwrite the instrumentation with no-op instructions if the instruction references private data. The advantage is that subsequent executions of the instrumented instruction are delayed only by cost of executing no-op instructions, rather that the cost of executing instrumentation code that includes additional memory references. Modifyingcode at runtime requires that the text segment be writable. We unprotect the entire text segment at the beginning of an application's execution using ATOM routines to obtain the size of the application's instrumented text segment. The reply sync request Figure 4. A single diff describes all modifications to page x from both oe 1 2 and oe 2 2 because of lazy diffing. primary complication is caused by the separation of the data and instruction caches. We use data stores to overwrite instrumentation code. The new instructions are seen as data by the system and can be stalled in the (write-back) data cache. Problems remain even after the new instructions are written to memory, because stale copies might remain in the instruction cache. We solve this problem by issuing a special PAL IMB Alpha instruction that makes the caches coherent. A second complication is that naively overwriting the entire instrumentation call while inside the call causes the stack to become corrupted. We get around this problem by merely saving an indication that the affected instrumentation calls should be deleted, rather than performing the deletion immediately. The instrumentation calls are actually deleted by code at subsequent synchronization points. This technique is applicable only if we assume that each memory access instruction exclusively references either private or shared data. We modified our system to detect instructions that access both shared and non-shared data at runtime. This information is only anecdotal in that it provides no guarantees of behavior during other runs. Nonetheless, this technique can be useful if applied with caution. We used our modified version of CVM to detect instructions that access both shared and non-shared data in two of our applications. We eliminated the offending instructions by manually cloning [6] the routines that contained them. 4.5.4 Diffs One optimization that we do not exploit is the use of diffs to capture write behavior. Diffs are summaries of changes made to a single page during an interval. They are created by comparing the current copy of a page with a twin, which is a copy saved before the modifications were begun. Hence, this diff seemingly has the same information as the write bitmaps, and use of the diffs could allow us to dispense with instrumentation of write accesses. However, diffs are created lazily, meaning that shared writes might be assigned to the wrong portion of a process's execution. For example, consider the process in Figure 4. We assume that x 1 and x 2 are on the same page. Lazy diff creation means that the diff describing P 2 's first write is not immediately created at the end of interval oe 1 . The problem is that the subsequent write to x 1 is then folded into the same diff, which is associated with the earlier interval. This merging does not violate consistency guarantees because LRC systems require applications to be free of data races. However, the merging will cause the system to incorrectly believe that a data-race exists between oe 1 1 and oe 1 . Another disadvantage of this approach is that the use of diffs would slightly weaken our race detection technique. Diffs contain only modifications to shared data. Locations that are overwritten with the same value do not appear in diffs, even Apps Input Set Sync. Memory Intervals Slowdown Intervals Bitmaps Msg Used Used Ohead Barnes barrier 32768 1 2.42 4% 47% 11% Spatial 512 mols, 5 iters lock, barrier 824 Water 512 mols, 5 iters lock, barrier 344 84 3.18 5% 19% 31% Table 1. Application Characteristics though their use might constitute a race. Performance We evaluated the performance of our prototype by searching for data races in five common shared-memory applications: Barnes (Barnes-Hut algorithm the from Splash2 [27] benchmark suite) FFT (Fast Fourier Transform), SOR (Jacobi relaxation), Water (a molecular dynamics simulation; from the Splash2 suite, and Spatial (the same problem as Water, but different algorithm; also from Splash2, but optimized for reduced synchronization). All applications were run on DECstations with four 275 MHz Alpha processors, connected by a 155 MBit ATM. All performance numbers are measured on data-race free applications, i.e., we first detected, identified, and removed data-races from Water, Barnes, and Spatial, and then measured the numbers to be shown. Table 1 summarizes the application inputs and runtime characteristics. "Memory size" is the size of the shared data segment. "Intervals / Barrier" is the average number of intervals created between barriers. As the number of interval comparisons is potentially proportional to the square of the number of intervals, this metric gives an approximate idea of the worst-case cost of running the comparison algorithm. Roughly speaking, a new interval is created for each synchronization acquire. Hence, barrier-only applications will have only a single interval per barrier epoch. "Slowdown" is the runtime slowdown for each of the applications withoutany of the optimizations described in Section 4.5, compared with an uninstrumented version of the application running on an unaltered version of CVM. The first iteration of each application is not timed because we are interested in the steady-state behavior of long-running applications. However, slowdowns would be even smaller if the first iteration were counted. Over the five applications, non-optimized execution time slows only by an average factor of 3.8. This number compares quite favorably even with systems that exploit extensive compiler analysis [17, 7]. The last three columns are discussed in Section 5.2. Figure 5 breaks down the application slowdown into five categories (again, without the optimizations described in Section 4.5). "CVM Mods" is the overhead added by the modifications to CVM, primarily setting up the data structures necessary for proper data-race detection and the additional bandwidth used by the read and write notices. "Bitmaps" describes the overhead of the extra barrier round required to retrieve bitmaps, together with the cost of the bitmap comparisons. "Intervals" refers to the time spent using the interval comparison algorithm to identify concurrent interval pairs with Barnes FFT SOR Spatial Water Slowdown CVM Mods Bitmaps Access Check Proc Call Orig Figure 5. Breakdown of overhead for unoptimized instrumentation techniques. overlapping page accesses. "Access" is the time spent inside the instrumentation's procedure calls determining whether accesses are to shared memory, and setting the proper bits if so. "Proc Call" is the procedure call overhead for our instrumentation. The base version of ATOM does not currently inline instrumentation; only procedure calls can be inserted into existing code. Section 5.5 describes the performance impact of an experimental version of ATOM that can inline instrumentation. "Orig" refers to the original running time. Access check time dominates the overhead of the two applications that slow the most: SOR and Spatial. Neither application has significant false sharing, or frequent synchronization. There are therefore few interval creations, and few opportunities to run the interval comparison algorithm. Barnes has the highest proportion of overhead spent in the interval comparison algorithm. The reason is that the process of determining whether a given pair of intervals access the same pages is expensive. Each Barnes process accesses a significant fraction of the entire shared address space during each interval. Our current representation of read and write notices as lists of pages is not efficient for large numbers of pages. This overhead could be reduced by changing the representation to bitmaps for intervals with many notices. The following subsections describe the above overheads in more detail. 5.1 Instrumentation Costs We instrumented each load and store that could potentially be involved in a data race. The instrumentation consists of a procedure call to an analysis routine, and hence adds "Proc Call" and "Access Check" overheads. By summing these columns from Figure 5, we can see that instrumentation accounts for an average of 64.6% of the total race-detection overhead. This overhead can be reduced by instrumenting fewer instructions. This goal is difficult because shared and private data are all accessed using the same addressing modes, and sometimes even share the same base registers. However, we eliminate most stack accesses by checking for use of the stack pointer as a base register. The fact that all shared data in our system is dynamically allocated allows us to eliminate instructions that access data through the the "base register", which points to the start of the statically-allocated data segment. Finally, we do not instrument any instructions in shared libraries because none of our applications pass segment pointers App Load and Store Instructions Stack Static Library CVM Inst. Barnes 558 320 118057 15759 933 FFT 308 207 118057 15759 358 Spatial 758 506 118057 15782 1043 Water 613 503 118057 15759 940 Table 2. Categorization of memory access instructions. "Inst" shows the number of instructions that are actually instrumented. to any libraries. This is the case with the majority of the scientific programs where data race detection is the most important. We can, however, easily instrument "dirty" library functions, if necessary. Table breaks down load and store instructions into the categories that we are able to statically distinguish for the base case, i.e., without optimizations applied. The first five columns show the number of loads and stores that are not instrumented because they access the stack, statically-allocated data, or are in library routines, including CVM itself. The sixth column shows the remainder. These instructions could not be eliminated, and are therefore possible data-race participants. We use ATOM to instrument each such access with a procedure call to an access check routine that is executed whenever the instruction is executed. On average, we are able to statically determine that over 99% of the loads and stores in our applications are to non-shared data. As an example, the FFT binary contains 134993 load and store instructions. Of these, 118057 instructions are in libraries. A further 308 instructions access data through the stack pointer, and hence reference stack data. Another 15759 are in the CVM system itself. Finally, 207 instructions access data through the global pointer, a register pointing to the base of statically allocated global memory. We can eliminate these instructions as well, since CVM allocates all shared memory dynamically. In the entire binary, there remain only 358 memory access instructions that could possibly reference shared memory, and hence might be a part of a data race. Nonetheless, Section 5.3 will show that the majority of run-time calls to our analysis routines are for private, not shared, data. 5.2 The Cost of the Comparison Algorithm The comparison algorithm has three tasks. First, the set of concurrent interval pairs must be found. Second, this list must be reduced to those interval pairs that access at least one page in common (e.g., one interval has a read notice for page x and the other interval has a write notice for page x). Each such pair of concurrent intervals exhibits unsynchronized sharing. However, the sharing may be either false sharing, i.e., the loads and stores to the page x reference different locations in x (not a data race), or true sharing, when the loads and stores reference at least one common location at the page x (data race). The column labeled "Intervals Used" in Table 1 shows the percentage of intervals that are involved in at least one such concurrent interval pair. This number ranges from zero for SOR, where there is no unsynchronized sharing (true or false), to 86% for Spatial, where there is a large amount of both true and false sharing. Note that the number of possible interval pairs is quadratic with respect to the number of intervals, so even if this stage eliminates only 14% of all intervals, as we do for Spatial, we may be eliminating a much higher percentage of interval pairs. The column labeled "Bitmaps Used" shows that an average of only 27% of all bitmaps must be retrieved from constituent processors in order to identify data races by distinguishing false from true sharing. As page access lists of concurrent intervals will only overlap in cases of false sharing or actual data races, the percentage of intervals and bitmaps involved in comparisons is fairly small. Note, however, the effect of bitmap and interval comparisons on Barnes. Although the absolute amount of overhead added by the comparisons is not large, it is larger relative to the rest of the overhead than for any other application. There is also a seeming disparity between the utilization of intervals and bitmaps for Barnes. Only 4% of intervals are used, but 47% of the bitmaps are used. This implies that the majority of the shared pages are accessed in only a small number of intervals, probably one or two phases of a timestep loop that has many phases. In fact, this is exactly the case for our version of Barnes. Most of the work is done in the force and position computation phases, which are separated by a barrier. However, each processor accesses the same bodies during both phases, and the sets of bodies accessed by different processors are disjoint. Hence, there is no true sharing between processors across the barrier for these computations. Since the bodies assigned to each processor during any iteration are scattered throughout the address space, a large amount of false sharing occurs. The barrier serves to double the effect of the false sharing by splitting each of the intervals in half, causing bitmaps to be requested twice for each page instead of once. Note that the barrier can not be removed without a slight reorganization of code because it synchronizes updates to a few scalar global variables. We tested this interpretation of the results by implementing the above reorganization. Removal of the barrier effectively reduced the interval and bitmap overheads in half, reducing the overall overhead by approximately 30%. The final column of Table 1 shows the amount of additional data needed by the race detection technique compared with the uninstrumented system. 5.3 The Effect of optimizations Table 3 shows the effect of our optimizations on the number of instructions actually instrumented. "No Opt." refers to the base case with no optimizations, "DF" is dataflow, "Batching" is self-explanatory, "Code Mod" refers to dynamic code modification, and "All" includes all three. "DF+Batching" is included to show the synergy between dataflow and batching in the absence of code modification. Code modification actually increases the number of instrumented sites in FFT and Water because of cloning. The last three columns of Table 3 show the number of times we were able to apply each batching method. Here, "2-3" represents combining two or three instructions of the same type with consecutive addresses. "Same" represents combining instructions of the same type and address. Finally, "Mix" is combining instructions with the same address, but different access types. These numbers imply that batching is most applicable for applications with complex data structures and access patterns. Apps Instrumented Instructions Batching Opt. Data Flow Batching Code Mod DF+Batching All 2-3 Same Mix Barnes 933 854 730 933 655 655 63 26 76 Spatial 1043 844 904 1043 725 725 Water 940 776 747 1156 604 733 39 38 58 Table 3. Static instrumentation statistics. Apps Millions of Instrumented Instructions Flow Batching Code Mod DF+Batching All Barnes 435.8 415.0 434.3 88.5 413.5 87.0 FFT 5.8 5.3 5.8 1.5 5.3 1.5 Spatial 108.3 39.5 88.2 28.5 22.1 20.0 Water 124.9 51.8 105.0 21.6 32.6 7.5 Table 4. Dynamic Optimization Statistics Table 4 shows the effect of our optimizations on the number of instrumented instructions executed at runtime. Although data flow analysis and batching together eliminate 29.8% and 22.9% of local reference instrumentations for Barnes and FFT, respectively, this accounts for only 5.1% and 8.8% of instrumented references at runtime. On the other hand, only 30.5% of instrumentationsand 35.8% of Water's instrumentationsare eliminated. Yet the eliminationof these instrumentations accounts for 80.0% and 73.9% of runtime references, respectively. Clearly the effectiveness of these optimizations is heavily application-dependent. Figure 6 shows the effect of these optimizations on the overall slowdown of the applications. The average slowdown when all three optimizations are applied is 2.8, which is an improvement of 26%. FFT has the lowest overhead at 22%. Figure 6 also includes bars for the inlining optimization discussed in Section 5.5. 5.4 The Cost of CVM Modifications Figure 5 shows that almost 15.8% of our overhead comes from "CVM Mods", or modifications made to the CVM system in order to support the race-detection algorithm. This overhead consists of the cost of setting up additional data structures for data-race detection and the cost of the additional bandwidth consumed by read notices. The last column of Table 1 shows the bandwidth overhead of adding read and additional write notices to synchronization messages. Individual read and write notices are the same size, but there are typically at least five times as many reads as writes, and read notices consume a proportionally larger amount of space than write notices. Additional write notices are needed because even the notices are no longer created lazily, even though diffs still are. Barnes FFT SOR Spatial Water Slowdown Base Data Flow Batch Code Mod Inlining All-Inlining Figure 6. Optimizations The bandwidth overhead for Water is quite large because the fine-grained synchronization means that many intervals and notices are created. By contrast, the primary cost for Spatial is that false sharing is quite prevalent, leading to a large number of bitmap requests. Inlining To verify our assumption that the procedure call and access check overhead can be significantly reduced by inlining, we used an unreleased version of ATOM, called XATOM, to inline read and write access checks. Our implementation decreases the cost of the inlined code fragments by using register liveness analysis to identify dead registers. Dead registers are used whenever possible to avoid spilling the contents of registers needed by the instrumentation code. Table 5 shows the effect of inlining on overall performance, relative to the base case with no optimizations. The column labeled "Runtime" shows the effect as a percentage of overall running time, while "Overhead" shows the same quantity as a percentage of instrumentation overhead. The "Static" column shows the percent of inlined instructions eliminated by register liveness analysis (mostly load and store instructions), and "Dynamic" shows the corresponding dynamic quantity. Elimination of these instructions is very useful because the majority are memory access instructions, and hence relatively expensive. The improvements roughly correlate with the procedure call overhead shown in Figure 5, where an average of 13.7% of total overhead is caused by procedure calls. However, inlining can also eliminate some of the access check overhead because the liveness analysis can reduce register spillage. This is particularly important in the case of SOR, where most of the overhead is in access checks. An important question that we have not answered is how effective the other optimizations are in combination with inlining. Inlining certainly decreases the potential of the other techniques because all of them work by decreasing the cost or number of access checks. However, they should still be effective in combination with inlining because the remaining overhead is still significant. Runtime code modification, in particular, would still be useful because inlining has no effect on the total number of instructions instrumented. However, the code modification mechanism would probably need to change slightly in order to address the fact that the inserted instrumentation is no longer just a few bytes. Inserting unconditional branches to the end of Apps Improvement Register Liveness Runtime Overhead Static Dynamic Barnes 15.0% 28.5% 21.7% 35.5% FFT 13.8% 23.5% 16.8% 15.8% Spatial 1.3% 1.9% 13.5% 7.3% Water 14.0% 25.7% 17.8% 24.4% Table 5. Inlining the instrumentation code might be more effective than overwriting the inlined instrumentation with a large number of no-op instructions. 6 Discussion 6.1 Reference Identification The system currently prints the shared segment address together with the interval indexes for each detected race condition. In combination with symbol tables, this information can be used to identify the exact variable and synchronization context. Identifying the specific instructions involved in a race is more difficult because it requires retaining program counter information for shared accesses. This information is available at runtime, but such a scheme would require saving program counters for each shared access until a future barrier analysis phase determined that the access was not involved in a race. The storage requirements would generally be prohibitive, and would also add runtime overhead. A second approach is to use the conflicting address and corresponding barrier epoch from an initial run of the program as input to a second run. During the second run, program counter information can be gathered for only those accesses to the conflicted address that originate in the barrier epoch determined to involve the data race. While runtime overhead and storage requirements can thereby be drastically reduced, the data race must occur in the second run exactly as in the first. This will happen if the application has no general races [20], i.e., synchronization order is deterministic. This is not the case in Water, the application for which we found data races. A solution is to modify CVM so as to save synchronization ordering information from the first run, and to enforce the same ordering in the second run. This is done in the work on execution replay in TreadMarks, a similar DSM. The approach of the Reconstruction of Lamport Timestamps (ROLT) [23] technique keeps track of minimal ordering information saved during an initial run to enforce exactly the same interleaving of shared accesses and synchronization in a second run. During the second run, a complete address trace can be saved for post-mortem analysis, although the authors do not discuss race detection in detail. The advantage of this approach is that the initial run incurs minimal overhead, ensuring that the tracing mechanism does not perturb the normal interleaving of shared accesses. The ROLT approach is complementary to the techniques described in this paper. Our system could be augmented to include an initial synchronization-tracing phase, allowing us to eliminate our perturbation of the parallel computation in the second, i.e., race-reference identification phase. Currently, we use the shared page of the variable involved in the data-race from the initial run as the target page for which we save program counters during the identification run. 6.2 Global Synchronization The interval comparison algorithm is run only at global synchronization operations, i.e., barriers. The applications and input sets in this study use barriers frequently enough, or otherwise synchronize infrequently enough, that the number of intervals to be compared at barriers is quite manageable. Nonetheless, there certainly exist applications for which global synchronization is not frequent enough to keep the number of interval comparisons to a small number. Ideally, the system would be able to incrementally discard data races without global cooperation, but such mechanisms would increase the complexity of the underlying consistency protocol [10]. If global synchronization is either not used, or not used often enough, we can exploit CVM routines that allow global state to be consolidated between synchronizations. Currently, this mechanism is only used in CVM for garbage collection of consistency information in long-running, barrier-free programs. 6.3 Accuracy Adve [2] discusses three potential problems in the accuracy of race detection schemes in concert with weak memory systems, or systems that support memory models such as lazy release consistency. The first is whether to return all data races, or only "first" data races [18, 2]. First races are essentially those that are not caused or affected by any prior race. Determining whether a given race is affected by any other effectively consists of deciding whether the operations of any other race precede (via hb1 !) the operations of the race in question. Our system currently reports all data races. However, we could easily capture an approximation of first races by turning off reporting of any races for which both accesses occur after (via hb1 !) the accesses of other data races. The second problem with the accuracy of dynamic race-detection algorithms is the reliability of information in the presence of races. Race conditions could cause wild accesses to random memory locations, potentially corrupting interval ordering information or access bitmaps. This problem exists in any dynamic race-detection algorithm, but we expect it to occur infrequently. A final accuracy problem identified by Adve is that of systems that attempt to minimize space overhead by buffering only limited trace information, possibly resulting in some races remaining undetected. Our system only discards trace information when it has been checked for races, and hence does not suffer this limitation. 6.4 Limitations We expect this technique to be applicable for a large class of applications. The applications in our test suite range from SOR, which is bandwidth-limited, to Water, which both synchronizes and modifies data at a fine granularity. These applications stress different portions of the race-detection technique: the raw cost of access instrumentation for SOR, and the complexity and cost of dealing with large numbers of intervals for Water. Furthermore, our applications (with the exception of SOR), have not been > modified in order to reduce false sharing. Multi-writer LRC tolerates false sharing much better than most other > protocols. > Applications that have been tuned for LRC tend not to have false sharing > removed. > Water, Spatial, and Barnes all have large amounts of false sharing. Nonetheless, our methodology is clearly not applicable in all situations. Chaotic algorithms, for example, tolerate races as a means of eliminating synchronization and improving performance. The false positives caused by tolerated races can obscure unintended races, rendering this class of applications ill-suited for our techniques. Similarly, protocol-specific optimization techniques may cause spurious races to be reported. An earlier version of Barnes had a barrier that enforced only anti-dependences. This barrier has been removed in our version of Barnes. The application is still correct because LRC delays the propagation of both consistency information and data. However, these anti-dependences are flagged by our system as data races, and can interfere with the normal operation of our technique. The techniques that we discuss in this paper are not necessarily limited to LRC systems and applications. Our approach is essentially to use existing synchronization-ordering information to reduce the number of comparisons that have to be made at runtime. This information could easily be collected in other distributed systems and memory models by putting appropriate wrappers around synchronization calls, and appending a small amount of additional information to synchronization messages. Multiprocessors could be supported by using wrappers and appending a small amount of data to synchronization state. Application of the rest of the techniques should be straightforward. 6.5 Further Performance Enhancements Performance of the underlying protocol could be improved by using our write instrumentation to create diffs, rather than using the twin and page comparison method. We did not investigate this option because of its complexity. Integrating this mechanism with the runtime code modification, for example, would be non-trivial. Compiler techniques could be used to expose more opportunities for batching. Loop unrolling and trace scheduling would be particularly effective for applications such as SOR, the application with the largest overhead in our application testbed. Finally, the interval comparison algorithm could be improved significantly. While the overhead added by the comparison algorithm was relatively small for our applications, better worst-case bounds would be desirable for a production system. One promising approach is the use of hierarchical comparison algorithms. For example, if two processes create a large number of intervals through exclusively pairwise synchronization, the number of intervals to be compared with other processes could be reduced by first aggregating the intervals that were created in isolation, and then using these aggregations to compare with intervals of other processes. 7 Related Work There has been a great deal of published work in the area of data race detection. However, most prior work has dealt with applications and systems in more specialized domains. Bitmaps have been used to track shared accesses before [7], but we know of no other language independent implementation of on-the-fly data-race detection for explicitly-parallel, shared-memory programs. We previously [21] described the performance of a preliminary form of our race-detection scheme that ran on top of CVM's single-writer LRC protocol [14]. This paper describes the performance of our race-detection scheme on top of CVM's multi-writer protocol. This protocol is a more challenging target because it usually outperforms the single-writer protocol significantly, making it more difficult to hide the race-detection overheads. Additionally, the work described in this paper includes several optimizations to the basic system (i.e., batching, data-flow analysis, runtime code modification, and inlining). Our work is closely related to work already alluded to in Section 6.3, a technique described (but not implemented) by Adve et al. [2]. The authors describe a post-mortem technique that creates trace logs containing synchronization events, information allowing their relative execution order to be derived, and computation events. Computation events correspond roughly to CVM's intervals. Computation events also have READ and WRITE attributes that are analogous to the read and page lists and bitmaps that describe the shared accesses of an interval. These trace files are used off-line to perform essentially the same operations as in our system. We differ in that our minimally-modified system leverages off of the LRC memory model in order to abstract this synchronization ordering information on-the-fly. We are therefore able to perform all of the analysis on-the-fly as well, and do away with trace logs, post-mortem analysis, and much of the overhead. Work on execution replay in TreadMarks could be used to implement race-detection schemes. The approach of the Reconstruction of Lamport Timestamps (ROLT) [23] technique is similar to the technique we described in Section 6.1 for identifying the instructions involved in races. Minimal ordering information saved during an initial run is used to enforce exactly the same interleaving of shared accesses and synchronization in the second run. During the second run, a complete address trace can be saved for post-mortem analysis, although the authors do not discuss race detection in detail. The advantage of this approach is that the initial run incurs minimal overhead, ensuring that the tracing mechanism does not perturb the normal interleaving of shared accesses. The ROLT approach is complementary to the techniques described in this paper. The primary thrust of our work is in using the underlying consistency mechanism to prune enough information on-the-fly so that post-mortem analysis is not necessary. As such, our techniques could be used to improve the performance of the second phase of the ROLT approach. Similarly, our system could be augmented to include an initial synchronization-tracing phase, allowing us to reduce perturbations of the parallel computation. Recently, work on Eraser [25] used verification of lock discipline to detect races in multi-threaded programs. Eraser's main advantage is that it can detect races that do not actually occur in the instrumented execution. However, it does not guarantee race-free behavior if no data-races are found, and can return false positives. Furthermore, the system does not support distributed execution. Finally, the overhead of Eraser's approach is an order of magnitude higher than ours. Work on detecting data-race detection for non-distributed multi-threaded programs has also been done for RecPlay [24], a Record/Replay system for multi-threaded programs. This work is similar to ROLT approach discussed here, but applied to multi-threaded programs. This work uses the happens-before relation to reconstruct and replay the execution of the initial run, and then in the second run perform access checks. Conclusions This paper has presented the design and performance of a new methodology for detecting data races in explicitly-parallel, shared-memory programs. Our technique abstracts synchronization ordering from consistency information already maintained by multiple-writer lazy-release-consistent DSM systems. We are able to use this information to eliminate most access comparisons, and to perform the entire data-race detection on-the-fly. We used our system to analyze five shared-memory programs, finding data races in three of them. Two of those data races, in standard benchmark programs, were bugs. The primary costs of data-race detection in our system are in tracking shared data accesses. We were able to significantly reduce these costs by using three optimization techniques: register data-flow analysis, batching, and inlining. Nonetheless, the majority of the runtime calls to our library are for non-shared accesses. We therefore used runtime code-modification to dynamically rewrite our instrumentation in order to eliminate access checks for instructions that accessed only non-shared data. By combining all of the optimizations except inlining, we were able to reduce the average slowdown for our applications to approximately 2.8, and to only 1.2 for one application. We expect that combining inlining with the other optimizations would reduce the slowdown even further. While the implementation described above is specific to LRC, our general approach is not. Our system exploits synchronization ordering to eliminate the majority of shared accesses without explicit comparison. Fine-grained comparisons are made onlywhere the coarse-grained comparisons fail to rule data races out. This approach could be used on systems supporting other programming models by using "wrappers" around synchronization accesses to track synchronization ordering. We believe that the utility of our techniques, in combination with the generality of the approach that we present, can help data-race detection to become more widely used. --R A unified formalization of four shared-memory models Detecting data races on weak memory systems. Debugging fortran on a shared memory machine. Race frontier: Reproducing data races in parallel program debugging. A fast instruction-set simulator for execution profiling A methodology for procedure cloning. An empirical comparison of monitoring algorithms for access anomaly detection. Memory consistency and event ordering in scalable shared-memory multiprocessors Parallel program debugging with on-the-fly anomaly detection Distributed Shared Memory Using Lazy Release Consistency. Lazy release consistency for software distributed shared memory. Treadmarks: Distributed shared memory on standard workstations and operating systems. The Coherent Virtual Machine. The relative importance of concurrent writers and weak consistency models. How to make a multiprocessor computer that correctly executes multiprocess programs. Improving the accuracy of data race detection. On the complexity of event ordering for shared-memory parallel program executions What are race conditions? Online data-race detection via coherency guarantees Instrumentation and optimization of win32/intel executables using etch. Execution replay for TreadMarks. Work in progress: An on-the-fly data race detector for recplay A dynamic data race detector for multi-threaded programs ATOM: A system for buildingcustomized program analysis tools. The SPLASH-2 programs: Characterization and methodological considerations --TR --CTR Edith Schonberg, On-the-fly detection of access anomalies, ACM SIGPLAN Notices, v.39 n.4, April 2004 Milos Prvulovic , Josep Torrellas, ReEnact: using thread-level speculation mechanisms to debug data races in multithreaded codes, ACM SIGARCH Computer Architecture News, v.31 n.2, May Min Xu , Rastislav Bodk , Mark D. Hill, A serializability violation detector for shared-memory server programs, ACM SIGPLAN Notices, v.40 n.6, June 2005 Chen Ding , Xipeng Shen , Kirk Kelsey , Chris Tice , Ruke Huang , Chengliang Zhang, Software behavior oriented parallelization, ACM SIGPLAN Notices, v.42 n.6, June 2007 Bohuslav Krena , Zdenek Letko , Rachel Tzoref , Shmuel Ur , Tom Vojnar, Healing data races on-the-fly, Proceedings of the 2007 ACM workshop on Parallel and distributed systems: testing and debugging, July 09-09, 2007, London, United Kingdom Sudarshan M. Srinivasan , Srikanth Kandula , Christopher R. Andrews , Yuanyuan Zhou, Flashback: a lightweight extension for rollback and deterministic replay for software debugging, Proceedings of the USENIX Annual Technical Conference 2004 on USENIX Annual Technical Conference, p.3-3, June 27-July 02, 2004, Boston, MA

shared memory;DSM;data races;on-the-fly

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Metric operations on fuzzy spatial objects in databases.

Uncertainty management for geometric data is currently an important problem for (extensible) databases in general and for spatial databases, image databases, and GIS in particular. In these systems, spatial data are traditionally kept as determinate and sharply bounded objects; the aspect of spatial vagueness is not and cannot be treated by these systems. However, in many geometric and geographical database applications there is a need to model spatial phenomena rather through vague concepts due to indeterminate and blurred boundaries. Following previous work, we first describe a data model for fuzzy spatial objects including data types for fuzzy regions and fuzzy lines. We then, in particular, study the important class of metric operations on these objects.

INTRODUCTION Representing, storing, quering, and manipulating spatial information is important for many non-standard database applications. But so far, spatial data modeling has implicitly assumed that the extent and hence the borders of spatial objects are precisely determined ("boundary syndrome"). Special data types called spatial data types (see [7] for a survey) have been designed for modeling these data in databases. We will denote this kind of entities as crisp spatial objects. In practice, however, there is no apparent reason for the whole boundary of a region to be determined. On the contrary, the feature of spatial vagueness is inherent to many geographic data [3]. Many geographical application examples illustrate that the boundaries of spatial objects (like geological, soil, and vegetation units) can be partially or totally indeterminate and blurred; e.g., human concepts like "the Indian Ocean" or "Southern England" are implicitly vague. In this paper we focus on a special kind of spatial vagueness called fuzziness. Fuzziness captures the property of many spatial objects in reality which do not have sharp boundaries or whose boundaries cannot be precisely determined. Examples are natu- ral, social, or cultural phenomena like land features with continuously changing properties (such as population density, soil quality, vegetation, pollution, temperature, air pressure), oceans, deserts, English speaking areas, or mountains and valleys. The transition between a valley and a mountain usually cannot be exactly ascertained so that the two spatial objects "valley" and "mountain" cannot be precisely separated and defined in a crisp way. We will designate this kind of entities as fuzzy spatial objects. The goal of this paper is to deal with the important class of metric operations on fuzzy spatial objects. Examples are the area operation on fuzzy regions or the length operation on fuzzy lines. It turns out that their definition is not as trivial as the definition of their crisp counterparts. The underlying formal object model follows the au- thor's previous work in [8] and offers fuzzy spatial data types like fuzzy regions and fuzzy lines in two-dimensional Euclidean space. Our concept to integrate fuzzy spatial data types into databases is to design them as abstract data types whose values can be embedded as complex entities into databases [9] and whose definition is independent of a particular DBMS data model. They can, e.g., be employed as attribute types in a relation. The future design of an SQL-like fuzzy spatial query language will profit from the abstract data type approach since it makes the integration of data types, predicates, and operations into SQL easier. The metric operations described in this paper are part of a so-called abstract model for fuzzy spatial objects. This model focuses on the nature of the problem and on its realistic description and solution with mathematical notations; it employs infinite sets and does not worry about finite representations of objects as they are needed in computers. This is done by the so-called discrete model which transforms the infinite representations into finite ones and which realizes the abstract operations as algorithms on these finite representations. It is thus closer to implementation. Section 2 discusses related work. Section 3 introduces fuzzy spatial objects. It gives some basic concepts of fuzzy set theory and then informally presents the design of fuzzy regions and fuzzy lines. Section 4 describes and formalizes metric operations on fuzzy spatial objects and identifies the two classes of real-valued and fuzzy-valued metric operations. Section 5 draws some conclusions and discusses future work. 2. RELATED WORK Mainly two kinds of spatial vagueness can be identified: uncertainty is traditionally equated with randomness and chance occurrence and relates either to a lack of knowledge about the position and shape of an object with an existing, real boundary (positional uncertainty) or to the inability of measuring such an object precisely (measurement uncertainty). Fuzziness, in which we are only interested in this paper, is an intrinsic feature of an object itself and describes the vagueness of an object which certainly has an extent but which inherently cannot or does not have a precisely definable boundary (e.g., between a mountain and a valley). This kind of vagueness results from the imprecision of the meaning of a con- cept. Models based on fuzzy sets have, e.g., been proposed in [1, 2, 4, 8]. 3. FUZZY SPATIAL OBJECTS In this section we very briefly and informally present the basic elements of an abstract model for fuzzy spatial objects as it has been formalized in [8]. The model is based on fuzzy set theory (and fuzzy topology) whose main concepts are introduced first, as far as they are needed in this paper. Afterwards the design of spatial data types for fuzzy regions and fuzzy lines is shortly discussed. 3.1 Crisp Versus Fuzzy Sets Fuzzy set theory [10] is an extension and generalization of Boolean set theory. It replaces the crisp boundary of a classical set by a gradual transition zone and permits partial and multiple set mem- bership. Let X be a classical (crisp) set of objects. Membership in a classical subset A of X can then be described by the characteristic function f0;1g such that for all x 2 X holds c A and only if x 2 A and c A This function can be generalized such that all elements of X are mapped to the real interval [0,1] indicating the degree of membership of these elements in the set in question. We call - the membership function of - A, and the set - A called a fuzzy set in X . A [strict] a-cut of a fuzzy set - A for a specified value a is the crisp set A a [A a 1g. The strict a-cut for a = 0 is called support of - . A fuzzy set is convex if and only if each of its a-cuts is a convex set. A fuzzy set - A is said to be connected if its pertaining collection of a-cuts is connected, i.e., for all points P;Q of - A, there exists a path lying completely within - A such that - A (R) - min(- A (P);- A (Q)) holds for any point R on the path. 3.2 Fuzzy Regions We first describe some desired properties of fuzzy regions and also discuss some differences in comparison with crisp regions. After that, we informally outline a data type for fuzzy regions. 3.2.1 Generalization of Crisp to Fuzzy Regions A very general model defines a crisp region as a regular closed set [5, 6, 7] in the Euclidean space IR 2 . This model is closed under (ap- propriately defined) geometric union, intersection, and difference. Similar to the generalization of crisp sets to fuzzy sets, we strive for a generalization of crisp regions to fuzzy regions on the basis of the point set paradigm and fuzzy concepts. Crisp regions are characterized by sharply determined boundaries enclosing and grouping areas with equal properties or attributes and separating different regions with different properties from each other; hence qualitative concepts play a central role. For fuzzy regions, besides the qualitative aspect, also the quantitative aspect becomes important, and boundaries in most cases disappear (between a valley and a mountain there is no boundary!). The distribution of attribute values within a region and transitions between different regions may be smooth or continuous. This important feature just characterizes fuzzy regions. A classification of fuzzy regions from an application point of view together with application examples is given in [8]. 3.2.2 Definition of Fuzzy Regions We now briefly give an informal description of a data type for fuzzy regions. A detailed formal definition can be found in [8]. A value of type fregion for fuzzy regions is a regular open fuzzy set whose membership function - predominantly continuous. F is defined as open due to its vaguenessand its lack of boundaries. The property of regularity avoids possible "geometric anomalies" (e.g., isolated or dangling line or point features, missing lines and points) of fuzzy regions. The property of - F to be predominantly continuous models the intrinsic smoothness of fuzzy regions where a finite number of exceptions ("continuity gaps") are allowed. 3.2.3 Fuzzy Regions As Collection of a-Level Regions A "semantically richer" characterisation of fuzzy regions describes them as collections of crisp a-level regions [8]. Given a fuzzy region F , we represent a region F a for an a 2 [0; 1] as the regular crisp set of points whose membership values in - F are greater than or equal to a. F a can have holes. The a-level regions of - F are nested, i.e., if we select membership values a n ?a a n+1 . 3.3 Fuzzy Lines In this section we informally describe a data type for fuzzy lines whose detailed formal definition can be found in [8]. We start with a simple fuzzy line - l which is defined as a continuous curve with smooth transitions of membership grades between neighboring points of - l , i.e., the membership function of - l is continuous too. The end points of - l may coincide so that loops are allowed. Self-intersections and equality of an interior with an end point, however, are prohibited. If - l is closed, the first end point must be the leftmost point to ensure uniqueness of representation. Let S be the set of fuzzy simple lines. An S-complex T is a finite subset of S such that the following conditions are fulfilled. First, the elements of T do not intersect or overlap within their interior. Sec- ond, they may not be touched within their interior by an endpoint of another element. Third, isolated fuzzy simple lines are disallowed (connectivity property). Fourth, each endpoint of an element of T must belong to exactly one or more than two incident elements of T to support the requirement of maximal elements and hence to achieve minimality of representation. Fifth, the membership values of more than two elements of T with a common end point must have the same membership value; otherwise we get a contradiction saying that a point of an S-complex has more than one membership value. All conditions together define an S-complex as a connected planar fuzzy graph with a unique representation. A value of the data type fline for fuzzy lines is then given as a finite set of disjoint S-complexes. 4. METRIC OPERATIONS ON FUZZY A very important class of operations on spatial objects are metric operations usually getting one or two spatial objects as arguments and yielding a numerical result. They compute metric (i.e., mea- surable) properties such as area and perimeter and are commonly used in the analysis of spatial phenomena. While their definitions are well-known, clear, and relatively easy for crisp spatial objects, it is not always obvious how to measure metric properties of fuzzy spatial objects and hence how to define corresponding operations. A central issue is whether the result of such a metric operation is a crisp number or rather a "fuzzy" number. From an application point of view both kinds of numerical results are acceptable and even desirable. A resulting single crisp number can be interpreted as an appropriately aggregated or weighted real value over all membership values of a fuzzy spatial object. An arising fuzzy number satisfies the expectation that, if the operands of a metric operation are fuzzy, then the numerical result should be fuzzy too. Therefore, we will consider the crisp (Section 4.1) and the fuzzy (Section 4.2) variant of several metric operations. Both variants have in common that they operate on fuzzy spatial objects and that they are reduced to the ordinary definitions in the crisp case. 4.1 Crisp-Valued Numerical Operations In this section we view operations on fuzzy spatial objects that yield crisp numbers. We first present a special view on membership functions which simplifies an understanding of the metric operations discussed afterwards. 4.1.1 Membership Functions Considered as Functions of Two Variables Essentially, a membership function for a spatial object s associates with each point which p belongs to s. A slightly modified view considers - as a function of the two variables x and y. This view has the benefit that we can visualize how - "works" in terms of its graph. The graph of - is the graph of the equation z = -(x;y) and comprises all three-dimensional points If s is a fuzzy region, the graph of the corresponding spatial membership function of two variables is a collection of disjoint surfaces (one for each fuzzy face) that lie above their domain s in the Euclidean plane. Each surface determines a solid or volume bounded above by the function graph and bounded below by a fuzzy face of s. If s is a fuzzy line, we obtain a collection of disjoint three-dimensional networks each consisting of a set of three-dimensional curves. As an example, Figure 1 shows the three-dimensional view of the membership function of a fuzzy region (showing the expansion of air pollution caused by a power station, for instance). Three-dimensional visualizations of membership functions of fuzzy spatial objects lead to an easier understanding of the metric operations discussed in the following. Figure 1: 3D representation of the membership function of a fuzzy region. 4.1.2 Metric Operations on Fuzzy Regions Metric operations on fuzzy regions usually yield a real value as a result and can be summarized as a collection of functions real with different function names for g and, of course, different semantics. Let - fregion where the - f i are the fuzzy faces of - F . The definition of an area operator applied to - F requires that - F is integrable. This condition is always fulfilled since our definition of a fuzzy region requires that - F is continuous or at least piecewise continuous. The area of - F can be defined as the volume under the membership function - area( - RR F (x;y) dx dy RR F (x;y) dx dy RR Thus, the integration can be performed either over the entire Euclidean plane, or equivalently over the support of - F , which is always bounded, or equivalently over the supports of the faces of - F . Note that - Crisp holes enclosed by fuzzy faces do not cause problems during the integration process; they simply do not contribute to the double integral. Evidently, if - G, we have area( - G). In particular, we obtain area( - RR F (x;y) dx dy RR A special case arises if - F is piecewise constant and - F consists of a finite collection fF a of crisp a-level regions. Then the area of - F is computed as the weighted sum of the areas of all a-level regions F a : area( - RR (x;y)2F a i a Next, we determine the height and the width of a fuzzy region. For computing the height (width) of a fuzzy region - F , all maximum membership values along the y-axis (x-axis) and parallel to the x-axis (y-axis) are aggregated and (their square roots are) added up, F is projected onto the y-axis (x-axis), and the maximum membership value is determined for each y-value (x-value). R F (x;y) dy and similarly we obtain R F (x;y) dx Both integrals are finite since - F has bounded support. Let - crisp. For the height operator we obtain max x2IR -2 1 for a fixed y 0 2 IR if the intersection of the horizontal line with F is non-empty. Otherwise, the expression yields Hence, R F (x;y) dy is the measure of the set of y 0 's such that F intersects In the case that F consists of several connected components, each component of F gives rise to a y-interval. Then height(F) is the union of these intervals. If F is connected, we only obtain one interval, and height(F) is just its length. Analogous thoughts hold for the width operator. So far, we have not explained the meaning and the necessity of the exponent 1 2 as part of the integrands. The introduction of this exponent is essential since it ensures our expectation that the area of a fuzzy region is equal or less than its height times its width, i.e., LEMMA 1. area( - PROOF. We can show this as follows: area( - RR F (x;y) dx dy R F (x;y) dy \Delta R F (x;y) dx F (x;y) dy \Delta R F (x;y) dx Without the exponent 1 2 we would have area( - F) which is a completely different result and which does not correspond to our intuition. Hence, metric operations on fuzzy spatial objects do not only dependon their geometric extent but also on the nature of their membership values so that a kind of "compen- sation" is necessary. In accordance with the definitions for height and width, the exponent 1 will also appear in the definitions of the following operators. If - F consists of a finite collection fF a of crisp a-level regions, we obtain R dy R The diameter of a spatial object is defined as the largest distance between any of its points. We will give here two definitions and distinguish between the outer diameter and the inner diameter. For the computation of the outer diameter we may leave the fuzzy re- gion; for the computation of the inner diameter we have to remain within its interior. The outer diameter of - F is defined as R F (u;v) du where u and v are any pair of orthogonal directions and where the maximum is evaluated over all possible directions u (we can also imagine that - F is smoothly rotated within the Cartesian coordinate system). If - is crisp, the value u yielding the maximum is the direction of the line along which the projection of - F has the largest size. Obviously, height( - F) and width( - F), and we can further conclude that area( - The inner diameter operator only relates to connected fuzzy re- gions. Let P and Q be any two points of - F , and let p PQ be a path from P to Q that lies completely in - F . Such a path must exist since F is assumed to be connected. The inner diameter is defined as min RR F (x;y) dx dy where the maximum is computed over all points P and Q of the Euclidean plane and where the minimum is determined over all paths between P and Q such that - F (P);- F (Q)) holds for any point R on p PQ . Since - F is connected, such a path always exists. If - path p PQ from P to Q will not yield the maximum. Otherwise, if both P and Q are in F , must lie completely in F so that min pPQ RR F (x;y) dx dy= length(p PQ ). In this case, the meaning of innerDiameter(F) amounts to its standard definition as the greatest possible distance between any two points in F where only paths lying completely in F are allowed. The relationship between inner and outer diameter is different for crisp and fuzzy regions. If - is crisp, we can derive two propositions: LEMMA 2. If F is a connected crisp region, then outerDiameter(F) - innerDiameter(F). PROOF. Select a line (which is not necessarily unique) upon which the projection of F onto the u-axis is largest. Since F is connected, this projection is an interval, and its length is given by outerDiameter(F). Let us assume that P and Q are those points of F that coincide with the end points of this interval. Then the shortest path p in F between P and Q is at least as long as the straight line segment s joining P and Q. Segment s is at least as long as the interval since the interval is a projection of s. F s Figure 2: Example of the relationship between inner (p) and outer diameter of a connected crisp region. An example of this relationship is that s crosses the exterior of F . Then path p is longer than s (Figure 2). LEMMA 3. If F is a convex crisp region, then PROOF. Since F is convex, F is connected so that outerDiameter(F) - innerDiameter(F) holds according to Lemma 2. Let P and Q be the end points of the shortest path yielding the maximum in the definition of innerDiameter(F). Since F is convex, the shortest path in F between P and Q is the straight line segment joining P and Q. The projection of F on this line segment thus has length at least innerDiameter(F) so that outerDiameter(F) - innerDiameter(F). For convex fuzzy regions the situation is different in the sense that the outer diameter can even be greater than the inner diameter. An illustration is given in Figure 3. Let c;d 2 IR ?0 . We consider a fuzzy region - F having the membership function a and thus consisting of two circular, concentric a-level regions F a 1 and F a 2 . Moreover, we assume that a 1 is much larger than a 2 and that d is only slightly larger than c. If R and S are two points on the boundary of F a 1 which are located on opposite sides so that their straight connection passes the center of F a 1 , then for F a 1 the inner and the outer diameter are equal according to Lemma 3, and we obtain innerDiameter(F a 1 and Q are two points on the boundary of F a 2 , they have the largest distance if they are located on opposite sides so that their straight connection passes the center of F a 2 . This is just the outer diameter of - F , and we have outerDiameter( - c). But the inner diameter of - F can be smaller if we consider a shortest path This path avoids F a 1 with its high membership value a 1 , and we can compute the value for a 1 so that innerDiameter( - F) holds. We must require that length(p PQ ) \Delta a c). This is the case if a 1 ? a 2 (length(p PQ )\Gamma2(d \Gammac)) 2c . The observation that the outer diameter of a fuzzy region - F can be larger than its inner diameter is only valid if - F is convex: LEMMA 4. If - F is a convex fuzzy region, then F). PROOF. If - F is a convex fuzzy region, we know due to the definition of connectedness that - F (P);- F (Q)) holds for any point R on the straight line segment PQ between any two R d c F a 1 F a 2 Figure 3: Example of the relationship between inner and outer diameter of a convex fuzzy region. points P and Q. Therefore, we obtain RR F (x;y) dx dy - RR F (x;y) dx dy for the shortest path p PQ between P and Q. A projection of - F onto the line segment PQ in any direction u yields R F (u;v) du - RR F (x;y) dx dy. Fi- nally, we obtain RR F (u;v) du RR F (x;y) dx dy RR F (x;y) dx dy In all other cases where - F is a more general fuzzy region, no general statement can be made about the relationship between inner and outer diameter. Based on the concept of outer diameter we can specify two other operators which characterize the shape of a fuzzy region. They rate the opposite geometric properties "elongatedness"and "roundness" and are defined in terms of the proportion of the minor outer diameter to the major outer diameter. The first operator is given as min R F (u;v) du The geometric property "roundness" can be regarded as the complement of "elongatedness": The next operator of interest computes the perimeter of a fuzzy region - F . Assuming that the membership function - F is continuous and that - -x and - -y denote the partial derivatives of - with respect to x and y, respectively, we can define the perimeter of - F as RR -x -2 F -y -2 F dx dy RR -y - 1- f dx dy One can prove that, if - G, we obtain perimeter( - G). If the membership function - F is piecewise constant so that - F consists of a finite collection fF a of crisp a-level regions, the perimeter of - F is defined as where calculates the length of the kth arc along which the discontinuity between the a-level regions with membership degrees a i and a j occurs and where this length is weighted by the absolute difference of a i and a j . 4.1.3 Metric Operations on Fuzzy Lines For fuzzy lines we consider operations g with the signature l n g be a fuzzy line. We start with the operation length measuring the size of - L and first determine the length of a simple fuzzy line - l. We know that the membership function of - l is given as - l just yields the support of - l, i.e., supp( - l ([0;1]). Hence RR l) F -y - 1- F dx dy Consequently, we obtain Another operation is strength which follows the principle that "a line is as strong as its weakest link". It computes the minimum membership value of a fuzzy line and is thus defined as 4.2 Fuzzy-Valued Metric Operations Another interpretation of metric operations on fuzzy spatial objects is that they yield a fuzzy numerical value as a result. This accords with the fuzzy character of the operand objects. We first explain the concept of fuzzy numbers needed for a description of the metric operations afterwards. 4.2.1 Fuzzy Numbers The concept of a fuzzy number arises from the fact that many quantifiable phenomena do not lend themselves to a characterisation in terms of absolutely precise numbers. For instance, frequently our watches are somewhat inaccurate, so we might say that the time is now "around five o'clock''. Or we might estimate the age of an elder man at "nearly seventy-five years". Hence, a fuzzy number is described in terms of a central value and a linguistic modifier like nearly, around, or approximately. Intuitively, a concept captured by such a linguistic expression is fuzzy, because it includes some number values on either side of its central value. Whereas the central value is fully compatible with this concept, the numbers around the central value are compatible with it to lesser degrees. Such a concept can be captured by a fuzzy number defined on IR. Its membership function should assign the degree of 1 to the central value and lower degrees to other numbers reflecting their proximity to the central value according to some rule. The membership function should thus decrease from 1 to 0 on both sides of the central value. Formally, a fuzzy number - A is a convex normalized fuzzy set of the real line IR such that (i) 9!x A is called the central of - A is piecewise continuous. The membership function of - A can also be expressed in a more explicit form. Let where a is a piecewise continuous function increasing to 1 at point b, and g is a piecewise continuous function decreasing from 1 at point b. We introduce freal as the type of all fuzzy (real) numbers. For the representation of the fuzzy-valued result of metric operations we introduce two restricted kinds of fuzzy numbers. The first kind contains numbers that are characterized by the property that function f is lacking, i.e., these numbers only have a right-sided membership function. The second kind comprises numbers that are characterized by the property that function g is lacking, i.e., these numbers only have a left-sided membership function. 4.2.2 Metric Operations on Fuzzy Regions For fuzzy regions we now consider fuzzy-valued operations g with the signature first confine ourselves to a subset of operations g 2 farea, perimeter, height, width, diame- terg, because their result can be expressed in a generic way by a restricted fuzzy number with a right-sided membership function, as we will see. To determine the result of g we switch to the view of - F as a collection of crisp a-level regions fF a where n is possibly infinite. Since the regions F a i are crisp, we can apply the corresponding known crisp operations g c to them. The relationship between g c and the membership values a i is given in the following lemma: LEMMA 5. a ? b , g c PROOF. From the definition of a fuzzy region as a collection of a-level regions we know that a ? b , F a ' F b . Since g c is a monotonically increasing function, we obtain F a ' F b , g c We now define g( - F) as the following fuzzy number: This fuzzy number has a right-sided membership function, because for the smallest a-level region F a 1 the membership value is a and for all other a-level regions F a increasing i and thus increasing g c the membership value a i decreases from 1 to 0. In particular, the support of g( - F) does not contain any smaller values than g c F is finite, we obtain a stepwise constant and hence piecewise continuous membership function for g( - F). Other- wise, if - F) is continuous, L - F is infinite, and we get a continuous membership function for g( - F). Intuitively, this result documents the vagueness of the operator F), because with the increase of g c the certainty and knowledge about its correctness decreases. We can confirm with the membership value 1 that g c is the value of g( - F). Thus, c is the lower bound (or the core). But the value could be higher, and if so, we can only confirm it with a lower membership value. The membership value here indicates the degree of imprecision of the operation g. Let now g 2 felongatedness;roundnessg. These two operations cannot be treated as fuzzy numbers with right-sided membership functions since they are not monotonically increasing, i.e., in general F a ' F b How to measure these two operations as fuzzy-valued numbers is currently an open issue. It is even doubtful whether they can be represented as general fuzzy numbers at all. 4.2.3 Metric Operations on Fuzzy Lines Each operation g 2 flength;strengthg on fuzzy lines has the signature L be a fuzzy line, and let fL a be a collection of crisp a-cuts where n is possibly infinite. For both operations we pursue a similar strategy as for the operations on fuzzy regions. The main difference in the definition of both operations is that length is an increasing function whereas strength is a decreasing function. Hence, analogously to Lemma 5 we can conclude that a But since L a ' L b , strength c (L a ) - strength c (L b ), we obtain a Nevertheless, we can define the value of g( - L) in the same manner for both operations as the fuzzy number For length this leads to a fuzzy number with a right-sided membership function, and for strength we obtain a fuzzy number with a left-sided membership function. 5. CONCLUSIONS AND FUTURE WORK As part of an abstract model for fuzzy spatial objects in the Euclidean space, we have defined metric operations that can have either real-valued or fuzzy-valued results. All operations considered have been unary functions. For future work we will also have to consider binary metric operations, and here we will have, in par- ticular, to deal with fuzzy distance and fuzzy direction operations. Having achieved a formal and rather complete data model for fuzzy spatial objects, we can transform it to a discrete model where we have to think about finite representations for the objects and algorithms for the operations. These are efforts that should later lead to an efficient implementation. 6. --R Fuzzy Set Theoretic Approaches for Handling Imprecision in Spatial Analysis. Natural Objects with Indeterminate Boundaries Geographic Objects with Indeterminate Boundaries. Qualitative Spatial Reasoning: A Semi-Quantitative Approach Using Fuzzy Logic Vague Regions. Spatial Data Types for Database Systems - Finite Resolution Geometry for Geographic Information Systems Uncertainty Management for Spatial Data in Databases: Fuzzy Spatial Data Types. Inclusion of New Types in Relational Database Systems. Fuzzy Sets. --TR

fuzzy sets;fuzzy-valued metric operation;fuzzy line;real-valued metric operation;fuzzy region;fuzzy spatial data type

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The Volumetric Barrier for Semidefinite Programming.

We consider the volumetric barrier for semidefinite programming, or "generalized" volumetric barrier, as introduced by Nesterov and Nemirovskii. We extend several fundamental properties of the volumetric barrier for a polyhedral set to the semidefinite case. Our analysis facilitates a simplified proof of self-concordance for the semidefinite volumetric barrier, as well as for the combined volumetric-logarithmic barrier for semidefinite programming. For both of these barriers we obtain self-concordance parameters equal to those previously shown to hold in the polyhedral case.

Introduction This paper concerns the volumetric barrier for semidefinite programming. The volumetric barrier for a polyhedral set A is an m \Theta n matrix, was introduced by Vaidya (1996). Vaidya used the volumetric barrier in the construction of a cutting plane algorithm for convex programming; see also Anstreicher (1997b, 1999a, 1999b). Subsequently Vaidya and Atkinson (1993) (see also Anstreicher (1997a)) used a hybrid combination of the volumetric and logarithmic barriers for P to construct an O(m 1=4 n 1=4 L)-iteration algorithm for a linear programming problem defined over P, with integer data of total bit size L. For m AE n this complexity compares favorably with O( mL), the best known iteration complexity for methods based on the logarithmic barrier. Nesterov and Nemirovskii (1994, Section 5.5) proved self-concordance results for the volu- metric, and combined volumetric-logarithmic, barriers that are consistent with the algorithm complexities obtained in Vaidya and Atkinson (1993). In fact Nesterov and Nemirovskii (1994) obtain results for extensions of the volumetric and combined barriers to a set of the form and C are m \Theta m symmetric matrices, and - denotes the semidefinite ordering. The set S is a strict generalization of P, since P can be represented by using diagonal matrices in the definition of S. Optimization over a set of the form S is now usually referred to as semidefinite programming; see for example Alizadeh (1995) or Vandenberghe and Boyd (1996). It is well-known (see Nesterov and Nemirovskii (1994)) that an extension of the logarithmic barrier to S obtains an m-self-concordant barrier. In Nesterov and Nemirovskii (1994) it is also shown that semidefinite extensions of the volumetric, and combined volumetric-logarithmic barrier are O( mn), and O( mn), self-concordant barriers for S, respectively. The self-concordance proofs in Nesterov and Nemirovskii (1994, Section 5.5) are extremely technical, and do not obtain the constants that would be needed to actually implement algorithms using the barriers. Simplified proofs of self-concordance for the volumetric and combined barriers for P are obtained in Anstreicher (1997a). In particular, it is shown these barriers are 225 mn, and 450 mn self-concordant barriers for P, respectively. The proofs of these self-concordance results use a number of fundamental properties of the volumetric barrier established in Anstreicher (1996, 1997a). Unfortunately, however, the analysis of Anstreicher (1997a) does not apply to the more general semidefinite constraint defining S, as considered in Nesterov and Nemirovskii (1994). With the current activity in semidefinite programming the extension of results for the volumetric and combined barriers to S is of some interest. For example, in Nesterov and Nemirovskii (1994, p.204) it is argued that with a large number of low-rank quadratic constraints, the combined volumetric-logarithmic barrier applied to a semidefinite formulation obtains a lower complexity than the usual approach of applying the logarithmic barrier directly to the quadratic constraints. The purpose of this paper is to extend the analysis of the volumetric and combined barriers in Anstreicher (1996, 1997a) to the semidefinite case. This analysis is by necessity somewhat complex, but in the end we obtain semidefinite generalizations for virtually all of the fundamental results in Anstreicher (1996, 1997a). These include: ffl The semidefinite generalization of the matrix Q(x) having where V (\Delta) is the volumetric barrier. ffl The semidefinite generalization of the matrix \Sigma, which in the polyhedral case is the diagonal matrix Representations of rV (x) and Q(x) in terms of \Sigma clearly show the relationship with the polyhedral case (see Table 1, at the end of Section 3). ffl Semidefinite generalizations of fundamental inequalities between Q(x) and the Hessian of the logarithmic barrier (see Theorems 4.2 and 4.3). ffl Self-concordance results for the volumetric, and combined, barriers identical to those obtained for the polyhedral case. In particular, we prove that these barriers are 225 mn, and 450 mn self-concordant barriers for S, respectively. The fact that we obtain self-concordance results identical to those previously shown to hold in the polyhedral case is somewhat surprising, because one important element in the analysis here is significantly different than in Anstreicher (1997a). In Anstreicher (1997a), self- concordance is established by proving a relative Lipschitz condition on the Hessian r 2 V (\Delta). This proof is based on Shur product inequalities, and an application of the Gershgorin circle theorem. The use of the Lipschitz condition is attractive because it eliminates the need to explicitly consider the third directional derivatives of the volumetric barrier. We have been unable to extend this proof technique to the semidefinite case, however, and consequently here we explicitly consider the third directional derivatives of V (\Delta). The proof of the main result concerning these third derivatives (Theorem 5.1) is based on properties of Kronecker products. Despite the fact that on this point the analytical techniques used here and in Anstreicher (1997a) are quite different, the final self-concordance results are identical. An outline of the paper follows. In the next section we briefly consider some mathematical preliminaries. The most significant of these are well-known properties of the Kronecker prod- uct, which we use extensively throughout the paper. In Section 3 we define the logarithmic, volumetric, and combined barriers for S, and state the main self-concordance theorems. The proofs of these results are deferred until Section 5. Section 4 considers a detailed analysis of the volumetric barrier for S. We first obtain Kronecker product representations for the gradient and Hessian of V (\Delta), which are then used to prove a variety of results generalizing those in Anstreicher (1996, 1997a). Later in the section the matrix \Sigma is defined, and alternative representations of rV (x) and Q(x) in terms of \Sigma are obtained (see Table 1). Section 5 considers the proofs of self-concordance for the volumetric and combined barriers. The main work here is to obtain Kronecker product representations for the third directional derivatives of V (\Delta), and then prove a result (Theorem 5.1) relating the third derivatives to Q(x). Preliminaries In this section we briefly consider several points of linear algebra and matrix calculus that will be required in the sequel. To begin, let A and B be m \Theta m matrices. We use tr(A) to denote the trace of A, and A ffl B to denote the matrix inner product Let oe A denote the vector of singular values of A, that is, the positive square roots of the eigenvalues of A T A. The Frobenius norm of A is then and the spectral norm is We say that a matrix A is positive semidefinite (psd) if A is symmetric, and has all non-negative eigenvalues. We use - to denote the semidefinite ordering for symmetric matrices: A - B if A \Gamma B is psd. For a vector is the n \Theta n diagonal matrix with Diag(v) for each i. We will make frequent use of the following elementary properties of tr(\Delta). Parts (1) and (3) of the following proposition are well-known, and parts (2) and (4) follow easily from (1) and (3), respectively. Proposition 2.1 Let A and B be m \Theta m matrices. Then 1. 2. If A is symmetric, then 3. If A and B are psd, then A ffl B - 0, and A ffl 4. If A - 0 and Let A and B be m \Theta n, and k \Theta l, matrices, respectively. The Kronecker product of A and B, denoted A\Omega B, is the mk \Theta nl block matrix whose block is a ij B, n. For our purposes it is also very convenient to define a "symmetrized" Kronecker product: For an m \Theta n matrix A, mn is the vector formed by "stacking" the columns of A one atop another, in the natural order. The following properties of the Kronecker product are all well known, see for example Horn and Johnson (1991), except for (2), which follows immediately from (1) and the definition of\Omega S . Proposition 2.2 Let A, B, C, and D be conforming matrices. Then 1. (A\Omega B)(C\Omega AC\Omega BD; 2. 3. 4. If A and B are nonsingular, then A\Omega B is nonsingular, and 5. 6. If A and B are psd, then A\Omega B is psd. Lastly we consider two simple matrix calculus results. Let X be a nonsingular matrix with (see for example Graham (1981, p.75)), @ and also (see for example Graham (1981, p.64)), @ where e i denotes the ith elementary vector. 3 Main Results Let G be a closed convex subset of ! n , and let F (\Delta) be a C 3 , convex mapping from Int(G) to !, where Int(\Delta) denotes interior. Then (Nesterov and Nemirovskii (1994)) F (\Delta) is called a #- self-concordant barrier for G if F (\Delta) tends to infinity for any sequence approaching a boundary point of G, for every x 2 sup As shown by Nesterov and Nemirovskii (1994, Theorem 3.2.1), the existence of a #-self- concordant barrier for G implies that a linear, or convex quadratic, objective can be minimized on G to within a tolerance ffl of optimality using O( iterations of Newton's method. Consider a set S ae ! n of the form where A i , and C are m \Theta m symmetric matrices. We assume throughout that the matrices fA i g are linearly independent, and that a point x with S(x) - 0 exists. It is then easy to show that 0g. The logarithmic barrier for S is the function defined on the interior of S. As shown by Nesterov and Nemirovskii (1994, Proposition 5.4.5), f(\Delta) is an m-self-concordant barrier for S, implying the existence of polynomial-time interior-point algorithms for linear, and convex quadratic, semidefinite programming. The volumetric barrier V (\Delta) for S, as defined in Nesterov and Nemirovskii (1994, Section 5.5), is the function The first main result of the paper is the following improved characterization of the self- concordance of V (\Delta). Theorem 3.1 each A i , and C, are m \Theta m symmetric matrices. Then 225 m 1=2 V (\Delta) is a #-self-concordant barrier for S, for Theorem 3.1 generalizes a result for the polyhedral volumetric barrier (Anstreicher (1997a, Theorem 5.1)), and provides an alternative to the semidefinite self-concordance result of Nesterov and Nemirovskii (1994, Theorem 5.5.1). It is worthwhile to note that in fact the analysis in Nesterov and Nemirovskii (1994, Section 5.5) does not apply directly to the barrier V (\Delta) for S as given here, because Nesterov and Nemirovskii assume that the "right-hand side" matrix C is zero. In practice, this assumption can be satisfied by extending S to the cone and then intersecting K with the linear constraint x recover S. The analysis in Nesterov and Nemirovskii (1994) would then be applied to the volumetric barrier - V (\Delta) for K. The advantage of working with K is that some general results of Nesterov and Nemirovskii can then be applied, because - V (\Delta) is (n 1)-logarithmically homogenous; see Nesterov and Nemirovskii (1994, Section 2.3.3). (For example Theorem 4.4, required in the analysis of Section 5, could be replaced by the fact that r - and Nemirovskii (1994, Proposition 2.3.4).) Our analysis shows, however, that the homogeneity assumption used in Nesterov and Nemirovskii (1994) is not needed to prove self-concordance for the semidefinite volumetric barrier. The combined volumetric-logarithmic barrier for S is the function where V (\Delta) is the volumetric barrier, f(\Delta) is the logarithmic barrier, and ae is a positive scalar. The combined barrier was introduced for polyhedral sets in Vaidya and Atkinson (1993), and extended to semidefinite constraints in Nesterov and Nemirovskii (1994). Our main result on the self-concordance of V ae (\Delta) is the following. Theorem 3.2 Let each A i , and C, are m \Theta m symmetric matrices. Assume that n ! m, and let ae (\Delta) is a #-self-concordant barrier for S, for Theorems 3.1 and 3.2 imply that if m AE n, then the self-concordance parameter # for the volumetric or combined barrier for S (particularly the latter) can be lower than m, the parameter for the logarithmic barrier. It follows that for m AE n the complexity of interior-point algorithms for the minimization of a linear, or convex quadratic, function over S may be improved by utilizing V (\Delta) or V ae (\Delta) in place of f(\Delta). 4 The Volumetric Barrier Let f(\Delta) be the logarithmic barrier for S, as defined in the previous section. It is well known (see for example Vandenberghe and Boyd (1996)) that the first and second partial derivatives of f(\Delta) at an interior point of S are given by: where throughout we use possible to reduce notation. Let A be the m 2 \Theta n matrix whose ith column is where the second equality uses Proposition 2.2 (5), the Hessian matrix can be represented in the form (see Alizadeh (1995)) Note that positive definite under the assumptions that and that the matrices fA i g are linearly independent. Our first goal in this section is to obtain Kronecker product representations for the gradient and Hessian of V (\Delta). To start, it is helpful to compute where the second equality uses (2). In addition, using (4) and the definitions of\Omega and\Omega S , it is easy to see that @ \Gamma1\Omega Now applying the chain rule, (1), and (5), we find that '- A \Gamma1\Omega A \Gamma1\Omega \Gamma1=2\Omega S I where the last equality uses Proposition 2.2 (1), S \Gamma1=2 is the unique positive definite matrix having (S \Gamma1=2 \Gamma1=2\Omega S \Gamma1=2 ]A A T [S A T [S is the orthogonal projection onto the range of [S \Gamma1=2\Omega S \Gamma1=2 ]A. Note that the jth column of [S \Gamma1=2\Omega S \Gamma1=2 ]A is exactly [S using Proposition 2.2 (5). It follows that P is a representation, as an m 2 \Theta m 2 matrix, of the projection onto the subspace of R m\Thetam spanned by fS \Gamma1=2 A j S \Gamma1=2 ng. We will next compute the second partial derivatives of V (\Delta). To start, using (2) and (5), we obtain \Gamma1\Omega \Gamma1\Omega Also, using (4) and the definition @ \Gamma1\Omega '\Omega \Gamma1\Omega j\Omega \Gamma1\Omega Combining (6), (9), and (10), and using Proposition 2.1 (2), we obtain are the n \Theta n matrices having \Gamma1\Omega \Gamma1\Omega \Gamma1\Omega \Gamma1\Omega Theorem 4.1 For any x having \Gamma1\Omega 2\Omega I \Gamma1\Omega \Gamma1\Omega Note that from Proposition 2.1 (3) and Proposition 2.2 (6) we immediately have - T Q- 0 are all psd. Since - is arbitrary, it follows that Q - 0, T - 0. In addition, the fact that P is a projection implies that 2\Omega where the last equality uses Proposition 2.2 (2). Applying Proposition 2.1 (4), we conclude that 2\Omega which is exactly is arbitrary, we have shown that T - (1=2)(Q +R), which together with (11), Q - 0, and T - 0 implies that To complete the proof we must show that R(x) - Q(x). Let - i , eigenvectors of - B, with corresponding eigenvalues - i , m. Then (see Horn and Johnson (1991, Theorem 4.4.5)) - 2\Omega S I has orthonormal eigenvectors - with corresponding eigenvalues (1=2)(- 2 Johnson (1991, Theorem B has the same eigenvectors - corresponding eigenvalues - i - j . It then follows from (- 2\Omega and Proposition 2.1 (4) then implies that P ffl ( - 2\Omega B), which is exactly is arbitrary we have shown that Q - R, as required. 2 Theorem 4.1 generalizes a similar result (Anstreicher (1996, Theorem A.4)) for the polyhedral volumetric barrier. It follows from Theorem 4.1 that V (\Delta) is convex on the interior of S. In the next theorem we demonstrate that in fact V (\Delta) is strictly convex. Theorem 4.2 is also a direct extension of a result for the polyhedral volumetric barrier; see Anstreicher (1996, Theorem A.5). Theorem 4.2 Let x have S(x) - 0. Then Q(x) - (1=m)H(x). are the eigenvalues of - B, with corresponding orthonormal eigenvectors As described in the proof of Theorem 4.1, - 2\Omega I then has a full set of orthonormal eigenvectors - corresponding eigenvalues (1=2)(- 2 It follows from (13) that On the other hand, P B) implies that together imply that - T Q- k-k, from which it follows that - T Q- (1=m)k-k For a given - the conclusion of Theorem 4.2 is that A strengthening of (23), using j - Bj in place of k - Bk, is a key element in our analysis of the self-concordance of V (\Delta), in the next section. The next theorem gives a remarkably direct generalization of a result for the polyhedral volumetric barrier; see Anstreicher (1996, Proposition 2.3). Theorem 4.3 Let x have be orthonormal eigenvectors of - B, with corresponding eigenvalues B) can be written P j we have loss of generality (scaling - as needed, and re-ordering indeces) we may assume that 1. Then (24) implies that i , from (21), we are naturally led to consider the optimization problem e: For fixed the constraint in (25) implies that so the objective value in (25) can be no lower than A straightforward differentiation shows that the minimal value for (26) occurs when v 2 1= m, and the value is We have thus shown that if j - Next we will obtain alternative representations of rV (x) and Q(x) that emphasize the connection between the semidefinite volumetric barrier and the volumetric barrier for a polyhedral set. For fixed x with the linear span of fU ng is equal to the span of f - ng. (Such fU i g may be obtained by applying a Gram-Schmidt procedure to f - U be the m 2 \Theta n matrix whose ith column is vec(U i ), and let k . Then from (8), can be written in the form It follows, from (7), that A A A A Similarly, from (12) we have A A A The characterizations of rV (x) and Q(x) given in (27) and (28) are very convenient for the proof of the following theorem, which will be required in the analysis of self-concordance in the next section. Theorem 4.4 Let x have Proof: From (28) we have using Proposition 2.2 (5). Letting - A be the m 2 \Theta n matrix whose ith column is vec( - can then write A T In addition, it follows from (27) that A T vec(\Sigma): (30) Combining (29) and (30), we obtain A A T A A T vec(\Sigma) A A T A A T because k , and tr(U 2 each k, by construction. 2 One final point concerning the matrix \Sigma is the issue of uniqueness, for a given Since \Sigma is defined above in terms of fU i g, and the fU i g are not unique, it is not at all obvious that \Sigma is unique. We will now show that \Sigma is invariant to the choice of fU i g, and is therefore unique. To see this, note that by definition denotes the ith column of U k (recall that each U k is symmetric by construction). denote the ith elementary vector, and let I be an m \Theta m identity matrix. By inspection [e i\Omega I] T U is then the m \Theta n matrix whose kth column is (U k ) i . It follows from (31) that j\Omega I][e where P is the projection from (8). Since P is uniquely determined by fA i g and is also unique, as claimed. In the following table we give a summary of first and second order information for the logarithmic and volumetric barriers, for polyhedral and semidefinite constraints. For the polyhedral case we have A is an m \Theta n matrix whose ith column is a i . Given x with be the vector whose components are those of the diagonal of P , and the volumetric barrier, in both the polyhedral and semidefinite cases, the matrix Q satisfies 3Q(x). Note that, as should be the case, all formulas for the semidefinite case also apply to the polyhedral case, with - Table 1: Comparison of Logarithmic and Volumetric Barriers Polyhedral Semidefinite Logarithmic Volumetric 5 Self-concordance In this section we obtain proofs for the self-concordance results in Theorems 3.1 and 3.2. We begin with an analysis of the third directional derivatives of V (\Delta). Let x have S(x) - 0, and Using (4), (9), and (13), we immediately obtain @ \Gamma1\Omega \Gamma1\Omega [S \Gamma1\Omega . Moreover, from (4) it is immediate that @ [S \Gamma1\Omega j\Omega \GammaS \Gamma1\Omega Combining (32) and (33), and using Proposition 2.1 (2), we conclude that the first directional derivative of - T Q(x)-, in the direction -, is given by @ \Gamma1\Omega \Gamma1\Omega \Gamma3AH \Gamma1\Omega \GammaAH \Gamma1\Omega 2\Omega 3\Omega 2\Omega and P is defined as in (8). Very similar computations, using (14) and (15), result in 2\Omega 2\Omega Combining (11) with (34), (35), and (36), we obtain the third directional derivative of V (\Delta): Theorem 5.1 Let x have Proof: Using the fact that 2\Omega 3\Omega 2\Omega from Proposition 2.2 (2), (37) can be re-written as 2\Omega 2\Omega We will analyze the two terms in (38) separately. First, from (17) we have 2\Omega 2\Omega B]: Using (18), and the similar relationship [ - 2\Omega S I], it follows that 2\Omega 2\Omega 2\Omega be the eigenvalues of - B. Then (see Horn and Johnson (1991, Theorem 4.4.5)) the eigenvalues of [ - are of the form (1=2)(- Bj I Using (39), (40), the fact that [ - 2\Omega Proposition 2.1 (4), we then obtain 2\Omega 2\Omega In addition, the fact that [ - 2\Omega have the same eigenvectors implies that 2\Omega 2\Omega 2\Omega and therefore fi fi fi fi 2\Omega 2\Omega The proof is completed by combining (38), (41), (42), and (13). 2 Using Theorem 5.1 we can now proof the first main result of the paper, characterizing the self-concordance of V (\Delta). Proof of Theorem 3.1: The fact that V (x) !1 as x approaches the boundary of S follows from (3), and the fact that S(x) is singular on the boundary of S. Combining the results of Theorems 4.3 and 5.1, we obtain using the fact that - T Q(x)- T r 2 V (x)[-], from Theorem 4.1. In addition, (see Horn and Johnson (1985, Corollary 7.7.4)), so Theorem 4.4 implies that The proof is completed by noting the effect on (43) and (44) when V (\Delta) is multiplied by the Next we consider the self-concordance of the combined volumetric-logarithmic barrier V ae (\Delta), as defined in Section 3. We begin with some well-known properties of the logarithmic barrier f(\Delta). Lemma 5.2 Let x have Proof: From (3) and (5) we obtain \Gamma1\Omega \Gamma1=2\Omega A for each i. It follows easily from (45) that A and therefore It is then immediate from the fact that j - Bj (see the proof of Theorem 5.1) that where the final equality uses (19). 2 It follows from Lemma 5.2, the fact that j - Bk, and that f(\Delta) is an m-self-concordant barrier for S, as shown by Nesterov and Nemirovskii (1994). Using Lemma 5.2 we immediately obtain the following generalization of Theorem 5.1. Corollary 5.3 Let x have Proof: Combining Theorem 5.1 and Lemma 5.2, we obtain Next we require a generalization of Theorem 4.3 that applies with Q(x)+aeH(x) in place of Q(x). The following theorem obtains a direct extension of a result for the polyhedral combined barrier (Anstreicher (1996, Theorem 3.3)) to the semidefinite case. To prove the theorem we will utilize the matrices fU k g, as defined in Section 4, to reduce the theorem to a problem already analyzed in the proof of Anstreicher (1996, Theorem 3.3). Theorem 5.4 Let x have be orthonormal eigenvectors of - B, with corresponding eigenvalues . By the definition of fU k g, there is a vector - so that and therefore k - -k. It follows from (46) that for each and therefore -, where W is the m \Theta n matrix with Let U be the m 2 \Theta n matrix whose kth column is vec(U k ), and let V be the m 2 \Theta m matrix whose ith column is - From (47) we then can then write the ith row of W . Then where P is the projection matrix from (8). Using (19), (21), and (48) we then have (w T Moreover it is clear that (w T -k1 . We are now exactly in the structure of the proof of Anstreicher (1996, Theorem 3.3), with U of that proof replaced by the matrix W . In that proof it is shown that the solution objective value of the problem min (w T (w T can be no lower thanq It follows that - T (Q(x) The final ingredient needed to prove the self-concordance of V ae (\Delta) is the following simple generalization of Theorem 4.4. Theorem 5.5 Let x have Proof: From the representations in Table 1 we easily obtain A: Let It follows that ae ae A A[I\Omega A A T [I\Omega \Sigma 1=2 ae ae ae Using the above results we can now prove the second main result of the paper, characterizing the self-concordance of the combined volumetric-logarithmic barrier for S. Proof of Theorem 3.2: Combining the results of Corollary 5.3 and Theorem 5.4, with mp (D using the fact that - T Q(x)- T r 2 V (x)[-], from Theorem 4.1. In addition, (1985, Corollary 7.7.4)), so Theorem 5.5 implies that The proof is completed by noting the effect on (49) and (50) when V ae (\Delta) is multiplied by the --R Interior point methods in semidefinite programming with applications to combinatorial optimization. Large step volumetric potential reduction algorithms for linear pro- gramming Volumetric path following algorithms for linear programming. On Vaidya's volumetric cutting plane method for convex program- ming Towards a practical volumetric cutting plane method for convex programming. Ellipsoidal approximations of convex sets based on the volumetric barrier. Kronecker Products and Matrix Calculus: with Applications Matrix Analysis Topics in Matrix Analysis A new algorithm for minimizing convex functions over convex sets. A technique for bounding the number of iterations in path following algorithms. programming. --TR

self-concordance;volumetric barrier;semidefinite programming

355443

General Interior-Point Maps and Existence of Weighted Paths for Nonlinear Semidefinite Complementarity Problems.

Extending the previous work of Monteiro and Pang (1998), this paper studies properties of fundamental maps that can be used to describe the central path of the monotone nonlinear com-plementarity problems over the cone of symmetric positive semidefinite matrices. Instead of focusing our attention on a specific map as was done in the approach of Monteiro and Pang (1998), this paper considers a general form of a fundamental map and introduces conditions on the map that allow us to extend the main results of Monteiro and Pang (1998) to this general map. Each fundamental map leads to a family of "weighted" continuous trajectories which include the central trajectory as a special case. Hence, for complementarity problems over the cone of symmetric positive semidefinite matrices, the notion of weighted central path depends on the fundamental map used to represent the central path.

Introduction section1 In a series of recent papers (see Kojima, Shida and Shindoh 1997, 1998, 1999, Kojima, Shindoh and Hara 1997, Shida and Shindoh 1996, Shida, Shindoh and Kojima 1997, 1998), Hara, Kojima, Shida and Shindoh have introduced the monotone complementarity problem in symmetric matrices, studied its properties, and developed interior-point methods for its solution. A major source where this problem arises is a convex semidefinite program which has in the last couple years attracted a great deal of attention in the mathematical programming literature (see for example Alizadeh 1995, Alizadeh, Haeberly and Overton 1997, 1998 Luo, Sturm and Zhang 1996, 1998, Monteiro 1997, 1998, Monteiro and Tsuchiya 1999, Monteiro and Zhang 1998, Nesterov and Nemirovskii 1994, Nesterov This work has been partly supported by the National Science Foundation under grants CCR-9700448, CCR-9902010 and INT-9600343. y School of Industrial and Systems Engineering, Georgia Tech, Atlanta, GA 30332. (monteiro@isye.gatech.edu) and Todd 1997, 1998, Potra and Sheng 1998, Ramana, Tun-cel and Wolkowicz 1997, Shapiro 1997, Vandenberghe and Boyd 1996, Zhang 1998). denote the m-dimensional real Euclidean space, S n denote the space of n \Theta n real symmetric matrices, S n ++ denote the subsets of S n consisting of the positive semidefinite and positive definite matrices, respectively, and C(-) j We denote the closure of a subset E of a metric space by cl E. Given a continuous map the complementarity problem which we shall study in this paper is to find a triple (X; Y; z) 2 S n \Theta S n \Theta ! m such that eq:psd cp F (X; Y; It is known (see the cited references) that there are several equivalent equations to represent the complementarity condition (X; Y problem. Associated with each of these equivalent equations, we can define an interior-point map that not only provides the foundation for developing path-following interior point methods for solving problem (1) but also serves to generalize the notion of weighted central path from linear programming to the context of the complementarity problem (1). In the work of Monteiro and Pang (1998), they focus on the equivalent complementarity equation (XY +Y several properties of the associated interior-point map. The main goal of this paper is to generalize the work of Monteiro and Pang (1998) to other equivalent complementarity equations having the general form \Phi(X; Y is a continuous map such that: ++ \Theta S n In terms of \Phi, problem (1) becomes equivalent to finding a triple (X; Y; z) 2 D \Theta ! m such that H(X; Y; m is the fundamental interior-point map (associated with \Phi) defined by F (X; Y; z) For the map \Phi(X; Y and the open convex cone ++ , Monteiro and Pang (1998) establish under some monotonicity conditions on the map F that the system sys.H H(X; Y; A have the following properties: (P1) it has a solution for every (A; B) 2 cl V \Theta F++ , where F++ j F (S n ++ \Theta S n (P2) the solution, denoted (X(A; B); Y (A; B); z(A; B)), is unique when (A; if a sequence converges to a limit (A1 ; B1 then the sequence converges to (X(A1 ; B1 if a sequence converges to a limit (A1 cl V \Theta F++ , then the sequence ))g is bounded, and that any of its accumulation point Note that when 0 2 F++ , (P1) implies the existence of a solution of (1), and (P2) implies the well-definedness of the weighted central paths passing through points in V , that is paths consisting of the unique solutions of system (3) with is a fixed point in V . Under appropriate conditions on the map \Phi and the open convex cone V , we show in this paper that the above properties also hold with respect to the general interior-point map (2). We also illustrate how our general framework applies to the following specific central-path maps: where LX is the lower Cholesky factor of X , U Y is the upper Cholesky factor of Y and W is the unique symmetric positive definite matrix such that be easily verified to be a special case of our general framework using the results of Monteiro and Pang 1998.) Observe that the third and fourth maps are only defined for points (X; Y ) in the set ++ \Theta S n ++ ), and hence they illustrate the need for considering maps \Phi whose domain are not the whole set S n . The approach of this paper is based on the theory of local homeomorphic maps. One of the preliminary steps of our analysis is to establish that H restricted to the set U \Theta ! m is a local homeomorphism, where U (V). For this property to hold for the above cited maps, it is necessary to choose the set V to be an open convex cone smaller than S n ++ , which is the cone used in connection with the map \Phi(X; Y in the analysis of Monteiro and Pang (1998). In fact, the sets V associated with the four maps above are appropriate conical neighborhoods of the line contained in S n ++ . Sturm and Zhang (1996) define a notion of weighted center for semidefinite programming based on the central path map \Phi(X; Y denotes the eigenvalues of XY arranged in nondecreasing order. According to their definition, given a vector w ++ such that w a point (X; Y; z) 2 S n ++ \Theta S n is an w-weighted center if F (X; Y; For linear maps F associated with semidefinite programming problems, they show the existence of an w-center for any such w. However, their w-center is not unique and hence does not lead to the notion of weighted central paths as our approach does. Finally, we observe that interior-point algorithms for solving the complementarity problem (1) which makes use of the fundamental interior-point map (2) have been proposed in Wang et al. (1996) and Monteiro and Pang (1999). This paper is organized as follows. Section 2 contains an exposition of the main results of this paper whose proofs are given in the subsequent sections. Section 2 is divided into three subsections. Subsection 2.1 summarizes some important definitions and facts from the theory of local homeomorphic maps. Subsection 2.2 introduces the key conditions on the map F and the pair (\Phi; V), and gives a few examples of pairs (\Phi; V) which satisfy these conditions. In Subsection 2.3, we state the main result of this paper and some of its consequences, including its specialization to the context of the convex nonlinear semidefinite programming problem. In Section 3, we provide the proof of the main result. Finally, in Section 4, we develop some technical results which allow us to verify that the pairs introduced in Section 2 satisfy our basic assumptions. The following notation is used throughout this paper. The symbols - and - denote, respectively, the positive semidefinite and positive definite ordering over the set of symmetric matrices; that is, for positive semidefinite, and X - Y (or Y OE X) means positive definite. The set of all n \Theta n matrices with real entries is denoted by M n . Let M n and M n ++ denote the set of matrices ? denote the subspace of M n consisting of the skew-symmetric matrices. For A 2 M n , let A ii denote the trace of A. For inner product in M n . For any matrix A 2 M n , let - is an eigenvalue of A T Ag . For a matrix A 2 M n with all real eigenvalues, we denote its smallest and largest eigenvalues by - min (A) and - max (A), respectively. Finally, define the sets Main Assumptions and Results section2 In this section, we state the assumptions imposed on the map F and the pair (\Phi; V) and give a few examples of pairs (\Phi; V) satisfying the required conditions. We also state the main result and its consequences, including its specialization to the context of the convex nonlinear semidefinite programming problem. This section is divided into three subsections. The first subsection summarizes a theory of local homeomorphic maps defined on metric spaces; the discussion is very brief. We refer the reader to Section 2 and the Appendix of Monteiro and Pang (1996), Chapter 5 of Ortega and Rheinboldt (1970), and Chapter 3 of Ambrosetti and Prodi (1993) for a thorough treatment of this theory. The second subsection introduces the key conditions on the map F and the pair (\Phi; V), and gives a few examples of pairs (\Phi; V) which satisfy these conditions. The third subsection states the main result and its corollaries. 2.1 Local homeomorphic maps subsection2.1 This subsection summarizes some important definitions and facts from the theory of local homeomorphic maps. If M and N are two metric spaces, we denote the set of continuous functions from M to N by C(M;N) and the set of homeomorphisms from M onto N by Hom(M;N ). For G 2 C(M;N ), Eg. The set simply denoted by G \Gamma1 (v). Given G 2 C(M;N ), that G(D) ' E, the restricted map ~ defined by ~ denoted by then we write this ~ G simply as GjD . We will also refer to Gj (D;E) as "G restricted to the pair (D; E)", and to GjD as "G restricted to D". The closure of a subset E of a metric space will be denoted by cl E. Any continuous function from a closed interval of the real line ! into a metric space will be called a path. We say that partition of the set V if space M is said to be connected if there exists no partition of M for which both O 1 and O 2 are non-empty and open. In the rest of this subsection, we will assume that M and N are two metric spaces and that 1 The map G 2 C(M;N) is said to be proper with respect to the set E ' N if G M is compact for every compact set K ' E. If G is proper with respect to N , we will simply say that G is proper. Our analysis rely upon the following two well-known results. The first one is a classical topological result whose proof can be found in Chapter 3 of Ambrosetti and Prodi (1993) and in the Appendix of Monteiro and Pang (1996). Proposition 1 main Assume that G : M ! N is a local homeomorphism. If N is connected G is proper if and only if the number of elements of G \Gamma1 (v) is finite and constant for v 2 N . The next result is derived in Section 2 of Monteiro and Pang (1996) and its proof is an immediate consequence of classical topological results. Proposition 2 cor1 Let M 0 ' M and N 0 ' N be given sets satisfying the following conditions: is a local homeomorphism and ; . Assume that G is proper with respect to some set E such that N 0 ' E ' N . Then G restricted to the local homeomorphism. If, in addition, N 0 is connected, then G(M 0 cl N 0 . 2.2 Fundamental assumptions and examples of central-path maps subsection2.2 In this subsection, we state the conditions that will be imposed on the map F and the pair (\Phi; V) during our analysis of the properties of the fundamental map (2). We also give a few examples of pairs (\Phi; V) which satisfy the required conditions. We start by giving a few definitions. defined on a subset dom (J) of M n \Theta M n \Theta ! m is said to be (X; Y )-equilevel-monotone on a subset H ' dom (J) if for any (X; Y; z) 2 H and dom (J), we will simply say that J is (X; Y )-equilevel-monotone. In the following two definitions, we assume that W , Z and N are three normed spaces and that /(w; z) is a function defined on a subset of W \Theta Z with values in N . z-boundedness The function /(w; z) is said to be z-bounded on a subset H ' dom (/) if for every sequence f(w k ; z k )g ' H such that fw k g and f/(w k ; z k )g are bounded, the sequence fz k g is also bounded. When dom (/), we will simply say that / is z-bounded. Definition 4 z-injective The function /(w; z) is said to be z-injective on a subset H ' dom (/) if the following implication holds: (w; z) dom (/), we will simply say that / is z-injective. We now state the main assumptions that will be made on the map F and the pair (\Phi; V). Assumption 1 ass1 The monotone, z-injective on S n ++ \Theta S n Assumption 2 ass2 We impose the following conditions on the pair (\Phi; V): is a continuous map such that C(0) [ (S n ++ \Theta S n is an open connected set such that (b) for any (X; Y ) 2 D and - 0, we have is a continuous strictly increasing function such that (c) if the sequence f(X k ; Y k )g ' D is such that f\Phi(X k ; Y k )g is bounded, then fX k ffl Y k g is (d) the set U j \Phi \Gamma1 (V) is contained in S n ++ \Theta S n is a closed set; (e) \Phi is differentiable on S n ++ \Theta S n ++ and for every (X; Y the following implication holds: We now make a few remarks regarding the above assumptions. First, condition (a) is stated so as to cover some relevant examples of maps \Phi which are defined only in the set C(0) [ (S n ++ \Theta S n (see Examples 4 and 5 below). Second, Assumption 2(c) implies that the map ' is onto, and hence that its inverse is defined everywhere in !+ . Third, conditions (a), (b) and (d) of Assumption 2 imply that U is a nonempty open set. Fourth, condition (b) implies that the curve consisting of the solutions of the systems eee as - ? 0 varies, is a parametrization of the central path associated with the complementarity problem (1), that is the set of solutions of the systems condition (e) is probably the most crucial one. Under Assumption 1, this condition is equivalent to the well-definedness of the Newton direction with respect to system (4) for any point (X; Y; z) in U . We next give some examples of pairs (\Phi; V) which satisfy Assumption 2. Example 1: Let D j S n be the map defined by \Phi(X; Y ++ . Example 2: Let D j S n be the map defined by \Phi(X; Y Example 3: Let D j S n be the map defined by \Phi(X; Y Example 4: Let D ++ \Theta S n be the map defined by ++ \Theta S n where LX is the lower Cholesky factor of X , that is LX is the unique lower triangular matrix with positive diagonal elements satisfying Example 5: Let D ++ \Theta S n be the map defined by ++ \Theta S n where LX is the lower Cholesky factor of X and U Y is the upper Cholesky factor of Y , that is U Y is the unique upper triangular matrix with positive diagonal elements satisfying U Y U T Example ++ \Theta S n be the map defined by ++ \Theta S n is the unique symmetric positive definite matrix satisfying The map of Example 1 was extensively studied in Monteiro and Pang (1998), and is a special case of the general framework studied in this paper. The verification that these examples satisfy Assumption 2 will be given in Section 4. The next result shows that each map above generates a whole family of maps that satisfy Assumption 2. Proposition 3 scale-family Let (\Phi; V) be a pair satisfying Assumption 2. Then: i) for every ff ? 0, the pair (ff\Phi; ffV) satisfies Assumption 2; ii) for every nonsingular matrix P , the pair (\Phi P ; V) satisfies Assumption 2, where \Phi S n is the map defined as \Phi P (X; Y Proof. The proof is just a simple verification. In all six examples above, the domain D P of the map \Phi P remains invariant, that is D every P . For a pair (V ; \Phi) and a nonsingular matrix P , let U P (V). In general, U P may differ from U , but for the maps of Examples 2 and 4, we have U every nonsingular matrix P . Moreover, the maps \Phi P corresponding to Examples 1-6 become e e Y e Y respectively, where L e X is the lower Cholesky factor of e Y is the upper Cholesky factor of e In the last equation, we used the fact that f is the unique symmetric matrix such that f Y . 2.3 The main result and its consequences subsection2.3 In this subsection we state the main result of this paper. We also state some of its consequences, including its specialization to the context of the convex nonlinear semidefinite programming problem. The main result extends Theorem 2 of Monteiro and Pang (1998) to fundamental maps (2) associated with pairs (\Phi; V) satisfying Assumption 2. It essentially states that the fundamental map (2) has some nice homeomorphic properties. Theorem 1 main2 Assume that F : S n Assumption 1 and the Assumption 2. Then the following statements hold for the map H: (a) H is proper with respect to cl V \Theta F++ ; (b) H maps U \Theta ! m homeomorphically onto V \Theta F++ , where U cl V \Theta F++ . Theorem 1 establishes the claimed properties (P1)-(P4) of the map H stated in the Introduction. Indeed, (P1) follows from conclusion (c); (P2) and (P3) follow from (b); and (P4) follows from (a). In what follows, we give two important consequences of the above theorem, assuming that 0 2 F++ . The first one, Corollary 1, has to do with the central path for the semidefinite complementarity problem (1); the second one, Corollary 2, is a solution existence result for the same problem. Corollary 1 co:path existence Suppose that F : S n Assumption 1 and the pair (\Phi; V) satisfies Assumption 2. Assume that (0; cl V \Theta F++ and let be paths such that P every t 2 (0; 1]. Then there exist (unique) paths ++ such that Moreover, every accumulation point of (X(t); Y (t); z(t)) as t tends to 0 is a solution of the complementarity problem (1). Corollary 2 co:solution existence Assume that F : S n Assumption 1 and the pair (\Phi; V) satisfies Assumption 2. If there exists (X that F (X cl V, the system cl U has a solution, which is unique when A 2 V. We now discuss the specialization of Theorem 1 to the context of the convex nonlinear semidefinite program. The complementarity problem (1) arises as the set of first-order necessary optimality conditions for the following nonlinear semidefinite program (see for example Shapiro 1997): eq:psd nlp minimize '(x) subject to G(x) 2 \GammaS n are given smooth maps. Indeed, it is well-known (see for example Shapiro 1997) that, under a suitable constraint qualification, if x is a local optimal solution of the semidefinite program, then there must exist j such that is the Lagrangian function defined by Letting r x L(x; U; we see that the first-order necessary optimality conditions for problem (5) are equivalent to the complementarity problem (1) with The is said to be positive semidefinite convex (psd-convex) if The following result follows as an immediate consequence of Theorem 1 and the results of Section 5 of Monteiro and Pang (1996b). Theorem 2 properties of F Suppose that (\Phi; V) is a pair satisfying Assumption 1, the function continuously differentiable and convex, G continuously differentiable and is an affine function such that the (constant) gradient matrix rh(x) has full column rank and the feasible set X j fx any one of the following conditions holds: (a) ' is strictly convex; (b) for every U 2 S n ++ , the map x strictly convex; (c) each G ij is an analytic function, then the following statements hold for the maps F and H given by (8) and (2), respectively: (i) H is proper with respect to cl V \Theta F (S n ++ \Theta S n homeomorphically onto V \Theta F (S n ++ \Theta S n cl V \Theta F (S n ++ \Theta S n (iv) the set F (S n ++ \Theta S n Proof. The assumptions of the theorem together with Proposition 4, Lemmas 7, 8 and 9 of Monteiro and Pang (1996b) imply that the map F given by (8) is (U; V )-equilevel-monotone on S n (x; j)-injective on S n ++ \Theta S n j)-bounded on S n Hence, the conclusions (i), (ii) and (iii) of the theorem follows directly from Theorem 1. It was shown in Theorem 4(iii) of Monteiro and Pang (1996b) that the set F ( e U \Theta ! m+p ) is convex, where e ++ \Theta S n g. Since C(-) ' e U ++ \Theta S n ++ for all - ? 0, it follows from Lemma 1(b) that U \Theta ! m+p ++ \Theta S n holds. 3 Proof of the Main Results section3 The goal of this section is to give the proof of Theorem 1. In the process of doing this, we will first establish Theorem 1 for the case in which the map F is affine. Apart from the fact that an affine considerably simplifies the analysis, we obtain through this case an important technical result (Lemma 5) which plays an important role in establishing Theorem 1 for the case of a nonlinear map F satisfying Assumption 1. We first establish two important technical lemmas that hold for nonlinear maps F satisfying Assumption 1. m be a map satisfying Assumption 1. Then, the following statements hold: (a) F restricted to S n ++ \Theta S n is an open map; in particular, F++ is an open set; Proof. We first establish (a). Let ~ m be defined by ~ F (X; Y; . Using the fact that F satisfies Assumption 1, it is easy to see that ~ F is (X; Y )-equilevel-monotone, z-injective on M n ++ \Theta S n Then, it follows from Lemma 11 of Monteiro and Pang (1998) that the map ~ defined by ~ ~ F (X; Y; z) maps M n ++ \Theta S n homeomorphically onto ~ ++ \Theta S n By the domain invariance theorem, this implies that ~ H , and hence ~ F , restricted to M n ++ \Theta S n is an open map. This conclusion together with a simple argument shows that F restricted to S n ++ \Theta S n is also an open map. We now show (b). Clearly, F (C(-) To prove the other inclusion, let B be an arbitrary element of F++ . Since by Corollary 3 of Monteiro and Pang (1998), the system F (X; Y; has a (unique) solution in S n ++ \Theta S n conclude that A consequence of Lemma 1(b) is as follows. Assumption 2(a) implies the existence of such that - 0 I 2 V . This together with (b) and (d) of Assumption 2 imply that C(' ++ \Theta S n ++ . Hence, in view of Lemma 1(b), it follows that F (U \Theta ! m properness m be a map satisfying Assumption 1, (\Phi; V) be a pair satisfying Assumption 2, and m be the map defined in (2). Then, H is proper with respect to cl V \Theta F++ . Proof. Let K be a compact subset of cl V \Theta F++ . We will show that H \Gamma1 (K) is compact from which the result follows. We first show that H \Gamma1 (K) is closed. Since K is closed and H is continuous, closed with respect to dom Hence, the closeness of H \Gamma1 (K) follows if we show that H \Gamma1 (K) is contained in a closed subset of dom (H). Indeed, the definition of H implies that H \Gamma1 (K) is contained in \Phi \Gamma1 (cl V) \Theta ! m , which by Assumption 2(d), is a closed set. We next show that H \Gamma1 (K) is bounded. Indeed, suppose for contradiction that there exists a sequence is compact and we may assume without loss of generality that there exists F 1 2 F++ such that Clearly, we have F ++ \Theta S n ++ and Y 1 - 0, there exists j ? 0 such that the set ++ \Theta S n contains clearly have that N1 is open, and using Lemma 1(a), it follows that is an open set. Thus, by (10) and the fact that F 1 2 F (N1 ), we conclude that for all k sufficiently large, say k This inequality together with fact that ( e imply for every k - k 0 . Using the fact that fH(X k ; Y k ; z k )g ' K and K is bounded, we conclude that Using Assumption 2(c) we have that fX k ffl Y k g is bounded. This fact together with the above inequality implies that the sequences fX k g and fY k g are bounded. Since must have lim k!1 kz k 1. From the fact that F is z-bounded, we conclude that lim k!1 contradicting (10). We now turn our attention to the case of an affine map F and derive a version of Theorem 1 in this context (see Theorem 3 below). We begin with a result that contains some elementary properties of affine maps. Lemma 3 easy Assume that G is an affine map and let G its linear part. Then the following statements hold: (a) G is (X; Y )-equilevel-monotone if and only if iii1 (b) G is z-injective if and only if (c) G is z-injective if and only if G is z-bounded. Proof. See Lemma 3 in Monteiro and Pang (1998). The next lemma is an important step towards establishing the main result for the case in which F is affine. m be an affine map which is (X; Y )- equilevel-monotone and z-injective, and let (\Phi; V) be a pair satisfying Assumption 2. Then the map H defined in (2) restricted to U \Theta ! m is a local homeomorphism. Proof. Since U \Theta ! m is an open set, it is sufficient to show that the derivative map H 0 (X; is an isomorphism for every (X; Y; z) 2 U \Theta ! m . For this purpose, fix any (X; Y; z) 2 U \Theta ! m . Since H 0 (X; Y; z) is linear and is a map between identical spaces, it is enough to show that Indeed, assume that the left-hand side of the above implication holds. By the definition H , we have By (14) and Lemma 3(a), we have \DeltaX ffl \DeltaY - 0. This together with (15) and Assumption 2(e) imply that Hence, by (14) and Lemma 3(b) we conclude that \Deltaz = 0. We have thus shown that the implication (13) holds. We are now ready to establish a version of the main result for the affine case. Theorem 3 main1 Assume that F is an affine map which is (X; Y )-equilevel-monotone and z-injective, and (\Phi; V) is a pair satisfying Assumption 2. Then, the following statements hold: (a) H maps U \Theta ! m homeomorphically onto V \Theta F++ ; cl V \Theta F++ . Proof. Let cl V \Theta F++ . We will show first that these sets and the map G j H satisfy the assumptions of Proposition 2. Indeed, first observe that H j M 0 is a local homeomorphism due to Lemma 4. The assumption that F is (X; Y )-equilevel-monotone and z-injective together with Lemma 3(c) imply that F is z-bounded. Hence, it follows from Lemma 2 that H is proper with respect to E. By the definition of M 0 , To show that H note that by Assumption 2(a), there exists Letting we have by Assumption 2(b) that \Phi(- 0 I By Assumption 2(a) and the fact that a continuous map carries connected sets onto connected sets, we conclude that N 0 j V \Theta F++ is a connected set. By Proposition 2, we conclude that H restricted to the pair (H local homeomorphism, H(M 0 cl N 0 . Clearly, the last inclusion implies the second inclusion in b). The first inclusion follows from the fact that \Phi \Gamma1 (cl V) being closed by Assumption 2(d) implies that cl U ' \Phi \Gamma1 (cl V) ' D. We now show (a). We have already shown that H maps M 0 onto N 0 and that ~ is a proper local homeomorphism. It remains to show that ~ H is one-to-one. Since N 0 is connected, by Proposition 1 it remains to show that for some (A; B) 2 N , the inverse image ~ has at most one element. Indeed, let I and take B 2 F++ arbitrarily. Assume that ( - is such that ~ by the definition of H , we have F ( - I , which by Assumption 2(b) implies that by Corollary 3 of Monteiro and Pang (1998), the system F (X; Y; has a unique solution in S n ++ \Theta S n ++ \Theta S n we conclude that ( - is the only solution of ~ We use the above result to prove the following very important technical lemma. Lemma 5 technical Let (\Phi; V) be a pair satisfying Assumption 2. Let (X be elements of U such that Proof. Assume for contradiction that (X It has been shown in the proof of Lemma 5 of Monteiro and Pang (1998) that there exists an affine which is (X; Y )- equilevel-monotone and satisfies F (X Assumption 1 with variable z is present). By Theorem 3(a), it follows that the associated map H (with restricted to U is one-to-one. Moreover, (16) and the relation F (X imply that The last two conclusions together with the fact that (X then imply that (X We are now ready to give the proof of Theorem 1. Proof of Theorem 1: The proof is close to the one given for Theorem 3. It consists of showing that the sets M j D cl V \Theta F++ , together with the map G j H , satisfy the assumptions of Proposition 2. But instead of using Lemma 4 to show that H j M 0 is a local homeomorphism, we use Lemma 5 to prove that H j M 0 maps homeomorphically onto H(M 0 is a continuous map from an open subset of the vector space S n \Theta S n \Theta ! m into the same space, by the domain invariance theorem it suffices to show that is one-to-one. For this purpose, assume that H( - U Then, by the definition of H , we have F ( - z) and Y ). Since F is (X; Y )-equilevel-monotone, we conclude that ( - Hence, by Lemma 5, we have ( - Y ). This implies that F ( - z), and by the z-injectiveness of F it follows that - z = ~ z. We have thus proved that H j M 0 maps M 0 homeomorphically is clearly a local homeomorphism and F is (X; Y )-equilevel-monotone and z-bounded by assumption, it follows from Lemma 2 that a) holds. The proofs of b) and c) use the same arguments as in the proof of Theorem 3. 4 Verification of Assumption 2 for Several Central-Path Maps section4 In this section we verify that the maps of Examples 1 to 6 satisfy Assumption 2. Verification for Example 1: Let D j S n be the map defined by \Phi(X; Y ++ . Conditions (a), (b) and (c) of Assumption 2 are straightforward, and clearly we have that U is contained in S n ++ \Theta S n ++ . Since \Phi is continuous and its domain is closed, the set \Phi \Gamma1 (cl V) is closed, and hence Assumption 2(d) holds. Assumption 2(e) follows from Theorem 3.1(iii) in Shida, Shindoh and Kojima (1998). Verification for Example 2: Let D j S n be the map defined by \Phi(X; Y Conditions (a), (b) and (c) of Assumption 2 are straightforward. As in Example 1, we easily see that \Phi \Gamma1 (cl V) is closed. Hence, Assumption 2(d) holds. Finally, it follows from the proof of Lemma 2.3 of Monteiro and Tsuchiya that the implication of Assumption 2(e) holds. Verification for Example 3: Let D j S n be the map defined by \Phi(X; Y Conditions (a), (b) and (c) of Assumption 2 are straightforward. As in Example 1, we easily see that \Phi \Gamma1 (cl V) is closed. Hence, Assumption 2(d) holds. Finally, Assumption 2(e) follows from Proposition 4 below with Verification for Example 4: Let D ++ \Theta S n be the map defined by ++ \Theta S n where LX is the lower Cholesky factor of X . Also, let Observe that the continuity of the map \Phi required in Assumption 2(a) follows from Lemma 6. The remaining requirements in (a) and conditions (b) and (c) of Assumption 2 are straightforward. We easily see that ++ \Theta S n which, in view of Lemma 6, is a closed set. Hence, Assumption 2(d) holds. Finally, Assumption 2(e) follows from Lemma 2.4 of Monteiro and Zanj'acomo (1997) with Verification for Example 5: Let D ++ \Theta S n be the map defined by ++ \Theta S n where LX and U Y are the lower Cholesky factor of X and the upper Cholesky factor of Y , respectively. Also, let 0g. Observe that the continuity of the map \Phi required in Assumption 2(a) follows from Lemma 6. The remaining requirements in (a) and conditions (b) and (c) of Assumption 2 are straightforward. We easily see that ++ \Theta S n -pfor some - ? 0 which, in view of Lemma 6, is a closed set. Hence, Assumption 2(d) holds. Finally, Assumption 2(e) follows from Proposition 5 below with Verification for Example ++ \Theta S n be the map defined by ++ \Theta S n is the unique positive definite matrix such that 0g. Observe that the continuity of the map \Phi required in Assumption 2(a) follows from Lemma 6. The remaining requirements in (a) and conditions (b) and (c) of Assumption 2 are straightforward. We easily see that ae ++ \Theta S n oe which, in view of Lemma 6, is a closed set. Hence, Assumption 2(d) holds. Assumption 2(e) follows from Proposition 6 below. Note that all the sets V in the above examples are convex cones containing the line f-I We now present the technical results needed for the verification of the Examples 1 to 6 described above. Lemma 6 sets Let jjj \Delta jjj be an arbitrary norm in S n such that jjjH jjj - kHk for all H 2 S n . ++ \Theta S n denote the map of Example 4, 5 or 6 described above. Then, \Phi is a continuous map and, for any fl 2 (0; 1), the set defined by ++ \Theta S n is a closed set. Proof. First note that for (X; Y ++ \Theta S n ++ , we have and where the first equality of (19) follows from the fact that W 1=2 XYW \Gamma1=2 is a symmetric matrix. To establish the continuity of the three maps, it is sufficient to show that if f(X ++ \Theta S n converges to a point ( - then the sequences f(W k converge to 0. Indeed, ( - implies that fX k ffl Y k g converges to 0, which, in view of (18) and (19), implies that these three sequences converge to 0. We now establish the closedness of the set (17), which we denote by - N fl . It is sufficient to show that if f(X converges to ( - . and hence due to the fact that jjj \Delta jjj - k \Delta k. Using (18), we easily see that fE k g is bounded. In view of (21), this implies that f- k g is bounded. Without loss of generality, we may assume that f- k g converges to some - We now consider the two possible cases: - implies that equivalently lim k!0 \Phi(X For the map of Example 4, this means that lim k!0 L T and for the one in Example 5, that lim k!0 U T Using the fact that U T lower triangular, the last limit implies that lim k!0 U T both maps \Phi, it follows from (18) that fX k ffl Y k g converges to 0, and hence that ( - Assume now that - ? 0. We will show that ( - ++ \Theta S n ++ , from which the conclusion that letting k tend to infinity in (20) and using the fact that - is continuous on S n ++ \Theta S n ++ . Indeed, assume for contradiction that ( - ++ \Theta S n ++ . Without loss of generality, we may assume that - X is singular. Consider first the map \Phi of Example 4. We have and since lim k!0 - min (X k Y k we conclude that lim k!0 - min By (21) this implies that - contradicting the assumption that - Consider now the map \Phi of Example 5. We have where the inequality follows from the fact that the real part of the spectrum of a real matrix is contained between the largest and the smallest eigenvalues of its Hermitian part (see p. 187 of Horn and Johnson (1991), for example) and the fact that U T being lower triangular has only real eigenvalues. Note that the sequences fU Y k g and fL X kg are bounded, since kU Y k U and - L be accumulation points of fU Y k g and fL X kg, respectively. Clearly, X, and since - X is singular, it follows that - L is also singular. Letting k tend to infinity in (22), we conclude that which as above yields the desired contradiction. Finally, consider the map \Phi of Example 6. We have and since lim k!0 - min (X k Y k we conclude that lim k!0 - min By (21) this implies that - contradicting the assumption that - The next three lemmas are used in the proof of Proposition 4. Lemma 7 ineqtr Let G 2 M n be such that tr Proof. Defining ? , and using the identities easily see that tr The result now follows by adding the two identities and using the assumption that tr be a matrix whose eigenvalues are all real and let (0; 1= be given. Then, the following implication holds: Proof. Assume that the left hand side of the implication holds. Using the fact that E, and hence I , has only real eigenvalues, it is easy to see that tr and hence the first inequality of the right hand side of the implication holds. The second inequality follows from Lemma 2.3.3 of Golub and Van Loan [5] and the fact that Lemma 9 Lyap The following statements hold: a) for every A 2 S n ++ and H 2 S n , the equation AU has a unique solution U moreover, this solution satisfies kAUk F - kHk ++ denote the square root function is an analytic function is the unique solution of X (X) is the derivative of ' at X and ' 0 (X)H is the linear map ' 0 (X) evaluated at H. Proof. See Lemmas 2.2 and 2.3 of Monteiro and Tsuchiya (1999). Proposition 4 tech-bound2 Suppose that the map of Example 3. If (X; Y ++ is a point such that k(X 1=2 Y 1=2 +Y 1=2 X 1=2 )=2\Gamma-I k F - fl- for some - ? 0, then F where g Proof. Using Lemma 9(a), it is easy to see that the equation \Phi 0 (X; Y )(\DeltaX; \DeltaY to where U and V are the unique solutions of the Lyapunov equations Multiplying the first (resp., second) equation on the left and on the right by X \Gamma1=2 (resp., Y \Gamma1=2 ) and using Lemma 9(a), we obtain Using the definition of g \DeltaX and g \DeltaY , it follows from (24) and (25) that Taking the Frobenius norm of both sides of the last equation and using (26), Lemma 8 with the triangular inequality for norms and the fact that g obtain F F \DeltaY k F F from which (23) follows. The next result has a proof similar to the one of Proposition 4, and hence we omit it. Instead of Lemma 9, its proof relies on lemma 2.2 of Monteiro and Zanj'acomo (1997). Proposition 5 tech-bound3 Suppose that ++ \Theta is the map in Example 5. If (X; Y ++ \Theta S n ++ is a point such that k(U T F where d Y \DeltaX L \GammaT X and d Y \DeltaY LX . Proposition 6 proposNT Let \Phi : C(0)[ (S n ++ \Theta S n denote the map of Example 6 and ++ \Theta S n ++ be a point such that k\Phi(X; Y Then, Proof. Using the identity easily see that \Phi(X; Y Letting we have that \Phi(X; Y Hence, the directional derivative of denote the directional derivative of the function V at the point (X; Y ) along the direction (\DeltaX; \DeltaY ++ , the condition \Phi 0 (X; Y )(\DeltaX; \DeltaY is equivalent to V denote the directional derivative of the function W at the point (X; Y ) along the direction (\DeltaX; \DeltaY ). Then, T satisfies the following equation obtained from the identity Using Lemma 9(b) and the definition of V , it is easy to see that the condition is equivalent to where U is the unique solution of the Lyapunov equation Letting g \DeltaX \DeltaY the identities (29), (30) and (31) become ~ \DeltaY tec2NT2 (32) ~ ~ Subtracting (33) from (32) and using (34), we obtain that g U T . This identity together with (33) and the fact that imply that F last relation implies that ~ Hence, by (32)-(34), we conclude that equivalently --R Interior point methods in semidefinite programming with application to combinatorial optimization. Complementarity and nondegeneracy in semidefinite programming. A Primer of Nonlinear Analysis Matrix Computations: Second Edition Topics in Matrix Analysis Reduction of monotone linear complementarity problem over cones to linear programs over cones. Local convergence of predictor-corrector infeasible- interior-point algorithms for SDPs and SDLCPs A predictor-corrector interior-point algorithm for the semidefinite linear complementarity problem using the Alizadeh-Haeberly-Overton search direction Duality and self-duality for conic convex program- ming Superlinear convergence of a symmetric primal-dual path following algorithm for semidefinite programming Polynomial convergence of primal-dual algorithms for semidefinite programming based on Monteiro and Zhang family of directions Properties of an interior-point mapping for mixed complementarity problems On two interior-point mappings for nonlinear semidefinite complementarity problems A potential reduction Newton method for constrained equations. Polynomial convergence of a new family of primal-dual algorithms for semidefinite programming Implementation of primal-dual methods for semidefinite programming based on Monteiro and Tsuchiya Newton directions and their variants A unified analysis for a class of path-following primal-dual interior-point algorithms for semidefinite programming Interior Point Methods in Convex Programming: Theory and Application Iterative Solution of Nonlinear Equations in Several Variables A superlinearly convergent primal-dual infeasible-interior-point algorithm for semidefinite programming Strong duality for semidefinite programming. Convex Analysis First and second order analysis of nonlinear semidefinite programs. Monotone semidefinite complementarity problems Centers of monotone generalized complementarity problems. Existence of search directions in interior-point algorithms for the SDP and the monotone SDLCP On Weighted Centers for Semidefinite Programming programming. An interior point potential reduction method for constrained equations. On extending some primal-dual interior-point algorithms from linear programming to semidefinite programming --TR

interior point methods;maximal monotonicity;generalized complementarity problems;monotone maps;mixed nonlinear complementarity problems;nonlinear semidefinite programming;weighted central path;continuous trajectories

355485

Building tractable disjunctive constraints.

Many combinatorial search problems can be expressed as 'constraint satisfaction problems'. This class of problems is known to be NP-hard in general, but a number of restricted constraint classes have been identified which ensure tractability. This paper presents the first general results on combining tractable constraint classes to obtain larger, more general, tractable classes. We give examples to show that many known examples of tractable constraint classes, from a wide variety of different contexts, can be constructed from simpler tractable classes using a general method. We also construct several new tractable classes that have not previously been identified.

INTRODUCTION Many combinatorial search problems can be expressed as 'constraint satisfaction problems' [Montanari 1974; Mackworth 1977], in which the aim is to nd an assignment of values to a given set of variables subject to specied constraints. For ex- ample, the standard propositional satisability problem [Garey and Johnson 1979] may be viewed as a constraint satisfaction problem where the variables must be assigned Boolean values, and the constraints are specied by clauses. The general constraint satisfaction problem is known to be NP-hard [Montanari 1974; Mackworth 1977]. However, by imposing restrictions on the constraint inter-connections [Dechter and Pearl 1989; Freuder 1985; Gyssens et al. 1994; Montanari 1974], or on the form of the constraints [Cooper et al. 1994; Jeavons et al. 1997; Jeavons and Cooper 1995; Kirousis 1993; Montanari 1974; van Beek and Dechter 1995; van Hentenryck et al. 1992], it is possible to obtain restricted versions of the problem that are tractable. Now that a number of tractable constraint types have been identied, it is of considerable interest to investigate how these constraint types can be combined, to yield more general problem classes that are still tractable. This paper presents the rst general results of this kind: we identify conditions under which dierent tractable constraint classes can be combined, in order to construct larger, more general, tractable constraint classes. We focus specically on 'disjunctive constraints', that is, constraints which have the form of the disjunction of two constraints of specied types. We show that whenever we are given tractable constraint types with certain properties, then the class of problems involving all possible disjunctions of constraints of these types is also tractable. This allows new tractable constraint classes to be constructed from simpler tractable classes, and so extends the range of known tractable constraint classes. We give examples to show that many known examples of tractable disjunctive constraints over both nite and innite domains can be constructed from simpler classes using these results. In particular, we demonstrate that ve out of the six tractable classes of Boolean constraints identied by Schaefer in [Schaefer 1978] can be obtained in this way (see Examples 5, 7 and 9). These include the standard Horn clauses and Krom clauses of propositional logic. Furthermore, we show that similar results hold for the 'max-closed' constraints rst identied in [Jeavons and Cooper 1995], the 'connected row-convex' constraints rst identied in [Deville et al. 1997] (see also [Jeavons et al. 1998]), the ORD-Horn constraints over temporal intervals described in [Nebel and Burckert 1995], the disjunctive linear constraints over the real numbers described in [Jonsson and Backstrom 1998; Koubarakis 1996], the 'extended Horn clauses' described in [Chandru and Hooker 1991], and the tractable set constraints described in [Drakengren 1997; Drakengren and Jonsson 1998; Drak- engren and Jonsson 1997]. In all of these cases our results lead to simplications of earlier proofs, and in many cases we are able to generalise the earlier results to Building Tractable Disjunctive Constraints 3 obtain larger families of tractable constraint classes. We also describe some new tractable classes of constraints that can be derived from the same results. The paper is organised as follows. In Section 2 we give the basic denitions for the constraint satisfaction problem, and dene the notion of a tractable set of constraints. In Section 3 we describe how sets of constraints can be combined to form disjunctive constraints, and identify a number of dierent conditions that are sucient to ensure tractability of these disjunctive constraints. In Section 4 we give examples to illustrate how these results can be used to establish the tractability of a wide variety of tractable constraint classes. 2. THE CONSTRAINT SATISFACTION PROBLEM The 'constraint satisfaction problem' was introduced by Montanari in 1974 [Mon- tanari 1974] and has been widely studied. Denition 1. An instance of a constraint satisfaction problem consists of: |A nite set of variables, |A set of values, D, (which may be nite or innite); |A nite set of constraints g. Each constraint c i is a pair (S i is a set of variables, called the constraint scope, and R i is a set of (total) functions from S i to D, called the constraint relation. The elements of a constraint relation indicate the allowed combinations of simultaneous values for the variables in the constraint scope. The number of variables in the scope of a constraint will be called the 'arity' of the constraint. In particular, unary constraints specify the allowed values for a single variable, and binary constraints specify the allowed combinations of values for a pair of variables. There is a unique 'empty constraint', (;; ;), for which the scope and the constraint relation are both empty. Note that we are representing constraint relations as sets of functions rather than the usual representation as sets of tuples. These two representations are clearly equivalent, since by xing an ordering for the variables in the scope of a constraint, we can associate each function with a corresponding tuple of values. However, the use of the functional representation simplies some of the denitions below. Example 1. When the set of values D is the set of real numbers R, then a relation on some set of variables, say fu; v; wg, is a set of total functions from fu; v; wg to R. For example, the following is a typical relation If we x any ordering on the variables, say (w; then the same relation can be represented as a set of 3-tuples of real numbers: A solution to a constraint satisfaction problem instance is a function, f , from the set of variables of that instance to the set of values, such that for each constraint the restriction of f to S i , denoted f j S i , is an element of R i . 4 D.Cohen, P.Jeavons, P.Jonsson, and M. Koubarakis In order to simplify the presentation we shall make a number of simplifying assumptions throughout the paper about the way in which constraint satisfaction problem instances are specied. First, we shall assume that we have a xed countable universe of possible variable names, and that every variable occurring in a problem instance is a member of this set. Furthermore, we shall assume that every variable in a problem instance occurs in the scope of some con- straint. This means that for any problem instance, the set of variables, V , does not need to be specied explicitly, but is given by the union of the constraint scopes of that instance. Finally, we shall assume that the set of values, D, for any instance does not need to be specied explicitly, but will be understood from the context. With these assumptions, a constraint satisfaction problem instance can be specied simply by specifying the corresponding set of constraints. Hence, we shall talk about 'solutions to a set of constraints'. The set of all solutions to the set of constraints C will be denoted Sol(C). In order to determine the computational complexity of a constraint satisfaction problem we need to specify how the constraints are encoded in nite strings of symbols. We shall assume in all cases that this representation is chosen so that the complexity of determining whether a constraint allows a given assignment of values to the variables in its scope is bounded by a polynomial function of the length of the representation. Example 2. Constraint relations may be nite or innite. A constraint with a constraint relation can be represented simply by giving an explicit list of all the elements in that relation, but constraints with innite relations clearly cannot. In both cases it is possible to use a suitable specication language, such as logical formulas, or linear equations. For example, when the set of possible values for the variables is ftrue; falseg, representing the Boolean values 'true' and `false', then the logical formula 'x 1 _ x 2 _:x 3 ' can be used to specify the constraint with scope and relation Similarly, when the set of possible values is the real numbers, R, then the equation can be used to specify the constraint with scope fx 1 and relation Deciding whether or not a given set of constraints has a solution is known to be NP-hard in general [Montanari 1974; Mackworth 1977]. In this paper we shall consider how restricting the allowed constraints aects the complexity of this decision problem. We therefore make the following denition. Denition 2. For any set of constraints, , CSP() is dened to be the decision problem with Instance:. A nite set of constraints C . Question:. Does C have a solution? If there is some algorithm which solves every instance in CSP() in polynomial time, then we shall say that CSP() is 'tractable', and refer to as a tractable set of constraints. Building Tractable Disjunctive Constraints 5 Example 3. For any set of possible values D, and any pair of variables, x and y, the binary disequality constraint with scope fx; yg and set of values D is dened as follows: denotes the constraint (fx; Since we are assuming that we have a xed universe of possible variable names, we can consider the set of all possible binary disequality constraints over D, for all possible choices of a pair of variables. This set will be denoted 6= D For any nite set D, the decision problem CSP( 6= D corresponds precisely to the Graph jDj-Colourability problem [Garey and Johnson 1979]. This problem is well-known to be tractable when jDj 2 and NP-complete when jDj 3. 3. TRAC The remainder of the paper focuses on the complexity of constraint satisfaction problems involving disjunctive constraints. We rst dene how individual constraints can be combined disjunctively. Denition 3. Let c be two constraints with a common set of possible values D. The disjunction of c 1 and c 2 , denoted c 1 _ c 2 , is dened as follows: The idea behind this denition is that an assignment satises the disjunction of two constraints if it satises either one of them. Note that for any constraint c, the disjunction of c and the empty constraint, (;; ;), gives c. Now we dene how a set of disjunctive constraints can be obtained from two arbitrary sets of constraints over the same set of possible values. Denition 4. For any two sets of constraints and , with a common set of possible values, dene the set of constraints _ as follows: The set of constraints _ (read as ' or-cross ') contains the disjunction of each possible pair of constraints from and . In many cases of interest and will both contain the empty constraint, (;; ;), and in these cases _ [. In most cases of this kind _ will be much larger than [ , and will therefore allow a much richer class of constraint satisfaction problems to be expressed. The next example shows that when tractable sets of constraints are combined using the disjunction operation dened in Denition 4 the resulting set of disjunctive constraints may or may not be tractable. Example 4. Let be the set containing all Boolean constraints which can be specied by a formula of propositional logic consisting of a single literal (where a literal is either a variable or a negated variable). The set of constraints is clearly tractable, as it is straightforward to verify in linear time whether a collection of simultaneous literals has a solution. 6 D.Cohen, P.Jeavons, P.Jonsson, and M. Koubarakis Now consider the set of constraints _. This set contains all Boolean constraints specied by a disjunction of 2 literals. The problem CSP( _2 ) corresponds to the 2-Satisfiability problem, which is well-known to be tractable [Garey and Johnson 1979]. Finally, consider the set of constraints _. This set contains all Boolean constraints specied by a disjunction of 3 literals. The problem CSP( _3 ) corresponds to the 3-Satisfiability problem, which is well-known to be NP-complete [Garey and Johnson 1979]. In many of the examples below we shall be concerned with constraints that are specied by disjunctions of an arbitrary number of constraints from a given set. To provide a uniform notation for such constraints we make the following denition. Denition 5. For any set of constraints, , dene the set as follows: =[ _i where The nal piece of machinery that we shall need to deal with disjunctive sets of constraints is a uniform way to recover the separate components in the disjunction. Denition 6. For any disjunctive set of constraints _, we dene two operations such that for any c 2 _, We shall assume that the constraints in _ are represented in such a way that 1 and 2 can be computed in linear time. In the following sections we identify certain conditions on sets of constraints and which are sucient to ensure that _ is tractable. 3.1 The guaranteed satisfaction property The rst condition we identify is rather trivial, but it is included here for com- pleteness, and because it is sucient to show how two of the six tractable classes of Boolean constraints identied by Schaefer in [Schaefer 1978] can be constructed from simpler classes. Denition 7. A set of constraints, , has the guaranteed satisfaction property if every nite C has a solution. Theorem 1. For any sets of constraints and , if has the guaranteed satisfaction property, then CSP( _) also has the guaranteed satisfaction property, and is therefore tractable. Proof. Every constraint in _ is of the form c_d for some c 2 and some d 2 . Hence if has the guaranteed satisfaction property, then any problem instance in CSP( _) has a solution satisfying the rst disjunct of each constraint. Example 5. Recall the set of unary Boolean constraints, , dened in Example 4, which contains all constraints specied by a single literal. Building Tractable Disjunctive Constraints 7 Let be the subset of containing only the constraints specied by a single negative literal. It is clear that has the guaranteed satisfaction property, since any problem instance in CSP() has the solution which assigns the value false to all variables. Hence, by Theorem 1, _ has the guaranteed satisfaction property and CSP( _ ) is tractable. This tractable set contains all constraints specied by Boolean clauses containing at least one negative literal. Examples of such clauses include the following The constraint relations dened by conjunctions of clauses of this form are precisely the elements of the rst class of tractable Boolean relations identied by Schaefer in [Schaefer 1978] (which he calls '0-valid' relations). A symmetric argument shows that if contains only the constraints specied by a single positive literal, then CSP( _ ) is again tractable. This tractable set contains all constraints specied by Boolean clauses containing at least one positive literal. The constraint relations dened by conjunctions of clauses of this form are precisely the elements of the second class of tractable Boolean relations identied by Schaefer in [Schaefer 1978] (which he calls '1-valid' relations). Example 6. Let be the set of all constraints specied by a linear disequality over the real numbers, that is, an expression of the form where the a i and b are (real-valued) constants. It is clear that has the guaranteed satisfaction property, since any problem instance in CSP() only rules out a nite number of hyperplanes from R n . Hence, by Theorem 1, for any set of constraints, , over the real numbers, _ has the guaranteed satisfaction property and CSP( _ ) is tractable. For example, let be the set containing all the constraints in together with all constraints specied by a single (weak) linear inequality, that is, an expression of the form where the a i and b are (real-valued) constants. In this case _ contains constraints such as the following: This example should be compared with the similar, but much more signicant, tractable class dened in Example 13 below. 3.2 The independence property In this section we identify a rather more subtle condition which can be used to construct tractable disjunctive constraints. We rst need the following denition: Denition 8. For any sets of constraints and , dene CSPk( [ ) to be the subproblem of CSP( [ ) consisting of all instances containing at most k constraints which are members of . 8 D.Cohen, P.Jeavons, P.Jonsson, and M. Koubarakis Using this denition, we now dene what it means for one set of constraints to be 'k-independent' with respect to another. Denition 9. For any sets of constraints and , we say that is k-independent with respect to if the following condition holds: any set of constraints C in solution provided every subset of C belonging to CSPk( [ has a solution. The intuitive meaning of this denition is that the satisability of any set of constraints chosen from the set can be determined by considering those constraints k at a time, even in the presence of arbitrary additional constraints from . In the examples below we shall demonstrate that several important constraint types have the 1-independence property. A more restricted notion of 1-independence has been widely studied in the literature of constraint programming, where it has been called simply 'independence' (see [Lassez and McAloon 1989; Lassez and McAloon 1992; Lassez and McAloon 1991], for example). The earlier property applies to an individual constraint class containing positive constraints and negative constraints, and has been used in the development of consistency checking algorithms and canonical forms [Lassez and McAloon 1991; Lassez and McAloon 1992]. However, we will show below that the more general notion of 1-independence of one class with respect to another, introduced here, can be used to prove the tractability of a wide variety of disjunctive constraint classes for which the earlier notion of independence does not hold. Consider the algorithm shown in Figure 1 for a function Ind-Solvable which determines whether or not a nite set of constraints C _ has a solution. Ind-Solvable(C: finite subset of _) repeat return true else S endif return false Fig. 1. An algorithm for the function Ind-Solvable The next result shows that Ind-Solvable correctly determines whether or not a nite set of constraints chosen from _ has a solution in all cases where is 1-independent with respect to . Lemma 1. If C is a nite subset of _, and is 1-independent with respect to , then the function Ind-Solvable dened in Figure 1 correctly determines whether or not C has a solution. Building Tractable Disjunctive Constraints 9 Proof. The algorithm shown in Figure 1 clearly terminates, because C is nite. Assume that C is a nite subset of _ for some and such that is 1-independent with respect to . We rst prove by induction that after every assignment to S, all of the constraints in S must be satised in order to satisfy the original set of constraints C. This is vacuously true after the rst assignment to S, because S is then equal to the empty set. At each subsequent assignment, S is augmented with the constraints obtained by applying 1 to the constraints in X . Now, the elements of X are constraints c of C such that 2 (c) is incompatible with S. Hence the only way such a c can be satised together with S is to satisfy the other disjunct of c, that is, the constraint given by 1 (c). Hence, by the inductive hypothesis, each constraint added to S must be satised in order to satisfy the original set of constraints C, and the result follows, by induction. This result establishes that when Ind-Solvable(C) returns false, C has no solutions. Conversely, when Ind-Solvable(C) returns true, then we know that X is empty. This implies that for each constraint c in C either 1 (c) belongs to S, or else 2 (c) is compatible with the constraints in S. Now, using the fact that is 1-independent with respect to , we conclude that S [ f 2 (c) j c 2 C; 1 (c) 62 Sg has a solution, and hence C has a solution. By analysing the complexity of the algorithm in Figure 1 we now establish the following result: Theorem 2. For any two sets of constraints and , if CSP1( [ ) is tractable, and is 1-independent with respect to , then CSP( _) is tractable. Proof. By Lemma 1, it is sucient to show that the algorithm in Figure 1 runs in polynomial time. We can bound the time complexity of this algorithm as follows. First note that jSj increases on each iteration of the repeat loop, but jSj is bounded by jCj since each constraint in S arises from an element of C. Hence there can be at most jCj iterations of this loop. Now let l(C) be the length of the string specifying C. During each iteration of the loop the algorithm determines whether or not there is a solution to S[ 2 (c) for each c remaining in C. Since this set of constraints is a member of CSP1( [ which is assumed to be tractable, these calculations can each be carried out in polynomial time in the size of their input. Note also that the length of this input is less than or equal to l(C). Hence, the time complexity of each of these calculations is bounded by p(l(C)), for some polynomial p. At the end of each iteration of the loop the algorithm determines whether or not there is a solution to S. Since S is also an element of CSP1( [ ), and the length of the specication of S is less than or equal to l(C), this calculation can also be carried out in at most p(l(C)) time. Hence, the total time required to complete the algorithm is O(jCj(jCj+1)p(l(C))), which is polynomial in the size of the input. Finally, we show that this result can be extended to arbitrary disjunctions of constraints in . P.Jonsson, and M. Koubarakis Lemma 2. For any set of constraints , if is 1-independent with respect to , then is also 1-independent with respect to . Proof. Assume that is 1-independent with respect to and let C be an arbitrary nite subset of [ . We need to show that if every subset of C which belongs to CSP 1( [ ) has a solution, then so does C. Let C 0 be a maximal subset of C belonging to CSP 1( [ ) and let s be a solution to C 0 . Since C 0 is maximal, it contains the set C , consisting of all the constraints in C which are elements of n , so s is a solution to C . If C 0 also contains a constraint d 2 , then there must be at least one constraint d 0 in such that s is a solution to d 0 , by the denition of . Hence, we can replace d with a (possibly) more restrictive constraint d 0 2 , without losing the solution s. If we carry out this replacement for each C 0 , then we have a set of constraints in CSP( [ ) such that each subset belonging to CSP1( [ ) has a solution. Now, by the fact that is 1-independent with respect to , it follows that this modied set of constraints has a solution, and hence the original set of constraints C has a solution, which gives the result. Corollary 1. For any two sets of constraints and , if CSP1( [ ) is tractable, and is 1-independent with respect to , then CSP( _ ) is tractable. Proof. If each instance of CSP 1( [ ) only contains at most one disjunctive constraint belonging to , and each disjunct of this constraint can be considered separately in polynomial time. Furthermore, if is 1-independent with respect to , then by Lemma 2 is also 1-independent with respect to . Hence, Theorem 2 can be applied to and , giving the result. Example 7. Recall the set of unary Boolean constraints, , dened in Example 4, which contains all constraints specied by a single literal. Let be the subset of containing only the constraints specied by a single positive literal, and let ;)g. Let be the subset of containing only the constraints specied by a single negative literal. Note that the set of constraints , constructed according to Denition 5, is equal to the set of constraints specied by arbitrary nite disjunctions of negative literals (including the empty disjunction). Now it is easily shown that is 1-independent with respect to 0 (since any collection of positive and negative literals has a solution if and only if all subsets containing at most one negative literal have a solution). Also, tractable, since each instance is specied by a conjunction of zero or more positive literals together with at most one negative literal. Hence, by Corollary 1, we conclude that 0 _ is tractable. But 0 _ is the set of constraints specied by a disjunction of literals containing at most one positive literal. Examples of such clauses include the following: It is easy to see that CSP( 0 _ ) corresponds exactly to the Horn-Clause Satisfiability problem [Garey and Johnson 1979]. The constraint relations dened Building Tractable Disjunctive Constraints 11 by conjunctions of clauses of this form are precisely the elements of the third class of tractable Boolean relations identied by Schaefer in [Schaefer 1978] (which he calls 'weakly negative' relations). By a symmetric argument, it follows that the set of constraints specied by a disjunction of literals containing at most one negative literal is also a tractable set of constraints. The constraint relations dened by conjunctions of clauses of this form are precisely the elements of the fourth class of tractable Boolean relations identied by Schaefer in [Schaefer 1978] (which he calls 'weakly positive' relations). We have shown that 1-independence can be used to establish tractability, and further examples are given in Section 4 below. To conclude this section we show that, if a set of constraints is k-independent with respect to a set , for some value of larger than one, then this may not be sucient to ensure tractability of _. Example 8. In this example we construct two sets of constraints, and , such that CSP2( [ ) is tractable, and is 2-independent with respect to , but CSP( _) is NP-complete. Recall the set of unary Boolean constraints, , dened in Example 4, which contains all constraints specied by a single literal. With these denitions, CSP( [ ) is equivalent to the standard 2-Satisfiability problem, and hence CSP2( [ ) is tractable. Furthermore, the 2-Satisfiability problem has the property that 'path-consistency' guarantees global consistency [Jeav- ons et al. 1998], which implies that any minimal insoluble subset of clauses in an instance of 2-Satisifiability contains at most two single literals, and hence is 2-independent with respect to , However, as discussed in Example 4, CSP( _) corresponds to the 3-satisfiability problem, and is therefore NP-complete. 3.3 The Krom property In this section we identify a nal sucient condition for constructing tractable disjunctive constraints. Denition 10. A set of constraints, , has the Krom property if it is 2-independent with respect to the empty set. Note that has the Krom property if and only if for every nite C having no solution, there exists a pair of (not necessarily distinct) constraints c that has no solution. The name \Krom property" is chosen to emphasise the close connection with Krom clauses (i.e., Boolean clauses of length 2) [Denenberg and Lewis 1984], which will be demonstrated later. Consider the algorithm shown in Figure 2, for a function Krom-Solvable, which determines whether or not a nite set of constraints C _ has a solution. The next result shows that Krom-Solvable correctly determines whether or not a nite set of constraints chosen from _ has a solution in all cases when has the Krom property. Lemma 3. If C is a nite subset of _, and has the Krom property, then the function Krom-Solvable dened in Figure 2 correctly determines whether or not C has a solution. 12 D.Cohen, P.Jeavons, P.Jonsson, and M. Koubarakis Krom-Solvable(C: finite subset of _) Define a set of Boolean variables fqc j c 2 Pg A := f(:q c 0 _ :q c 00 then return true else return false Fig. 2. An algorithm for a function Krom-Solvable Proof. We show that the function Krom-Solvable dened in Figure 2 returns true when applied to C if and only if C has a solution. only-if:. Assume that Krom-Solvable returns true. This implies that there exists a satisfying truth assignment, , for A [ B. Dene the set of constraints We rst show that C 0 has a solution. Since has the Krom property, C 0 has no solution only if there exist c has no solution. However, this cannot happen because satises the formulae in the set A. Now, let the function f be a solution of C 0 . For each constraint c 2 C we know that at least one of 1 (c) and 2 (c) is a member of C 0 , because satises the formulae in B. Since f is a solution of C 0 , it follows that f can be extended (if necessary) to a solution of C (by assigning an arbitrary value to any variable not constrained by C 0 ). if:. Assume that C has a solution and let f be any such solution. Dene the truth assignment : fq c j c 2 Pg ! ftrue; falseg as follows: We show that is a satisfying truth assignment of A[B by considering the elements of A and B in turn. (1) For each formula (:q c 0 _ :q c 00 ) 2 A, we know that fc 0 ; c 00 g has no solutions. Hence it cannot be the case that (q c 0 true, which means that (:q c _ :q c 0 ) is satised by . (2) For each formula (q c 0 _ q c 00 know that there is a constraint c 2 C such that 1 is a solution to C, f satises either c, c 0 , or both. Hence, assigns true to at least one of q c 0 ; q c 00 , which means that satised by . It follows that A [ B is satisable, so Krom-Solvable returns true. By analysing the complexity of the algorithm in Figure 2 we now establish the following result: Theorem 3. For any set of constraints , if CSP() is tractable, and has the Krom property, then CSP( _) is tractable. Building Tractable Disjunctive Constraints 13 Proof. By Lemma 3, it is sucient to show that the algorithm in Figure 2 runs in polynomial time. We can bound the time complexity of this algorithm as follows. Let C be a nite subset of _ and let l(C) be the length of the string specifying C. Since computing 1 and 2 takes linear time, the set P can be computed in O(l(C)) time. It contains at most 2jCj elements. To compute the set A, the algorithm must determine whether or not there is a solution to fc for every pair of constraints c is an instance of CSP(), which is assumed to be tractable, these calculations can each be carried out in polynomial time in the size of their input. Hence the time required to compute the set A is O(jCj 2 p(l(C))), for some polynomial p. The set A contains at most jCj(2jCj 1) elements. The set B can clearly be computed in time O(jCjl(C)) and contains jCj elements. Finally, the algorithm must decide the satisability of a set of Krom clauses containing at most jCj(2jCj 1)+ jCj elements. By using the linear time algorithm for this problem given in [Aspvall et al. 1979], this step can be carried out in O(jCj 2 ) time. Hence, the total time required by the algorithm is O(jCj 2 p(l(C))), which is polynomial in the size of the input. The proof of Theorem 3 uses the well-known result about the tractability of solving Krom clauses. The next example shows that these two results are equivalent. Example 9. Recall the set of unary Boolean constraints, , dened in Example 4, which contains all constraints specied by a single literal. It is easily shown that has the Krom property (since any collection of positive and negative literals has a solution unless it contains two dierent literals involving the same variable). Hence, by Theorem 3, _ is tractable. However, the set _ contains all constraints specied by Boolean clauses containing at most two literals, that is, all Krom clauses. The constraint relations dened by conjunctions of clauses of this form are precisely the elements of the fth class of tractable Boolean relations identied by Schaefer in [Schaefer 1978] (which he calls 'bijunctive' relations). To conclude this section we show that if a set of constraints has a higher level of k-independence with respect to the empty set, then this may not be sucient to ensure tractability of _. Example 10. In this example we construct a tractable set of relations such that is 3-independent with respect to the empty set, but we show that CSP( _) is NP-complete. For all possible variables x and y, we dene the unary constraints zero(x); one(x) and the binary constraint 6= N (x; y) as follows: (1) zero(x) denotes the constraint (2) one(x) denotes the constraint (3) 6= N (x; y) denotes the constraint (fx; Dene to be the set containing all possible constraints of these three types. (Note that is well-dened since we are assuming that we have a xed universe of 14 D.Cohen, P.Jeavons, P.Jonsson, and M. Koubarakis possible variable names.) Let C be a nite subset of . We will show that if C has no solution, then (at least) one of the following constraint sets is a subset of C, for some variables x and y. To establish this fact, assume that none of the above stated sets of constraints are subsets of C. We will show that in this case it is always possible to construct a solution. Let the set of variables appearing in the constraints of C be g. Dene the function f For each constraint c 2 C we can reason as follows: |Since |Since c is the constraint 6= N |Since neither S3 nor S4 is a subset of C, if c is the constraint 6= N Hence, in all cases c is satised by f , so f is a solution to C. Since none of the sets S1,S2,S3,S4 contains more than 3 elements, we have established that is 3-independent with respect to the empty set, and that is tractable. To establish the NP-completeness of CSP( _), we construct a polynomial time reduction from the NP-complete problem 4-Colourability [Garey and Johnson 1979], as follows. be an arbitrary graph and construct an instance of CSP( _) as follows: for each v 2 V , introduce two variables v 0 and v 00 together with the constraints For each edge (v; w) 2 E, introduce the constraint This transformation can obviously be carried out in polynomial time and the resulting set of constraints is a subset of _. To see that the resulting instance has a solution if and only if the graph G is 4-colourable, identify colour 1 with and so on. For every pair of adjacent nodes v; w in G, the constraint imposed on the corresponding variables v ensures that v and w must be assigned dierent colours. Building Tractable Disjunctive Constraints 15 4. APPLICATIONS In this section we will use the results established above to demonstrate that many known tractable sets of constraints can be obtained by combining simpler tractable sets of constraints using the disjunction operation dened in Denition 4. We will also describe some new tractable sets of constraints which have not previously been identied. Example 11. [Max-closed constraints] The class of constraints known as 'max-closed' constraints was introduced in [Jeav- ons and Cooper 1995] and shown to be tractable. This class of constraints has been used in the analysis and development of a number of industrial scheduling tools [Le- saint et al. 1998; Purvis and Jeavons 1999]. Max-closed constraints are dened in [Jeavons and Cooper 1995] for arbitrary nite sets of values which are totally ordered. This class of constraints includes all of the 'basic constraints' over the natural numbers in the constraint programming language CHIP [van Hentenryck et al. 1992]. The following are examples of max- closed constraints when the set of possible values is any nite set of natural numbers: In this example we will show that the tractability of max-closed constraints is a simple consequence of Corollary 1. Furthermore, by using Corollary 1 we are able to generalise this result to obtain tractable constraints over innite sets of values. Max-closed constraints were originally dened in terms of an algebraic closure property on the constraint relations [Jeavons and Cooper 1995]. However, it is shown in [Jeavons and Cooper 1995] that they can also be characterised as those constraints which can be specied by a conjunction of disjunctions of inequalities of the following form: In this expression the x i are variables and the a i are constants. To apply the results of Section 3.2, let the set of possible values, D, be any totally ordered set. Dene to be the set of all constraints specied by a single inequality of the form x i < a i , for some a i 2 D, together with the empty constraint. Dene to be the set of all constraints specied by a single inequality of the form x i > a i , for some a i 2 D. Note that the set of constraints , constructed as in Denition 5, is equal to the set of constraints specied by arbitrary nite disjunctions of inequalities of the form x i > a i . It is easily shown that is 1-independent with respect to . Also, CSP1( [ is tractable, since each instance consists of a conjunction of upper bounds for individual variables together with at most one lower bound. Hence, by Corollary 1, _ is tractable. By the result quoted above, this establishes that max-closed constraints are tractable. Unlike the arguments used previously to establish that max-closed constraints P.Jonsson, and M. Koubarakis are tractable [Jeavons and Cooper 1995; Jeavons et al. 1995], the argument above can still be applied when the set of values D is innite. Example 12. [Connected row-convex constraints] The class of binary constraints known as 'connected row-convex' constraints was introduced in [Deville et al. 1997] and shown to be tractable. This class properly includes the 'monotone' relations, identied and shown to be tractable by Montanari in [Montanari 1974]. In this example we will show that the tractability of connected row-convex constraints is a simple consequence of Theorem 3. Furthermore, by using Theorem 3 we are able to generalise this result to obtain tractable constraints over innite sets of values. Let the set of possible values D be the ordered set fd 1 . The denition of connected row-convex constraints given in [Deville et al. 1997] uses a standard matrix representation for binary relations: the binary relation R over D is represented by the mm 0-1 matrix M , by setting if the relation contains the pair hd A relation is said to be connected row-convex if the following property holds: the pattern of 1's in the matrix representation (after removing rows and columns containing only 0's) is connected along each row, along each column, and forms a connected 2-dimensional region (where some of the connections may be diagonal). Here are some examples of connected row-convex An alternative characterisation of this class of constraints, in terms of an algebraic closure property was given in [Jeavons et al. 1998]. Here we obtain another alternative characterisation by noting that the corresponding matrices have a very restricted structure. If we eliminate all rows and columns consisting entirely of zeros, and then consider any remaining zero in the matrix, all of the ones in the same row as the chosen zero must lie one side of it (because of the connectedness condition on the row). Similarly, all of the ones in the same column must lie on one side of the chosen zero. Hence there is a complete path of zeros from the chosen zero to the edge of the matrix along both the row and column in one direction. But this means there must be a complete rectangular sub-matrix of zeros extending from the chosen zero to one corner of the matrix (because of the connectedness condition). This implies that the whole matrix can be obtained as the intersection (conjunc- tion) of 0-1 matrices that contain all ones except for a submatrix of zeros in one corner (simply take one such matrix, obtained as above, for each zero in the matrix to be constructed). There are four dierent forms of such matrices, depending on which corner sub- Building Tractable Disjunctive Constraints 17 matrix is zero, and they correspond to constraints expressed by disjunctive expressions of the four following forms: In these expressions x are variables and d i ; d j are constants. Finally, we note that a row or column consisting entirely of zeros corresponds to a constraint of the form for an appropriate choice of d 1 and Hence, any connected row-convex constraint is equivalent to a conjunction of expressions of these forms. To apply the results of Section 3.3, dene to be the set of all unary constraints specied by a single inequality of the form x i d i or x i d i , for some d i 2 D. It is easily shown that has the Krom property and CSP() is tractable, since each instance consists of a conjunction of upper and lower bounds for individual variables. Hence, by Theorem 3, _ is tractable. By the alternative characterisation described above, this establishes that connected row-convex constraints are tractable. Unlike the arguments used previously to establish that connected row-convex constraints are tractable [Deville et al. 1997; Jeavons et al. 1998], the argument above can still be applied when the set of values D is innite. Example 13. [Linear Horn constraints] The class of constraints over the real numbers known as 'linear Horn' constraints was introduced in [Jonsson and Backstrom 1998; Koubarakis 1996] and shown to be tractable. A linear Horn constraint is specied by a disjunction of weak linear inequalities and linear disequalities where the number of inequalities does not exceed one. The following are examples of linear Horn constraints: Linear Horn constraints form an important class of linear constraints with explicit connections to temporal reasoning [Jonsson and Backstrom 1998]. In particular, the class of linear Horn constraints properly includes the point algebra of [Vilain et al. 1989], the (quantitative) temporal constraints of [Koubarakis 1992; Koubarakis 1995] and the ORD-Horn constraints of [Nebel and Burckert 1995]. All these classes of temporal constraints can therefore be shown to be tractable using the framework developed here. Let the set of possible values be the real numbers (or the rationals). Dene to be the set of all constraints specied by a single (weak) linear inequality (e.g., together with the empty constraint. Dene to be the set of all constraints specied by a single linear disequality (e.g., x 1 P.Jonsson, and M. Koubarakis Note that is the set of constraints specied by a disjunction of disequalities, and the problem CSP( [ ) corresponds to deciding whether a convex polyhe- dron, possibly minus the union of a nite number of hyperplanes, is the empty set. It was shown in [Lassez and McAloon 1989] that the set [ is independent (us- ing their more restrictive notion of independence referred to in Section 3.2, above), and hence that this problem is tractable. However, the set of constraints specied by linear Horn constraints corresponds to the much larger set _ , and this set is not independent in the sense dened in [Lassez and McAloon 1989] (see [Koubarakis 1996]). In order to establish that this larger set of constraints is tractable we shall use the more general notion of 1-independence introduced in this paper. Consider any set of constraints C in CSP( [ (the subset of C which is specied by weak linear inequalities), and let C of C which is specied by linear disequalities). By considering the geometrical interpretation of the constraints as half spaces and excluded hyperplanes in R n , it is clear that C is consistent if and only if C 0 is consistent, and, for each c 2 C 00 , the set consistent (see [Koubarakis 1996]). Hence, is 1-independent with respect to . Now Lemma 2 and Lemma 1 imply that the function Ind-Solvable dened in Figure 1 can be used to determine whether CSP( _ ) has a solution. (In fact, the algorithm in Figure 1 can be seen as a generalisation of the algorithm Consistency which was developed specically for this problem in [Koubarakis 1996].) Finally, to establish tractability, we note that whether a set of inequalities, C 0 , is consistent or not can be decided in polynomial time, using Khachian's linear programming algorithm [Khachian 1979]. Furthermore, for any single disequality constraint, c, we can detect in polynomial time whether C 0 [ fcg is consistent by simply running Khachian's algorithm to determine whether C 0 implies the negation of c. Hence, CSP1( [ ) is tractable, so we can apply Corollary 1 and conclude that CSP( _ ) is tractable. Example 14. [Extended Horn clauses] It was shown by Chandru and Hooker in [Chandru and Hooker 1991] that the class of Horn clauses may be generalised to a much larger class of tractable sets of clauses, which they refer to as 'extended Horn clauses'. To establish that any extended Horn set of clauses can be solved in polynomial time, Chandru and Hooker give a very indirect argument based on a result from the theory of linear programming [Chandru and Hooker 1991]. In this example we shall establish the tractability of a slightly more general class of constraint sets using a much more direct argument, based on Corollary 1. In order to dene this new class of tractable constraint sets over Boolean variables, we rst need to describe how a set of Boolean constraints can be associated with a tree structure. Let T be a rooted, undirected, tree in which the edges are labelled with propositional literals in such a way that each variable name occurs at most once. Note that T may have an innite number of edges. If we select an edge of T , then it can be oriented in two dierent ways: either towards the root, or else away from the root. An edge of T , together with a particular selected orientation, will be called Building Tractable Disjunctive Constraints 19 an arc of T . For each possible arc, a, of T we dene an associated literal, in the following way. If a is oriented away from the root, then we dene the associated literal to be the label of a in T , otherwise, if a is oriented towards the root, then we dene the associated literal to be the negation of the label of a in T . x3 Fig. 3. A labelled tree For example, let T be the tree shown in Figure 3, with root node n 0 . and consider the arc from node n 0 to n 1 , denoted [n The literal associated with [n according to the denition just given, is x 1 . Now consider the arc from node n 5 to The literal associated with [n 5 Given any set of arcs, we dene an associated clause consisting of the disjunction of the associated literals. For example, the clause associated with the set of arcs Some sets of arcs have the special property that they form a path. For example, in the tree shown in Figure 3, the arcs [n a path of length 2. The arcs [n 6 a path of length 4. For any rooted, undirected, tree T , we dene T to be the set of all Boolean constraints specied by a clause associated with some path in T . For example, when T is the tree shown in Figure 3, then T includes the clauses but does not include the clause or the clause x 1 _ :x 2 . We rst note that for any tree T , including innite trees, the set of constraints T is tractable. This is because any instance of CSP( T ) can be solved by the following polynomial-time algorithm (adapted from [Chandru and Hooker 1991]): (1) If the instance contains any clauses associated with paths of length one (i.e., unit clauses), then assign values to the corresponding variables to satisfy these 20 D.Cohen, P.Jeavons, P.Jonsson, and M. Koubarakis clauses. If any variable receives contradictory assignments then report that the instance is insoluble and stop. (2) If any variables have been assigned values, then remove all clauses which are satised by these assignments, contract the corresponding edges in T (i.e., remove these edges and identify each pair of end points), and return to Step 1. (3) Report that the problem is soluble. (Since all remaining clauses correspond to paths of length two or more in T , there is guaranteed to be a solution which can be obtained by assigning the values true and false alternately along each branch of T , starting with true.) However, if we allow disjunctions of constraints in T , then we no longer have tractability. In fact, we will now show that CSP( T can be NP-complete. x3 Fig. 4. A labelled star This intractability result holds even when the trees are restricted to be stars (that is, a tree where every edge is incident to the root). Dene Sn to be a star with n edges labelled x . For example, the star S 5 is shown in Figure 4. Now consider the innite star, S1 , with edges labelled x :. The set of constraints consists of all constraints specied by clauses of the form x i , :x i , or x i _:x j , for all choices of i and j. Hence the class of constraints S1 _ S1 consists of all constraints specied by disjunctions of these clauses, or, in other words, by clauses of the form x possible choices of i; j; k; l. This set of clauses does not fall into any of the 6 tractable classes identied by Schaefer in [Schaefer 1978], and hence the corresponding satisability problem, CSP(S1 _ S1 ), is NP-complete. Building Tractable Disjunctive Constraints 21 In order to obtain tractable sets of disjunctive constraints, we now identify a restricted subset of T which is 1-independent with respect to T . For any rooted, undirected, tree T , we dene T to be the set of all Boolean constraints specied by a clause associated with some path in T which ends at the root node of T . For example, when T is the tree shown in Figure 3, then T includes the clauses but does not include the clause x 4 _ :x 3 , or the clause :x 5 _ x 1 . As a second example, when T is a star labelled with positive literals, such as the one shown in Figure 4, then T consists of all constraints specied by a single negative literal. Since T T , we can use the same algorithm as before to show that the problem To show that T is 1-independent with respect to T we note that any minimal incompatible subset of clauses can involve at most one clause from T . In view of these facts, we can apply Corollary 1 to conclude that T _ T is tractable. Notice that in the special case when T is a star labelled with positive literals the set of constraints T _ corresponds precisely to the set of Horn clauses over the variables labelling T . In the more general case when T is an arbitrary tree labelled with positive literals, the set of constraints T _ T includes the extended Horn set of clauses associated with T , as dened by Chandru and Hooker [Chandru and Hooker 1991]. Further- more, in the most general case when T is an arbitrary tree labelled with arbitrary literals, the set of constraints T _ T includes the hidden extended Horn clauses associated with T , as dened by Chandru and Hooker [Chandru and Hooker 1991]. Hence our results are sucient to show that extended Horn clauses and hidden extended Horn clauses are tractable. However, we point out that the extended Horn set associated with a tree T , as dened in [Chandru and Hooker 1991], is a little more restricted than the set of constraints T _ T , and hence our result represents a generalisation of the result in [Chandru and Hooker 1991]. This is because the literals in any clause from an extended Horn set are required to form an 'extended star-chain' pattern in the corresponding tree [Chandru and Hooker 1991]. Such a pattern consists of an arbitrary number of edge disjoint paths terminating at the root node, together with at most one other path. The literals of the clauses representing constraints in _ similar patterns, but there is no requirement for the paths into the root to be disjoint. For example, when T is the tree shown in Figure 3, then T _ T includes the clause :x 1 _:x 2 _:x 3 _x 6 (formed from the disjunction of :x 1 _:x 2 and :x 1 _:x 3 and x 6 ). However, this clause is not in the (hidden) extended Horn set associated with T . Example 15. [Extended Krom clauses] The ideas described in Example 14, together with Theorem 3, can be used to identify a new class of tractable Boolean constraint sets which we will call 'extended Krom' sets. As in Example 14, we associate propositional clauses with sets of arcs in a labelled tree. Let T be a rooted, undirected tree, whose edges are labelled with propositional 22 D.Cohen, P.Jeavons, P.Jonsson, and M. Koubarakis literals in such a way that each variable name appears at most once. Note that T may have an innite number of edges. We dene T to be the set of all Boolean constraints specied by a clause associated with some path in T which either starts or ends at the root node of T . For example, when T is the tree shown in Figure 3, then T includes the clauses but does not include the clause x 4 _:x 3 , or the clause As a second example, when T is a star labelled with positive literals, such as the one shown in Figure 4, then T consists of all constraints specied by a single positive or negative literal. The algorithm described in Example 14 can be used to show that CSP( T ) is tractable. Furthermore, it is easy to show that any set of clauses chosen from T is satisable, unless it contains the unit clauses x i and :x i for some i. Hence T has the Krom property. In view of these facts, we can apply Theorem 3 to conclude that T _ T is tractable. Notice that in the special case when T is a star the set of constraints T corresponds precisely to the set of Krom clauses over the variables labelling T . In the more general case when T is an arbitrary tree, the set of constraints includes a wider variety of clauses. For example, when T is the tree shown in Figure 3, then T includes the following clauses: Example 16. [Tractable set constraints] In [Drakengren and Jonsson 1998; Drakengren and Jonsson 1997], Drakengren and Jonsson identify a number of tractable classes of set constraints that are spec- ied by expressions involving set-valued variables and the relation symbols , disj (denoting disjointness) and 6=. In this example we will show that these tractability results can be obtained as a simple consequence of Theorem 2. In this way we provide a shorter proof and show that these results about set constraints conform to one of the general patterns for tractable disjunctive constraints described in this paper. We rst give the relevant denitions from [Drakengren and Jonsson 1998; Drak- engren and Jonsson 1997]. An atomic set constraint or atomic set relation is an expression of the form x i x disjunctive set constraint or disjunctive set relation (DSR) is a disjunction of atomic constraints. A DSR is called Horn if it consists of zero or more disjuncts of the form x i and at most one disjunct of the form x i x j . A DSR is called 2-Horn if it consists of zero or more disjuncts of the form x i 6= x j and at most one disjunct of the form . The following are examples of Horn DSRs. The latter two examples in this list are also 2-Horn. An -interpretation is a function that maps all set variables in a problem instance to (possibly empty) sets. An S-interpretation is a function that maps all Building Tractable Disjunctive Constraints 23 set variables to nonempty sets. An atomic constraint x 1 R x 2 is satised by an S ; -interpretation (S-interpretation), I , if and only if I(x 1 there exists an S ; -interpretation which satises some disjunct d i of d. A set C of DSRs is there exists an S ; -interpretation (S-interpretation) which satises all members of C. Such a satisfying S ; -interpretation (S-interpretation) is called an S ; -model (S-model) of C. The following decision problems are studied in [Drakengren and Jonsson 1998; Drakengren and Jonsson 1997]: Instance:. A nite set C of Horn DSRs. Question:. Does there exist an S ; -model of C? |HornDSRSat: Instance:. A nite set C of Horn DSRs. Question:. Does there exist an S-model of C? Instance:. A nite set C of 2-Horn DSRs. Question:. Does there exist an S ; -model of C? |2HornDSRSat: Instance:. A nite set C of 2-Horn DSRs. Question:. Does there exist an S-model of C? The problem HornDSRSat ; is NP-complete [Drakengren and Jonsson 1998]. To show that the problem HornDSRSat can be solved in polynomial time we dene to be the set of all constraints specied by expressions of the form x y (where x and y are set-valued variables which must be assigned non-empty sets), together with the empty constraint. We dene to be the set of all constraints specied by expressions of the form x 6= y or x disj y (where x and y are set-variables which must be assigned non-empty sets), together with the empty constraint. Then is the set of all constraints specied by arbitrary disjunctions of expressions of the form x 6= y or x disj y. In [Drakengren and Jonsson 1997] it is shown that the problem of deciding whether a set of atomic set constraints has an S-model can be solved in polynomial time. The algorithm presented in [Drakengren and Jonsson 1997] represents set constraints by a labeled directed graph and essentially consists of two tests. First, if there is a triple of set variables x; y and z for which the constraints z x; z y and x disj y are implied by the given constraint set, the input set is rejected (because variable z is forced to be equal to ;). Secondly, if there is a pair of set variables x and y for which the constraints x are implied by the given constraint set, then the input is rejected (because these constraints cannot all be satised). If a constraint set passes both of these tests then it is accepted. The existence of this simple algorithm implies that CSP( [ ) and hence also can be solved in polynomial time. It also implies that is 1- independent with respect to . Hence, by Corollary 1 we can immediately conclude that CSP( _ ), which corresponds to the decision problem HornDSRSat, can be solved in polynomial time. In fact the algorithm dened in Figure 1 can be P.Jonsson, and M. Koubarakis seen as a generalisation of the iterative version of Algorithm Horn-Sat presented in [Drakengren and Jonsson 1997]. The problem 2HornDSRSat is a subproblem of HornDSRSat so it can also be solved in polynomial time. To show that the decision problem 2HornDSRSat ; can be solved in polynomial time, dene to be the set of all constraints specied by an expression which is either of the form x y or else of the form x disj y (where x and y are set-valued variables which can be assigned arbitrary sets), together with the empty constraint. Dene to be the set of all constraints specied by an expression of the form x 6= y (where x and y are set-valued variables which can be assigned arbitrary sets), together with the empty constraint. In this case is the set of constraints specied by a disjunction of expressions of the form x 6= y. In [Drakengren and Jonsson 1998] it is shown that the problem of deciding whether a set of atomic set constraints has an S ; -model can be solved in polynomial time. As in the above case, set constraints are represented by a labeled directed graph. The algorithm proceeds in two steps. First, for any triple of set variables x; y and z for which the constraints z x; z y and x disj y are implied by the given constraint set, variable z is forced to be equal to ;. Then the constraints are examined to nd out whether a contradictory triple of constraints x implied by the given constraint set. If this is the case, then the set is rejected. Otherwise, it is accepted. The existence of this simple algorithm implies that CSP( [ ) and hence also can be solved in polynomial time. It also implies that is 1- independent with respect to . Hence, by Corollary 1 we can immediately conclude that CSP( _ ), which now corresponds to the decision problem 2HornDSRSat ; , can be solved in polynomial time. As in the above case, the algorithm dened in Figure 1 can be seen as a generalisation of the iterative version of Algorithm 2- Horn-Sat presented in [Drakengren and Jonsson 1998]. Having identied the key properties underlying all of these tractable classes, we are now in a position to identify new tractable classes simply by searching for appropriate sets of tractable constraints which have these properties. Example 17. [Disjunctive congruences] Constraints in the form of congruence relations over the integers are used in a variety of applications, including the representation of large integers in computer algebra systems [von zur Gathen and Gerhard 1999], and the representation of periodic events in temporal databases [Kabanza et al. 1995; Bertino et al. 1998]. One of the fundamental results of elementary number theory is the Chinese Remainder Theorem [Jackson 1975], which states that a collection of simultaneous linear congruences x a x a x an (mod mn ) is solvable if and only if the greatest common divisor of m i and m j divides a i a j , Building Tractable Disjunctive Constraints 25 for all distinct i and j. When this condition holds between a pair of congruences, we shall say that they are compatible (Note that compatibility can be decided in polynomial time.) Using this result, together with the results established earlier in this paper, we will construct a number of new tractable classes of constraints. Let the set of possible values be the set of integers. Dene to be the set of all unary constraints which are specied by a linear congruence of the form x a (mod m), for some natural number a, and some modulus m, together with the empty constraint. For each natural number b, dene b to be the subset of containing all unary constraints which are specied by a congruence of the form x b (mod m), for some m. A problem instance in CSP() is specied by a collection of simultaneous congru- ences. For example, a typical set of constraints in CSP(), involving the variables x 1 and x 2 , would be: The Chinese Remainder Theorem implies that any set of constraints in CSP() has a solution, unless it contains a pair of constraints which are incompatible. In view of this fact, the results obtained in this paper can be used to construct tractable disjunctive constraints in the following three ways: |For any natural number b, it is clear from the denition of compatibility above that every pair of constraints in b is compatible. Hence, b has the guaranteed satisfaction property, and by Theorem 1 we conclude that CSP( b _ ) is tractable. This means that there is an ecient way to solve any collection of simultaneous disjunctions of congruences which all have the property that at least one disjunct comes from b . For example, when have the following collection: |For any natural number b, CSP b 1( [ b ) is tractable, because we can determine whether or not any given instance has a solution in polynomial time by examining each pair of constraints to see whether they are compatible. Further- more, no pair of constraints which are both in b can be incompatible, so b is 1-independent with respect to . Hence, by Corollary 1, we conclude that b ) is also tractable, for any natural number b. This means that there is an ecient way to solve any collection of simultaneous disjunctions of congruences which all have the property that at most one disjunct comes from and the remainder (if any) from b . For example, when have the following collection of congruences: 26 D.Cohen, P.Jeavons, P.Jonsson, and M. Koubarakis |From the observations already made it is clear that the set has the Krom prop- erty. Hence, by Theorem 3 we conclude that CSP( _) is tractable. This means that there is an ecient way to solve any collection of simultaneous disjunctions of congruences which all contain at most two disjuncts. For example, we might have the following collection: Note that the new tractable disjunctive constraints sets constructed in these three distinct ways are all incomparable with each other. 5. CONCLUSION In this paper we have established three sucient conditions for tractability of disjunctive constraints. We have shown that these conditions account for a wide variety of known tractable constraint classes, over both nite and innite sets of values, and that they aid the search for new tractable constraint classes. The examples we have given of new tractable classes obtained in this way are as follows: |A generalisation of max-closed constraints to innite (ordered) domains (Exam- ple 11); |A generalisation of connected row-convex constraints to innite (ordered) domains (Example 12); |A (slight) generalisation of extended Horn Clauses (Example 14); |The new class of extended Krom clauses (Example 15); |Three new classes of tractable disjunctive congruences (Example 17). These results provide the rst examples of a constructive approach to obtaining tractable constraints, based on combining known tractable classes. This new approach to obtaining tractable classes leads to results of great gener- ality, as we have shown in this paper. 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A unifying framework for tractable con- straints Closure properties of constraints. Tractable constraints on ordered domains. Journal of Computer and System Sciences A polynomial time algorithm for linear programming. Fast parallel constraint satisfaction. Dense time and temporal constraints with 6 From local to global consistency in temporal constraint networks. A canonical form for generalized linear constraints. In TAPSOFT A constraint sequent calculus. A canonical form for generalized linear constraints. Journal of Symbolic Computation Engineering dynamic scheduler for work manager. Consistency in networks of relations. Networks of constraints: Fundamental properties and applications to picture processing. Reasoning about temporal relations: a maximal tractable subclass of Allen's interval algebra. 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Modern Computer Algebra. --TR A sufficient condition for backtrack-bounded search Tree clustering for constraint networks (research note) Constraint propagation algorithms for temporal reasoning: a revised report Extended Horn sets in propositional logic A canonical form for generalized linear constraints A generic arc-consistency algorithm and its specializations A constraint sequent calculus Fast parallel constraint satisfaction Decomposing constraint satisfaction problems using database techniques Characterising tractable constraints Reasoning about temporal relations On the minimality and global consistency of row-convex constraint networks Handling infinite temporal data Tractable constraints on ordered domains Closure properties of constraints Constraints, consistency and closure A unifying approach to temporal constraint reasoning An access control model supporting periodicity constraints and temporal reasoning Modern computer algebra Computers and Intractability Engineering Dynamic Scheduler for Work Manager Independence of Negative Constraints A Unifying Framework for Tractable Constraints From Local to Global Consistency in Temporal Constraint Networks The complexity of satisfiability problems --CTR D. A. Cohen, Tractable Decision for a Constraint Language Implies Tractable Search, Constraints, v.9 n.3, p.219-229, July 2004 Mathias Broxvall, A method for metric temporal reasoning, Eighteenth national conference on Artificial intelligence, p.513-518, July 28-August 01, 2002, Edmonton, Alberta, Canada David Cohen , Peter Jeavons , Richard Gault, New tractable constraint classes from old, Exploring artificial intelligence in the new millennium, Morgan Kaufmann Publishers Inc., San Francisco, CA, David Cohen , Peter Jeavons , Richard Gault, New Tractable Classes From Old, Constraints, v.8 n.3, p.263-282, July Mathias Broxvall , Peter Jonsson , Jochen Renz, Disjunctions, independence, refinements, Artificial Intelligence, v.140 n.1-2, p.153-173, September 2002 Claudio Bettini , X. Sean Wang , Sushil Jajodia, Solving multi-granularity temporal constraint networks, Artificial Intelligence, v.140 n.1-2, p.107-152, September 2002 Andrei Krokhin , Peter Jeavons , Peter Jonsson, Reasoning about temporal relations: The tractable subalgebras of Allen's interval algebra, Journal of the ACM (JACM), v.50 n.5, p.591-640, September Mathias Broxvall , Peter Jonsson, Point algebras for temporal reasoning: algorithms and complexity, Artificial Intelligence, v.149 n.2, p.179-220, October

constraint satisfaction problem;relations;complexity;NP-completeness;disjunctive constraints;independence

356483

The CREW PRAM Complexity of Modular Inversion.

One of the long-standing open questions in the theory of parallel computation is the parallel complexity of the integer gcd and related problems, such as modular inversion. We present a lower bound $\Omega (\log n)$ for the parallel time on a concurrent-read exclusive-write parallel random access machine (CREW PRAM) computing the inverse modulo certain n-bit integers, including all such primes. For infinitely many moduli, our lower bound matches asymptotically the known upper bound. We obtain a similar lower bound for computing a specified bit in a large power of an integer. Our main tools are certain estimates for exponential sums in finite fields.

Introduction . In this paper we address the problem of parallel computation of the inverse of integers modulo an integer M . That is, given positive integers M # 3 and x < M , with gcd(x, we want to compute its modular inverse inv M (x) # N defined by the conditions is the Euler function, inversion can be considered as a special case of the more general question of modular exponentiation. Both these problems can also be considered over finite fields and other algebraic domains. For inversion, exponentiation and gcd, several parallel algorithms are in the literature [1, 2, 3, 9, 10, 11, 12, 13, 14, 15, 18, 20, 21, 23, 28, 30]. The question of obtaining a general parallel algorithm running in poly-logarithmic time (log n) O(1) for n-bit integers M is wide open [11, 12]. Some lower bounds on the depth of arithmetic circuits are known [11, 15]. On the other hand, some examples indicate that for this kind of problem the Boolean model of computation may be more powerful than the arithmetic model; see discussions of these phenomena in [9, 11, 15]. In this paper we show that the method of [5, 26] can be adapted to derive non-trivial lower bounds on Boolean concurrent-read exclusive-write parallel random access machines (CREW PRAMs). It is based on estimates of exponential sums. Our bounds are derived from lower bounds for the sensitivity #(f) (or critical complexity) of a Boolean function . It is defined as the largest integer m # n such that there is a binary vector the vector obtained from x by flipping its ith coordinate. In other words, #(f) is the maximum, over all input vectors x, of the number of points y on the unit Hamming sphere around x with f(y) #= f(x); see e.g., [31]. Since [4], the sensitivity has been used as an e#ective tool for obtaining lower bounds of the CREW PRAM complexity, i.e., the time complexity on a parallel random access machine with an unlimited number of all-powerful processors, where each machine can read from and write to one memory cell at each step, but where no write conflicts are allowed: each memory cell may be written into by only one processor, at each time step. Fachbereich Mathematik-Informatik, Universitat Paderborn, 33095 Paderborn, Germany (gathen@uni-paderborn.de) School of MPCE, Macquarie University, Sydney, NSW 2109, Australia (igor@mpce.mq.edu.au) J. VON ZUR GATHEN AND I. E. SHPARLINSKI By [22], 0.5 log 2 (#(f)/3) is a lower bound on the parallel time for computing f on such machines, see also [6, 7, 8, 31]. This yields immediately the lower bound #(log n) for the OR and the AND of n input bits. It should be contrasted with the common CRCW PRAM, where write conflicts are allowed, provided every processor writes the same result, and where all Boolean functions can be computed in constant time (with a large number of processors). The contents of the paper is as follows. In Section 2, we prove some auxiliary results on exponential sums. We apply these in Section 3 to obtain a lower bound on the sensitivity of the least bit of the inverse modulo a prime. In Section 4, we use the same approach to obtain a lower bound on the sensitivity of the least bit of the inverse modulo an odd squarefree integer M . The bound is somewhat weaker, and the proof becomes more involved due to zero-divisors in the residue ring modulo M , but for some such moduli we are able to match the known upper and the new lower bounds. Namely, we obtain the lower bound #(log n) on the CREW PRAM complexity of inversion modulo an n-bit odd squarefree M with not 'too many' prime divisors, and we exhibit infinite sequences of M for which this bound matches the upper bound O(log n) from [11] on the depth of P-uniform Boolean circuits for inversion modulo a 'smooth' M with only `small' prime divisors; see (4.6) and (4.7). For example, the bounds coincide for moduli are any #s/ log s# prime numbers between s 3 and 2s 3 . We apply our method in Section 5 to the following problem posed by Allan Borodin (see Open Question 7.2 of [11]): given n-bit positive integers m, x, e, compute the mth bit of x e . Generally speaking, a parallel lower bound #(log n) for a problem with n inputs is not a big surprise. Our interest in these bounds comes from their following features: . some of these questions have been around for over a decade; . no similar lower bounds are known for the gcd; . on the common CRCW PRAM, the problems can be solved in constant time; . for some types of inputs, our bounds are asymptotically optimal; . the powerful tools we use from the theory of finite fields might prove helpful for other problems in this area. 2. Exponential sums. The main tool for our bounds are estimates of exponential sums. For positive integers M and z, we write eM The following identity follows from the formula for a geometric sum. Lemma 2.1. For any integer a, Lemma 2.2. For positive integers M and H, we have X is the remainder of H - 1 modulo M . THE CREW PRAM COMPLEXITY OF MODULAR INVERSION 3 Proof. We note that Thus X From Lemma 2.1 we see that the last sum is equal to MW , where W is the number of (x, y) with It is easy to see that and the result follows. Taking into account that (r we derive from Lemma 2.2 that the bound X holds for any H and M . Also, it is easy to see that for H # M , then the identity of Lemma 2.2 takes the form X Finally, we have X Indeed, this sum is smaller by the term corresponding to a = 0, which equals H 2 . 4 J. VON ZUR GATHEN AND I. E. SHPARLINSKI In the sequel, we consider several sums over values of rational functions in residue rings, which may not be defined for all values. We use the symbol P # to express that the summation is extended over those arguments for which the rational function is well-defined, so that its denominator is relatively prime to the modulus. We give an explicit definition only in the example of the following statement, which is known as the Weil bound ; see [19, 25, 32]. Lemma 2.3. Let f, g # Z[X] be two polynomials of degrees n, m, respectively, and p a prime number such that the rational function f/g is defined and not constant modulo p. Then X # (n +m- 1)p 1/2 . Let #(k) denote the number of distinct prime divisors of an integer k. The following statement is a combination of the Chinese Remainder Theorem and the Weil bound. Lemma 2.4. Let M # N be squarefree with M # 2, d a divisor of M , and f, g # Z[X] of degrees n, m, respectively, such that the rational function f/g is defined and not constant modulo each prime divisor of M . Then X Proof. In the following, p stands for a prime divisor of M . We define M p # N by the conditions Then, one easily verifies the identity We use the estimate of Lemma 2.3 for those p for which p | / d and p > max{n, m}, and estimate trivially by p the sum for each other p. Then X # Y p | (n +m- 1)p 1/2 Y p|d Since #(M/d) #(M), we obtain the desired estimate. Lemma 2.5. Let M # 2 be a squarefree integer, f, g # Z[X] of degrees n, m, respectively, such that f/g is defined and neither constant nor a linear function modulo each prime divisor p of M . Then for any N,H, d # N with H # M and d|M , we have X Proof. From Lemma 2.1 we obtain X X X X X From Lemma 2.4 we see that for each a < M the sum over u can be estimated as X d. Applying the estimate (2.2), we obtain the result. The following result is the particular case Theorem 1 of [29]. Lemma 2.6. There exists a constant c such that for all polynomials with gcd(a t , . , a 1 , X For a 0 , . , a k-1 # Z, not all zero, we define (a 0 , . , a k-1 ) to be the largest exponent e for which 2 e divides a 0 , . , a k-1 . Lemma 2.7. Let a 0 , . , a k-1 # Z not be all zero, and a Proof. We extend to Q by (a/b) = (a)-(b) and to nonzero matrices in Q kk by taking the minimum value at all nonzero columns. Then (U v) # (U)+(v) for a matrix U and a vector v such that Uv #= 0. . The determinant of this Vandermonde matrix has value 6 J. VON ZUR GATHEN AND I. E. SHPARLINSKI We consider an entry of the adjoint adC k of C k . Each of the summands contributing to the determinant expansion of that entry is divisible by so that (In fact, we have equality, since det C k-1 has the right hand side as its -value and is one entry of adC k .) Therefore Now from the inequality (b) the result follows. We also need an estimate on the number of terms in the sum of Lemma 2.5. For a polynomial g # Z[X] and M,H # Z, we denote by T g (M,H) the number of x # Z for which 0 # x < H and gcd (g(x), M) = 1. The following result is, probably, not new and can be improved via more sophisticated sieve methods. Lemma 2.8. Let M > 1 be squarefree and g # Z[x] of degree m such that gcd(g(x), Z. Then for all integers H # M , we have Proof. We denote by #(M, H) the number of x # {0, . , H - 1} such that and set is squarefree, the inclusion-exclusion principle yields #(d)=k #(d, H). For any divisor d of M we have d #(p). Therefore, #(d)=k #(d) #(d) By assumption, g takes a nonzero value modulo every prime divisor p of M . Thus #(p) # min{m, p - 1}, and the claim follows. Throughout this paper, log z means the logarithm of z in base 2, ln z means the natural logarithm, and 1, if z # 1. Lemma 2.9. For positive integers m and M , with M > 1 squarefree, we have Y Proof. We split the logarithm of the product as follows p#2m p>2m , and prove a lower bound on each summand. For the first one, we use that for x > 1 by [24], (3.15). Thus, for m > 1 p#2m p#2m It is easy to verify that for the sum on the left hand side does not exceed 3m as well. For the second summand, we use that (1 This implies that p>2m p>2mp From [24], (3.20), we know that s be the sth prime number, so that p s # s 2 for s # 2. Thus for s # 2 we have 2. The inequality between the first and last term is also valid for s = 1. Now (2.4), (2.5), and (2.6) imply the claim. 8 J. VON ZUR GATHEN AND I. E. SHPARLINSKI 3. PRAM complexity of the least bit of the inverse modulo a prime number. In this section, we prove a lower bound on the sensitivity of the Boolean function representing the least bit of the inverse modulo p, for an n-bit prime p. For x # N with gcd(x, we recall the definition of in (1.1). Furthermore, for x 0 , . , x n-2 # {0, 1}, we let We consider Boolean functions f with n - 1 inputs which satisfy the congruence for all x 0 , . , x n-2 # {0, 1} with no condition is imposed for the value of f(0, . , 0). Finally we recall the sensitivity # from the introduction. Theorem 3.1. Let p be a su#ciently large n-bit prime. Suppose that a Boolean function f(x 0 , . , x n-2 ) satisfies the congruence (3.2). Then log n - 1. Proof. We let k be an integer parameter to be determined later, with 2 # k # n - 3, and show that large enough. For this, we prove that there is some integer z with 1 # z # 2 n-k-1 and provided that p is large enough. We note that all these 2 k z and 2 k z are indeed invertible modulo p. We set e k. Then it is su#cient to show that there exist integers z, w 0 , . , w k with Next we set k. Then it is su#cient to find integers x, y, u 0 , . , Indeed from each solution of the system (3.4) we obtain a solution of the system (3.3) by putting On the other hand, the system (3.4) contains more variables and is somewhat easier to study. A typical application of character sum estimates to systems of equations proceeds as follows. One expresses the number of solutions as a sum over a # Z p , using Lemma 2.1, then isolates the term corresponding to a = 0, and (hopefully) finds that the remaining sum is less than the isolated term. Usually, the challenge is to verify the last part. In the task at hand, Lemma 2.1 expresses the number of solutions of (3.4) as THE CREW PRAM COMPLEXITY OF MODULAR INVERSION 9 a a a where the first summand corresponds to a R to the remaining sum, and we used (2.2). For other the sum over x, y satisfies the conditions of Lemma 2.5, with a where a i Therefore f/g is neither constant nor linear modulo p. Thus, Y X We have left out the factors |e p (-a i # i )|, which equal 1, transformed the summation index 2a i into a i , and used the identity (2.2). It is su#cient to show that H 2 K 2(k+1) is larger than |R|, or that Since K # (p - 6)/4, it is su#cient that (3. J. VON ZUR GATHEN AND I. E. SHPARLINSKI We now set log n)/6#, so that 6(k 2. Now (1 real z > 0, and 2. Furthermore, p 1/2 < 2 n/2 and 32n/3 < n 3/2 , and (3.6) follows from log n Hence the inequality (3.5) holds, and we obtain #(f) # k # n/6 - 0.5 log n - 1. From [22] we know that the CREW PRAM complexity of any Boolean function f is at least 0.5 log(#(f)/3), and we have the following consequence. Corollary 3.2. Any CREW PRAM computing the least bit of the inverse modulo a su#ciently large n-bit prime needs at least 0.5 log n - 3 steps. 4. PRAM complexity of inversion modulo an odd squarefree integer. In this section, we prove a lower bound on the PRAM complexity of finding the least bit of the inverse modulo an odd squarefree integer. To avoid complications with gcd computations, we make the following (generous) definition. Let M be an odd squarefree n-bit integer, and f a Boolean function with n inputs. Then f computes the least bit of the inverse modulo M if and only if for all x # {0, 1} n-1 with gcd(num(x), is the nonnegative integer with binary representation x, similar to (3.1). Thus no condition is imposed for integers x # 2 n or that have a nontrivial common factor with M . Theorem 4.1. Let M > 2 be an odd squarefree integer with #(M) distinct prime divisors, and f the Boolean function representing the least bit of the inverse modulo M , as above. Then 4Lnln #(M) +O(1) . Proof. We let k an integer parameter to be determined later. We want to show that there is some integer z with 1 # z # 2 n-k-1 for which As in the proof of Theorem 3.1 we see that in this case #(f) # k. We put e k. It is su#cient to show that there exist integers such that Next, we set As in the proof of Theorem 3.1 we see that it is su#cient to find integers x, y, u 0 , . , satisfying the following THE CREW PRAM COMPLEXITY OF MODULAR INVERSION 11 conditions for Lemma 2.1 expresses the number of solutions as a a a a where S d is the subsum over those 0 # a 0 , . , a k < M for which It is su#cient to show that d<M |S d |. First we note that SM consists of only one summand corresponding to a to be added equal 1, we only have to estimate the number of terms for which the argument of defined. For each y with 0 # y < H, we apply Lemma 2.8 to the polynomial of degree k + 1. We set using Lemmas 2.8 and 2.9, we deduce that The other |S d | are bounded from above by X a J. VON ZUR GATHEN AND I. E. SHPARLINSKI a Now let a i d Then a i d and f/g is neither constant nor linear modulo any prime divisor . Thus we can apply Lemma 2.5 and find that X a the hypothesis of the lemma is satisfied because M is squarefree. If d < M , then a M/d, with at least one b i #= 0. Then e M/d e M/d 1#b0<M/d X X Since M/d is odd, we may replace the summation index 2b i by b i . From the inequalities (2.3) and (2.1) we find 1#b0<M/d X X Combining these inequalities, we obtain therefore d<M d -3/2 where Using (4.1) and (4.2) it is now su#cient to prove that for some To do so we suppose that and will show that k satisfies the opposite inequality. Obviously, we may assume that We also recall that K # (M - 6)/4 and # M2 -k-3 . Now if immediately obtain (4.3). Otherwise, we derive from (4.4) that Comparing this inequality with the inequality (4.3) we obtain the desired statement. Our bound takes the form for an odd squarefree n-bit M with #(M) # ln M/LnlnM for some constant # < 0.5. We recall that almost all odd squarefree numbers M . 14 J. VON ZUR GATHEN AND I. E. SHPARLINSKI We denote by i PRAM (M) and i BC (M) the CREW PRAM complexity and the Boolean circuit complexity, respectively, of inversion modulo M . We know from [11, 21] that for any n-bit integer M . The smoothness #(M) of an integer M is defined as its largest prime divisor, and M is b-smooth if and only if #(M) # b. Then Since we are mainly interested in lower bounds in this paper, we do not discuss the issue of uniformity. Corollary 4.2. for any odd squarefree n-bit integer M with #(M) # 0.49 ln M/LnlnM . Theorem 4.3. There is an infinite sequence of moduli M such that the CREW PRAM complexity and the Boolean circuit complexity of computing the least bit of the inverse modulo M are both #(log n), where n is the bit length of M . Proof. We construct infinitely many odd squarefree integers M with #(M) # 0.34 ln M/LnlnM , thus satisfying the lower bound (4.8), and with smoothness thus satisfying the upper bound O(ln ln O(log n) of [11] on the depth of Boolean circuits for inversion modulo such M . For each integer s > 1 we select #s/ ln s# primes between s 3 and 2s 3 , and let M be the product of these primes. Then, M # s 3s/ large enough. 5. Complexity of one bit of an integer power. For nonnegative integers u and m, we let Btm (u) be the mth lower bit of u, i.e., Btm with each u i # {0, 1}. If u In this section, we obtain a lower bound on the CREW PRAM complexity of computing Btm (x e ). For small m, this function is simple, for example Bt 0 can be computed in one step. However, we show that for larger m this is not the case, and the PRAM complexity is #(log n) for n-bit data. Exponential sums modulo M are easiest to use when M is a prime, as in Section 3. In Section 4 we had the more di#cult case of a squarefree M , and now we have the extreme case Theorem 5.1. Let m and n be positive integers with f be the Boolean function with 2n inputs and Proof. We set , and consider so that #(f) #(g). Furthermore, k is an integer parameter with e # k # 2 to be determined later. To prove that #(g) # k, it is su#cient to show that there exists an integer x with The first equality holds for any such x because e 2 # m, and thus the conditions are equivalent to the existence of integers x, u 0 , . , u k-1 such that which is implied by the existence of x, u 0 . , u k-1 , v 0 , . , v k-1 with We set Lemma 2.1 expresses the number of solutions of (5.1) as a a a is the subsum over all integers 0 # a 0 , . , a k-1 < 2 m with It is su#cient to show that 0#<m Sm contains only one summand, for a Using the function from Section 2, we have for # < m that (a 0 ,.,a k-1 )=# X a J. VON ZUR GATHEN AND I. E. SHPARLINSKI Now let a 0 , . , a k-1 < 2 m . We set a 0#j#e so that a We put X where c is the constant from Lemma 2.6. This bound also holds for # m, because the sum contains terms with absolute value 1. Using the (crude) estimate e e, and noting that a from Lemma 2.7 we derive that for tuples with (a 0 , . , a k-1 provided that k # 2. Substituting this bound in (5.5), we obtain where (a THE CREW PRAM COMPLEXITY OF MODULAR INVERSION 17 We set 2. and as in the proof of Theorem 4.1, from Lemma 2.2 we find Next, we obtain 0#<m 0#<m ck 2 n+2mk-k-e-m/e-k 2 /2e+3 0#<m < ck 2 n+2mk-k-e-m/e-k 2 /2e+4 . We set -m 1/3 It easy to verify that the inequality (5.2) holds for this choice of k, provided that m is large enough. Corollary 5.2. Let . The CREW PRAM complexity of finding the mth bit of an n-bit power of an n-bit integer is at least 0.25 log m- o(log m). In particular, for 6. Conclusion and open problems. Inversion in arbitrary residue rings can be considered along these lines. There are two main obstacles for obtaining similar results. Instead of the powerful Weil estimate of Lemma 2.3, only essentially weaker (and unimprovable) estimates are available [17, 27, 29]. Also, we need a good explicit estimate, while the bounds of [17, 27] contain non-specified constants depending on the degree of the rational function in the exponential sum. The paper [29] deals with polynomials rather than with rational functions, and its generalization has not been worked out yet. Open Question 6.1. Extend Theorem 4.1 to arbitrary moduli M . Moduli of the form prime number, are of special interest because Hensel's lifting allows to design e#cient parallel algorithms for them [2, 11, 15]. Theorem 5.1 and its proof demonstrate how to deal with such moduli and what kind of result should be expected. Each Boolean function f(X 1 , . , X n ) can be uniquely represented as a multilinear polynomial of degree of the form 0#k#d 1#i1<.<ik #r We define its weight as the number of nonzero coe#cients in this representation. Both the weight and the degree can be considered as measures of complexity of f . In [5, 26], the same method was applied to obtain J. VON ZUR GATHEN AND I. E. SHPARLINSKI good lower bounds on these characteristics of the Boolean function f deciding whether x is a quadratic residue modulo p. However, for the Boolean functions of this paper, the same approach produces rather poor results. Open Question 6.2. Obtain lower bounds on the weight and the degree of the Boolean function f of Theorem 4.1. It is well known that the modular inversion problem is closely related to the GCD-problem. Open Question 6.3. Obtain a lower bound on the PRAM complexity of computing integers u, v such that relatively prime integers M # N > 1. In the previous question we assume that gcd(N, M) = 1 is guaranteed. Otherwise one can easily obtain the lower bound #(f) #(n) on the sensitivity of the Boolean function f which on input of two n-bit integers M and N , returns 1 if they are relatively prime, and 0 otherwise. Indeed, if is an n bit integer, then the function returns 0 for That is, the PRAM complexity of this Boolean function is at least 0.5 log n +O(1). Acknowledgment . This paper was essentially written during a sabbatical visit by the second author to the University of Paderborn, and he gratefully acknowledges its hospitality and excellent working conditions. --R 'Parallel Implementation of Schonhage's Integer GCD Algorithm' Joachim von zur Gathen and Daniel Panario Joachim von zur Gathen Joachim von zur Gathen Joachim von zur Gathen Joachim Von Zur Gathen Joachim von zur Gathen and Gadiel Seroussi Introduction to number theory Michal M Theory and computation Number theoretic methods in cryptography: Complexity lower bounds The complexity of Boolean functions --TR

parallel computation;CREW PRAM complexity;exponential sums;modular inversion

356488

On Interpolation and Automatization for Frege Systems.

The interpolation method has been one of the main tools for proving lower bounds for propositional proof systems. Loosely speaking, if one can prove that a particular proof system has the feasible interpolation property, then a generic reduction can (usually) be applied to prove lower bounds for the proof system, sometimes assuming a (usually modest) complexity-theoretic assumption. In this paper, we show that this method cannot be used to obtain lower bounds for Frege systems, or even for TC0-Frege systems. More specifically, we show that unless factoring (of Blum integers) is feasible, neither Frege nor TC0-Frege has the feasible interpolation property. In order to carry out our argument, we show how to carry out proofs of many elementary axioms/theorems of arithmetic in polynomial-sized TC0-Frege.As a corollary, we obtain that TC0-Frege, as well as any proof system that polynomially simulates it, is not automatizable (under the assumption that factoring of Blum integers is hard). We also show under the same hardness assumption that the k-provability problem for Frege systems is hard.

Introduction One of the most important questions in propositional proof complexity is to show that there is a family of propositional tautologies requiring superpolynomial size proofs in a Frege or Extended Frege proof system. The problem is still open, and it is thus a very important question to understand which techniques can be applied to prove lower bounds for these systems, as well as for weaker systems. In recent years, the interpolation method has been one of the most promising approaches for proving lower bounds for propositional proof systems and for bounded arithmetic. Here we show that this method is not likely to work for Frege systems and some weaker systems. The basic idea behind the interpolation method is as follows. Department of LSI, Universidad Polit'ecnica de Catalu~na, Barcelona, Spain, bonet@lsi.upc.es. Research partly supported by EU HCM network console, by ESPRIT LTR Project no. 20244 (ALCOM-IT), CICYT TIC98-0410-C02-01 and TIC98-0410-C02-01 y Department of Computer Science, University of Arizona, toni@cs.arizona.edu. Research supported by NSF Grant CCR-9457782, US-Israel BSF Grant 95-00238, and Grant INT-9600919/ME-103 from NSF and M - SMT (Czech Republic) z Department of Applied Math, Weizmann Institute, ranraz@wisdom.weizmann.ac.il. Research supported by US-Israel BSF Grant 95-00238 We begin with an unsatisfiable statement of the form F (x; z denotes a vector of shared variables, and x and y are vectors of private variables for formulas A 0 and A 1 respectively. Since F is unsatisfiable, it follows that for any truth assignment ff to z , either A 0 (x; ff) is unsatisfiable or A 1 (y; ff) is unsatisfiable. An interpolation function associated with F is a boolean function that takes such an assignment ff as input, and outputs only if A 0 is unsatisfiable, and 1 only if A 1 is unsatisfiable. (Note that both A 0 and A 1 can be unsatisfiable in which case either answer will suffice). How hard is it to compute an interpolation function for a given unsatisfiable statement F as above ? It has been shown, among other things, that interpolation functions are not always computable in polynomial time unless Nevertheless, it is possible that such a procedure exists for some special cases. In particular, a very interesting and fruitful question is whether one can find (or whether there exists) a polynomial size circuit for an interpolation function, in the case where F has a short refutation in some proof system S . We say that a proof system S admits feasible interpolation if whenever S has a polynomial size refutation of a formula F (as above), an interpolation function associated with F has a polynomial size circuit. Kraj'i-cek [K2] was the first to make the connection between proof systems having feasible interpolation and circuit complexity. There is also a monotone version of the interpolation idea. Namely, for conjunctive normal monotone if the variables of z occur only positively in A 1 and only negatively in A 0 . In this case, an associated interpolant function is monotone, and we are thus interested in finding a polynomial size monotone circuit for an interpolant function. We say that a proof system S admits monotone feasible interpolation if whenever S has a polynomial size refutation of a monotone F , a monotone interpolation function associated with F has a monotone polynomial size circuit. Beautiful connections exist between circuit complexity, and proof systems having feasible interpolation, in both (monotone and non-monotone) cases: In the monotone case, superpolynomial lower bounds can be proven for a (sufficiently strong) proof system that admits feasible interpolation. This was presented by the sequence of papers [IPU, BPR, K1], and was first used in [BPR] to prove lower bounds for propositional proof systems. (The idea is also implicit in [Razb2]). In short, the statement F that is used is the Clique interpolation formula, A 0 (g; x)-A 1 (g; y) , where A 0 states that g is a graph containing a clique of size k (where the clique is described by the x variables), and A 1 states that g is a graph that can be colored with (where the coloring is described by the y variables). By the pigeonhole principle, this formula is unsatisfiable. However, an associated monotone interpolation function would take as input a graph g , and distinguish between graphs containing cliques of size k from those that can be colored with , such a circuit is of exponential size. Thus, exponential lower bounds follow for any propositional proof system S that admits feasible monotone interpolation. Similar ideas also work in the case where S admits feasible interpolation (but not necessarily monotone feasible interpolation). The first such result, by [Razb2], gives explicit superpolynomial lower bounds for (sufficiently strong) proof systems S admitting feasible interpolation, under a cryptographic assumption. In particular, it was shown that a (non-monotone) interpolation function, associated with a certain statement expressing P 6= NP , is computable by polynomial size circuits only if there do not exist pseudorandom number generators. Therefore, lower bounds follow for any (sufficiently strong) propositional proof system that admits feasible interpolation (conditional on the cryptographic assumption that there exist pseudorandom number generators). It is also possible to prove nonexplicit superpolynomial lower bounds for a (sufficiently strong) proof system under the assumption that NP is not computable by polynomial sized circuits. Many researchers have used these ideas to prove lower bounds for propositional proof systems. In particular, in the last five years, lower bounds have been shown for all of the following systems using the interpolation method: Resolution [BPR], Cutting Planes [IPU, BPR, Pud, CH], generalizations of Cutting Planes [BPR, K1, K3], relativized bounded arithmetic [Razb2], Hilbert's Nullstellensatz [PS], the polynomial calculus [PS], and the Lovasz- Schriver proof system [Pud3]. 1.1 Automatizability and k-provability As explained in the previous paragraphs, the existence of feasible interpolation for a particular proof system S gives rise to lower bounds for S . Feasible interpolation, moreover, is a very important paradigm for proof complexity (in general) for several other reasons. In this section, we wish to explain how the lack of feasible interpolation for a particular proof system S implies that S is not automatizable. We say that a proof system S is automatizable if there exists a deterministic procedure D that takes as input a formula f and returns an S -refutation of f (if one exists) in time polynomial in the size of the shortest S -refutation of f . Automatizability is a crucial concept for automated theorem proving: in proof complexity we are mostly interested in the length of the shortest proof, whereas in theorem proving it is also essential to be able to find the proof. While there are seemingly powerful systems for the propositional calculus (such as Extended Resolution or even ZFC), they are scarce in theorem proving because it seems difficult to search efficiently for a short proof in such systems. In other words, there seems to be a tradeoff between proof simplicity and automatizability - the simpler the proof system, the easier it is to find the proof. In this section, we formalize this tradeoff in a certain sense. In particular, we show that if S has no feasible interpolation then S is not automatizable. This was first observed by Russell Impagliazzo. The idea is to show that if S is automatizable (using a deterministic procedure D ), then S has feasible interpolation. Theorem 1 If a proof system S does not have feasible interpolation, then S is not automatiz- able. Proof Suppose that S is automatizable, and suppose D is the deterministic procedure to find proofs, and moreover, D is guaranteed to run in time n c , where n is the size of the shortest proof of the input formula. Let A 0 (x; z) - A 1 (y; z) be the interpolant statement, and let ff be an assignment to z . We want to output an interpolant function for A 0 (x; ff) - A 1 (y; ff) . First, we run D on A 0 (x; z) - A 1 (y; z) to obtain a refutation of size s . Next, we simulate D on A 0 (x; ff) for T (s) steps, and return 0 if and only if D produces a refutation of A 0 (x; ff) will be chosen to be the maximum time for D to produce a refutation for a formula that has a refutation of size s ; thus T c in this case. This works because in the case where A 1 (y; ff) is satisfiable with satisfying assignment fl , we can plug fl into the refutation of A 0 (x; ff) - A 1 (y; ff) to obtain a refutation of A 0 (x; ff) of size s . Therefore S has feasible interpolation. ut Thus, feasible interpolation is a simple measure that formalizes the complexity/search trade- off: the existence of feasible interpolation implies superpolynomial lower bounds (sometimes modulo complexity assumptions), whereas the nonexistence of feasible interpolation implies that the proof system cannot be automatized. A concept that is very closely related to automatizability is k-provablity. The k-symbol provability problem for a particular Frege system S is as follows. The problem is to determine, given a propositional formula f and a number k , whether or not there is a k-symbol S proof of f . The k -line provability problem for S is to determine whether or not there is a k -line S proof of f . The k -line provability is an undecidable problem for first-order logic [B1]; and the first complexity result for the k-provability problem for propositional logic was provided by Buss [B2] who proved the rather surprising fact that the k-symbol propositional provability problem is NP -complete for a particular Frege system. More recently, [ABMP] show that the k-symbol and k -line provability problems cannot be approximated to within linear factors for a variety of propositional proof systems, including Resolution and all Frege systems, unless The methods in our paper show that both the k-symbol and k -line provability problems cannot be solved in polynomial time for any TC 0 -Frege system, Frege system, or Extended Frege system, assuming hardness of factoring (of Blum integers). More precisely, using the same idea as above, we can show that if there is a polynomial time algorithm A solving the k-provability problem for S , then S has feasible interpolation: Suppose that is the unsatisfiable statement. We first run A with first verifies that there is a size proof of F for some fixed value of c . Now let ff be an assignment to z . As above, we run A to determine if there is an O(s)-symbol (or O(s)-line) refutation of A 0 (x; ff) and return 0 if and only if A accepts. In fact, this proof can be extended easily to show that both the k-symbol and k -line provability problems cannot be approximated to within polynomial factors for the same proof systems (TC 0 -Frege, Frege, Extende Frege) under the same hardness assumption. 1.2 Interpolation and one way functions How can one prove that a certain propositional proof system S does not admit feasible interpolation ? One idea, due to Kraj'i-cek and Pudl'ak [KP], is to use one way permutations in the following way. Let h be a one way permutation and let A 0 (x; z); A 1 (y; z) be the following formulas. The formula A th bit of x is 0. The formula th bit of y is 1. Since h is one to one, A 0 (x; z) - A 1 (y; z) is unsatisfiable. Assume that A 0 ; A 1 can be formulated in the proof system S , and that in S there exists a polynomial size refutation for A 0 (x; z) -A 1 (y; z) . Then, if S admits feasible interpolation it follows that given an assignment ff to z there exists a polynomial size circuit that decides whether A 0 (x; ff) is unsatisfiable or A 1 (y; ff) is unsatisfiable. Obviously, such a circuit breaks the i th bit of the input for h . Since can be constructed for any i , all bits of the input for h can be broken. Hence, assuming that the input for h is secure, and that in the proof system S there exists a polynomial size refutation for A 0 - A 1 , it follows that S does not admit feasible interpolation. A major step towards the understanding of feasible interpolation was made by Kraj'i-cek and Pudl'ak [KP]. They considered formulas A 0 ; A 1 based on the RSA cryptographic scheme, and showed that unless RSA is not secure, Extended Frege systems do not have feasible interpolation. It has been open, however, whether or not the same negative results hold for Frege systems, and for weaker systems such as bounded depth threshold logic or bounded depth Frege. 1.3 Our results In this paper, we prove that Frege systems, as well as constant-depth threshold logic (referred to below as do not admit feasible interpolation, unless factoring of Blum integers is computable by polynomial size circuits. (Recall that Blum integers are integers P of the are both primes such that p 1 mod our result significantly extends [KP] to weaker proof systems. In addition, our cryptographic assumption is weaker. To prove our result, we use a variation of the ideas of [KP]. In a conversation with Moni Naor, he observed that the cryptographic primitive needed here is not a one way permutation as in [KP], but the more general structure of bit commitment. Our formulas A 0 ; A 1 are based on the Diffie-Hellman secret key exchange scheme [DH]. For simplicity, we state the formulas only for the least significant bit. (Our argument works for any bit). Informally, our propositional statement DH will be The common variables are two integers X;Y , and P and g . P represents a number (not necessarily a prime) of length n , and g an element of the group Z . The private variables for A 0 are integers a; b , and the private variables for A 1 are integers c; d . Informally, A 0 say that g a mod that ab mod P is even. Similarly, A 1 d) will say that and g cd mod P is odd. The statement A 0 - A 1 is unsatisfiable since (informally) if A 0 ; A 1 are both true we have ab mod We will show that the above informal proof can be made formal with a (polynomial size) proof. On the other hand, an interpolant function computes one bit of the secret key exchanged by the Diffie-Hellman procedure. Thus, if TC 0 -Frege admits feasible interpolation, then all bits of the secret key exchanged by the Diffie-Hellman procedure can be broken using polynomial size circuits, and hence the Diffie-Hellman cryptographic scheme is not secure. Note, that it was proved that for are both primes such that breaking the Diffie-Hellman cryptographic scheme is harder than factoring P ! [BBR] (see also [Sh, Mc]). It will require quite a bit of work to formalize the above statement and argument with a short proof. Notice that we want the size of the propositional formula expressing the Diffie Hellman statement to be polynomially bounded in the number of binary variables. And additionally, we want the size of the proof of the statement to also be polynomially bounded. A key idea in order to define the statement and prove it efficiently, is to introduce additional common variables to our propositional Diffie-Hellman statement. The bulk of the argument then involves showing how (with the aid of the auxillary variables) one can formalize the above proof by showing that basic arithmetic facts, including the Chinese Remainder Theorem, can be stated and proven efficiently within 1.4 Section description The paper is organized as follows. In Section 2, we define our system. In Section 3, we define the used for the proof. In Section 4, we define precisely the interpolation formulas which are based on the Diffie-Hellman cryptographic scheme. In Section 5 we show how to prove our main theorem, provided we have some technical lemmas that will be proved fully in section 7. In Section 6 there is a discussion and some open problems. Finally in section 7 we prove all the technical lemmas required for the main theorem. The unusual organization of the paper is due to the many very technical lemmas required to show the result, that are essential to the correctness of the argument, but not every reader might want to go through. Sections 1-6 give an exposition of the result, relying on the complete proofs in the technical part. systems For clarity, we will work with a specific bounded-depth threshold logic system, that we call reasonable definition of such a system should also suffice. Our system is a sequent-calculus logical system where formulas are built up using the connectives - , only if the number of 1's in x is at least k , and \Phi i (x) is true if and only if the number of 1's in x is i mod 2.) Our system is essentially the one introduced in [MP]. (Which is, in turn, an extension of the system PTK introduced by Buss and Clote [BC, Section 10].) Intuitively, a family of formulas f each formula has a proof of size polynomial in the size of the formula, and such that every line in the proof is a are built up using the connectives -, Th k , \Phi 1 , \Phi 0 , :. All connectives are assumed to have unbounded fan-in. Th k interpreted to be true if and only if the number of true A i 's is at least k ; \Phi j interpreted to be true if and only if the number of true A i 's is equal to j mod 2. The formula denotes the logical AND of the multi-set consisting of A and similarly for - , \Phi j and Th k . Thus commutativity of the connectives is implicit. Our proof system operates on sequents which are sets of formulas of the form A The intended meaning is that the conjunction of the A i 's implies the disjunction of the B j 's. A proof of a sequent S in our logic system is a sequence of sequents, S 1 ; :::; S q , such that each sequent S i is either an initial sequent, or follows from previous sequents by one of the rules of inference, and the final sequent, S q , is S . The size of the proof is its depth is The initial sequents are of the form: (1) A ! A where A is any formula; (2) . The rules of inference are as follows. Note that the logical rules are defined for n - 1 and k - 1 . First we have simple structural rules such as weakening (formulas can always be added to the left or to the right), contraction (two copies of the same formula can be replaced by one), and permutation (formulas in a sequent can be reordered). The remaining rules are the cut rule, and logical rules which allow us to introduce each connective on both the left side and the right side. The cut rule allows the derivation of \Gamma; The logical rules are as follows. 1. , we can derive 2. (Negation-right) From 3. (And-left) From A 1 ; 4. (And-right) From 5. (Or-left) From A 6. 7. (Mod-left) From A 1 ; \Phi 1\Gammai derive 8. (Mod-right) From A derive 9. (Threshold-left) From Th k derive derive A proof is a bounded-depth proof in our system of polynomial size. More formally we have the following definitions. Ng be a family of sequents. Then fR family of TC 0 proofs for F if there exist constants c and d such that the following conditions hold: (1) Each R n is a valid proof of (\Gamma our system, and 2) For all i, the depth of R n is at most d; and (3) For all n, the size of R n is at most We note that we have defined a specific proof system for clarity; our result still holds for any reasonable definition of a proof (it can be shown that our system polynomially simulates any Frege-style system). The difference between a polynomial size proof in our system and a polynomial size TC 0 proof is similar to the difference between NC 1 and TC 0 . 3 The In this section we will describe some of the needed to formulate and to refute the Diffie-Hellman formula. For simplicity of the description, let us assume that we have a fixed number N which is an upper bound for the length of all numbers used in the refutation of the Diffie-Hellman formula. The number N will be used to define some of the formulas below. After seeing the statement and the refutation of the Diffie-Hellman formula, it will be clear that it is enough to take N to be a small polynomial in the length of the number P used for the Diffie-Hellman formula. 3.1 Addition and subtraction We will use the usual carry-save AC 0 -formulas to add two n-bit numbers. Let be two numbers. Then x denote the following AC 0 -formula: There will be . The bit z i will equal the mod 2 sum of C i , x i and y i , where C i is the carry bit. Intuitively, C i is 1 if there is some bit position less than i that generates a carry that is propagated by all later bit positions until bit i . Formally, C i is computed by OR(R th bit position generates a carry, and P k is 1 if the k th bit position propagates but does not generate a carry.) As for subtraction, let us show how to compute z . Think of x; y as N -bit numbers. y is the complement (modulo 2) of the N bits of y , and x is the complement of the N bits of x . Denote note that s is equal to 2 N similarly t is equal to 2 N we know that x \Gamma y - 0 and thus we know that y and thus . Thus, for any i , we can compute z i by (s N+1 - s i 3.2 Iterated addition We will now describe the inputs m numbers, each n bits long, and outputs their sum x 1 We assume that m - N . The main idea is to reduce the addition of m numbers to the addition of two numbers. Let x i be x i;n ; :::; x i;1 (in binary representation). Let l = dlog 2 Ne . Let 2l , and assume (for simplicity) that r is an integer. Divide each x i into r blocks where each block has 2l bits, and let S i;k be the number in the k th block of x i . That is, 2l Now, each S i;k has 2l bits. Let L i;k be the low-order half of S i;k and let H i;k be the high order half. That is, S Denote r r Then, r r r Hence, we just have to show how to compute the numbers H;L . Let us show how to compute L , the computation of H is similar. Denote r Since each L i;k is of length l , each L k is of length at most l which is at most 2l . Hence, the bits of L are just the bits of the L k 's combined. That is, As for the computation of the L k 's, note that since each L k is a poly-size sum of logarithmic length numbers, it can be computed using poly-size threshold gates. 3.3 Modular arithmetic Next, we describe our that compute the quotient and remainder of a number z modulo p , where z is of length n . remainder and The inputs for the remainder and the quotient formulas are as follows: 1. the number z ; 2. numbers 3. numbers k i and r i for all 1 - i - n . The intended values for the variables k i and r i are such that 2 (for all 1 - i - n). The intended values for the variables p i are i \Delta p . Suppose that z assume that the input variables k i , r i and take the right values. Then our formula [z] p will output r , and our formula div p (z) will output k . The formulas are computed as follows. Suppose that the k i , r i and p i variables satisfy (p l be such that l \Delta and can therefore be computed by div p (z) is computed by SUM n Notice that if the k i , r i and p i 's are not such that 2 then the formulas are not required to compute the correct values of the quotient or remainder, and can give junk. 3.4 Product and iterated product We will write x \Delta y to denote the formula SUM i;j [2 i+j \Gamma2 x computing the product of two n-bit numbers x and y . By 2 i+j \Gamma2 x i y j we mean 2 i+j \Gamma2 if both x i and y j are true, and 0 otherwise. Lastly, we will describe our computing the iterated product of m numbers. This formula is basically the original formula of [BCH], and articulated as a TC 0 -formula in [M]. The iterative product, PROD[z gives the product of z 1 ; :::; z m , where each z i is of length n , and we assume that m;n are both bounded by N . The basic idea is to compute the product modulo small primes using iterated addition, and then to use the constructive Chinese Remainder Theorem to construct the actual product from the product modulo small primes. Let Q be the product of the first t primes is the first integer that gives a number Q of length larger than N 2 . Since q 1 ; :::; q t are all larger than 2, t is at most N 2 , and by the well known bounds for the distribution of prime numbers the length of each q j is at most O(log N) . For each q j , let g j be a fixed generator for Z . Also, for each q j , let u j - Q be a fixed number with the property that u j mod q (such a number exists by the Chinese Remainder Theorem). PROD[z computed as follows. 1. First we compute r , for all . This is calculated using the modular arithmetic described earlier. 2. For each 1 - a. Compute a ij such that (g a ij . This is done by a table lookup. b. Calculate c c. Compute r j such that g c j . This is another table lookup. 3. Finally compute We will hardwire the values . Thus, this computation is obtained by doing a table lookup to compute followed by an iterated sum followed by a mod Q calculation. 3.5 Equality, and inequality Often we will write are both vectors of variables or formulas: When we We apply the same conventions when writing 6=; !; - . 4 The Diffie-Hellman Formula We are now ready to formally define our propositional statement DH . DH will be the conjunction of A 0 and A 1 . The common variables for the formulas will be: a) P and g representing n-bit integers, and for every i - 2n , we will also add common variables for g 2 i mod P . b) X;Y , and for every i - 2n , we will also add common variables for X 2 i mod P , and for mod P . c) We also add variables for P These variables are needed to define arithmetic modulo P (see section 3.3). For the following: the common variable mod mod . The will be the conjunction of the following 1. (Which means g a mod 2. For every j - n , mod P: (Which means (g 2 j modP .) Note that from this it is easy to prove for 3. Similar formulas for g b mod mod P . 4. PROD i;j is even. (Which means g ab mod P is even.) 5. For every i - N , formulas expressing formulas are added to guarantee that the modulo P arithmetic is computed correctly). Similarly, the formula A 1 d) will be the conjunction of the above formulas, but with a replaced by c , b replaced by d , and the fourth item stating that g cd mod P is odd. Note that the definition of the iterated product (PROD ) requires the primes q 1 ; :::; q t (as well as their product Q , and the numbers u are fixed for the length n . So we are going to hardwire the numbers q well as the correct values for the r i 's and k i 's needed for the modulo q j arithmetic, for each one of these numbers. refutation for DH We want to describe a refutation for DH . As mentioned above the proof proceeds as follows. 1. Using A 0 , show that g ab mod 2. Using A 1 , show that X b mod 3. Show that g cb mod 4. Using A 0 , show that g bc mod 5. Using A 1 , show that Y c mod 6. Show that g dc mod We can conclude from the above steps that A 0 and A 1 imply that g ab mod but now we can reach a contradiction since A 0 states that g ab mod P is even, while A 1 states that g cd mod P is odd. We formulate g ab mod P as PROD i;j and X b mod P as Thus, step 1 is formulated as PROD i;j and so on. Steps 1,2,4,5 are all virtually identical. Steps 3 and 6 follow easily because our formulas defining g ab make symmetry obvious. Thus the key step is to show step 1; that is, to show how to prove g ab mod As mentioned above, this is formulated as follows. PROD i;j We will build up to the proof that g ab mod P equals X b mod P by proving many lemmas concerning our basic -formulas. The final lemma that we need is the following: Lemma 4 For every z 1;1 ; :::; z m;m 0 and p, there are TC 0 -Frege proofs of The proof of the lemma is given in section 7. Using Lemma 4 for the first equality, and point 2 from section 4 for the second equality, we can now obtain: PROD i;j which proves step 1. The main goal of Section 7 is hence to show that the statement has a short proof. This is not trivial because our are quite complicated (and in particular the formulas for iterated product and modular arithmetic). In order to prove the statement, we will need to carry out a lot of the basic arithmetic in Before we go on to the technical part, we will try to give some intuition on how the proof of the main lemma is built. We organized the proof as a sequence of lemmas that show how many basic facts of arithmetic can be formulated and proved in TC 0 -Frege (using our formulas). The proofs of these lemmas require careful analysis of the exact formula used for each operation. The proof of some of these lemmas is straightforward (using the well known while the proof of other lemmas require some new tricks. In short, the main lemmas that are used for the proof of the final statement (Lemma are the following: 1. (Lemma 38). For every x; y and p , there are TC 0 -Frege proofs of 2. (Lemma 41). For every z are -Frege proofs of 3. (Lemma 47). For every z 1 ; z 2 , there are TC 0 -Frege proofs of First, we prove some basic lemmas about addition, subtraction, multiplication, iterative- sum, less-than, and modular arithmetic. Among these lemmas will be Lemma 38. The proof of Lemma 41 is cumbersome, but it is basically straightforward, given some basic facts about modular arithmetic. Recall that to do the iterated product we have to first compute the product modulo small primes and then combine all these products to get the right answer using iterated sum. Therefore, many basic facts of the modulo arithmetic need to be proven in advance, as well as some basic facts of the iterated sum. Once this is done we need to obtain the same fact modulo p (Lemma 48). At this point it is easier to go through the regular product, where the basic facts of modular arithmetic are easier to prove. Therefore it is important to show that TC 0 -Frege can prove (Lemma 47). In our application z 1 and z 2 will themselves be iterated products. To show this fact we use the Chinese Remainder Theorem. We first prove the equality modulo small primes. This is relatively easy, since the sizes of these primes are sufficiently small (O(log N)), and we can basically check all possible combinations. Once this is done, we apply the Chinese Remainder theorem to obtain the equality modulo the product of the primes, and since this product is big enough, we obtain the desired result. Our proof of the Chinese Remainder Theorem, is different than the standard textbook one. The main fact that we need to show is that if for every j , [R] q j there are TC 0 -Frege proofs of [R] The usual proofs use some basic facts of division of primes, that would be hard to implement here. Instead we prove by induction on This method allows us to work with numbers smaller than the q i 's and again since these numbers are sufficiently small, we can verify all possibilities. 6 Discussion and open problems We have shown that TC 0 -Frege does not have feasible interpolation, assuming that factoring of Blum integers is not efficiently computable. This implies (under the same assumptions) that -Frege as well as any system that can polynomially-simulate -Frege is not automatizable. It is interesting to note that our proof and even the definition of the Diffie-Hellman formula itself is nonuniform, essentially due to the nonuniform nature of the iterated product formulas that we use. It would be interesting to know to what extent our result holds in the uniform TC 0 proof setting. A recent paper [BDGMP] extends our results to prove that bounded-depth Frege doesn't have feasible interpolation assuming factoring Blum integers is sufficiently hard (actually their assumptions are stronger than ours). As a consequence bounded-depth Frege is not automatizable under somewhat weaker hardness assumptions. An important question that is still open is whether resolution, or some restricted forms of it, is automatizable. A positive answer to this question would have important applied consequences. 7 Formal proof of main lemma The goal of this section is to prove Lemma 4. As mentioned earlier, we will build up to the proof of this lemma, by showing that basic facts concerning arithmetic, multiplication, iterated multiplication and modulus computations can be efficiently carried out in our proof system. Before we begin the formal presentation, we would like to note that we will be giving a precise description of a sequence of lemmas that are sufficient in order to carry out a full, formal proof Lemma 4. However, since there are many lemmas and many of them have obvious proofs, we will describe at a meta-level what is required in order to formalize the argument in TC 0 -Frege, rather than give an excessively formal -Frege proof of each lemma. In what follows, x , y and z will be numbers. Each one of them will denote a vector of n variables or formulas (representing the number), where n - N and x i (respectively y i , z i ) denotes the i th variable of x (representing the i th bit of the number x). When we need to talk about more than three numbers, we will write z 1 ; :::; z m to represent a sequence of m n-bit (where m;n - N ), and now z i;j is the j th variable of z i (representing the j th bit in the i th number). Recall that whenever we say below "there are TC 0 -Frege proofs" we actually mean to say "there are polynomial size TC 0 -Frege proofs". Some trivial properties like are not stated here. 7.1 Some basic properties of addition, subtraction and multiplication Lemma 5 For every x; y , there are TC 0 -Frege proofs of x Proof of Lemma 5 Immediate from the fact that the addition formula was defined in a symmetric way. ut Lemma 6 For every x; z , there are TC 0 -Frege proofs of x Proof of Lemma 6 By the definition of the addition formula, the i th bit of ((x equal to \Phi 1 (\Phi 1 is the carry bit going into the th position, when we add x and y , and C similarly defined to be the carry bit going into the i th position when we add Using basic properties of \Phi 1 and the above definitions, there is a simple proof that if \Phi 1 (C i (x; Thus it is left to show that for all i We will show how to prove the stronger equality (It can be verified that this is the strongest equality possible for the 4 quantities That is, all 6 assignments for that satisfy the above equality are actually possible.) We will prove this by induction on i . For the carry bits going into the first position are zero, so the above identity holds trivially. To prove the above equality for i , we assume that it holds for We will prove the equality by considering many cases, where a particular case will assume a fixed value to each of the following seven quantities: subject to the condition that C It is easy to check that the number of cases is 48 since there are 2 choices for x choices for y choices for z and 6 choices in total for C Each case will proceed in the same way. We will first show how to compute using the above seven values. Then we simply verify that in all 48 cases where the inductive hypothesis holds, the equality is true. First, we will show that This requires a proof along the following lines. If x then the left-hand side of the above statement is true since position a carry, and also the right-hand side of the statement is also true. Similarly, if x both sides of the above statement are false (since position carry). The last case is when x vice-versa). In this case, position propagates a carry, so the i th carry bit is 1 if and only if there exists a such that the j th position generates a carry, and all positions between carries - but this is exactly the definition of C . Thus, we have in this last case that both sides of the statement are true if and only if C Using the above fact, and also that only if z arguments show that C i (y; can also be computed as simple formulas of the seven pieces of information. ut Lemma 7 For every x; y , there are TC 0 -Frege proofs of Proof of Lemma 7 computed by taking the first N bits of that by the definition of the addition formula it follows easily that all bits of (y + y) are 1, and hence that ((y and hence the first N bits of this number are the same as the first N bits of x . ut Lemma 8 For every x; y , there are Proof of Lemma 8 computed by taking the first N bits of x By the definition of the addition formula, and since x - y , it can be proved that the (N th bit of hence that Therefore, as in Lemma 7, In particular, the first N bits of are the same as those of x Lemma 9 For every x; z , there are TC 0 -Frege proofs of x Proof of Lemma 9 Follows immediately from Lemma 7 and Lemma 6 as follows: For every z , there are TC 0 -Frege proofs of Proof of Lemma 10 We need to show that for every j , This is shown by a rather tedious but straightforward proof following the definition of the formula SUM for iterated addition. Namely, we show first that and similarly that Secondly, we show that [H using the definition of + . This second step is not difficult because all carry bits are zero. ut Lemma 11 For every z 1 ; :::; z m , and every fixed permutation ff , there are TC 0 -Frege proofs of (That is, the iterated sum is symmetric.) Proof of Lemma 11 Immediate from the fact that the formula SUM was defined in a symmetric way. ut Lemma 12 For every z , there are TC 0 -Frege proofs of Proof of Lemma 12 By definition of the iterated addition formula SUM , it is straightforward to prove that and similarly that Then it is also straightforward to show, using the definition of the formula for + , that for every j . (Again, all carry bits are zero.) ut Lemma 13 For every z 1 ; :::; z m , there are TC 0 -Frege proofs of Proof of Lemma 13 Recall that SUM [z computed by adding two numbers H;L . Recall that L is computed by first computing the numbers L is the low-order half of the k th block of z i . The first equality follows from Lemma 13, and Similarly, computed by H 0 +L 0 , where L 0 is computed by first computing the numbers In both L k ; L 0 k the sum is computed using poly-size threshold gates, e.g. by using the unary representation of each L i;k . It is therefore straightforward to prove for each k , L (e.g. by trying all the possibilities for L 0 proving the formula separately for each possibility). Now consider the formula Since in this addition there is no carry flow from one block to the next one, and since the bits of L; L in each block are just the bits of L k ; L 0 (respectively), we can conclude that in a similar way we can prove that we are now able to conclude ut Lemma 14 For every z 1 ; :::; z m , there are TC 0 -Frege proofs of Proof of Lemma 14 Can be proved easily from Lemma 13, and Lemma 6 as follows: Lemma 15 For every z 1 ; :::; z m , and every 1 - k - m, there are TC 0 -Frege proofs of Proof of Lemma 15 By Lemma 13, Lemma 14 and Lemma 11, we have The proof now follows by repeating the same argument m \Gamma k times, where Lemma 12 is used for the base case. ut Lemma 16 For every x; y , there are TC 0 -Frege proofs of x Proof of Lemma 16 Immediate from the fact that the product formula was defined in a symmetric way. ut Lemma 17 For every x; x is a power of 2, there are TC 0 -Frege proofs of x Proof of Lemma 17 It is straightforward to prove that 2 i \Delta y where y is any sequence of bits, consists of adding to the end of y , i 0's. The lemma easily follows. ut Lemma For every z 1 ; :::; z m , and every x where x is a power of 2, there are proofs of Proof of Lemma The proof of this lemma is like the proof of Lemma 21, but using Lemma 17 instead of 20. ut Lemma 19 For every x; are powers of 2 there are TC 0 -Frege proofs of x Proof of Lemma 19 Same as the proof of Lemma 17. ut The following three lemmas are generalizations of the previous three lemmas. Lemma 20 For every x; y; z there are TC 0 -Frege proofs of x Proof of Lemma 20 By definition of the product formula, x Similarly, x By iterative application of Lemma 15 (using also Lemma 11), Similarly (using also Lemma 13, and Lemma 14), and in the same way (using the same lemmas) Thus, we have to prove We will prove this by proving that for every i , this is trivial. Otherwise, x using Lemma 19, Lemma 18, and Lemma we have In the same way, (using also Lemma 17), ut Lemma 21 For every z 1 ; :::; z m , and every x, there are TC 0 -Frege proofs of Proof of Lemma 21 We will show that for every i , The lemma then follows by the combination of all these equalities. The case proven as follows: The first equality follows by applying Lemma 13, and the second equality by Lemma 12. For the general step: The first equality follows from Lemmas 13 and the second equality follows from Lemma 20; the third equality follows from Lemma 13. ut Lemma 22 For every x; z , there are TC 0 -Frege proofs of x Proof of Lemma 22 We will show that x \Delta (y \Delta z) is equal to SUM i;j;k [2 i+j+k\Gamma3 x . The same will be true for and the lemma follows. By the definition of the product, y Hence, by Lemma 10, and two applications of Lemma 21 (and using freely Lemma 16), x Since it can be easily verified that (2 , the above is equal to and by an iterative application of Lemma 15 (using also Lemma 11) the above is equal to 7.2 Some basic properties of less-than Lemma 23 For every x; y , there are also of Proof of Lemma 23 Either there is a bit i such that i is the most significant bit where x and y differ, or not. If all bits are equal, then But if there is i such that it is the most significant bit where they differ, then if x Lemma 24 For every x; y , there are Proof of Lemma 24 By lemma 23, Suppose for a contradiction that and we get x ? x (which is easily proved to be false). Lemma 25 For every x; y; z there are Proof of Lemma 25 If then the proof of the first statement is obvious. Otherwise, suppose that i is the most significant bit where x i 6= y i and that x Similarly, suppose that j is the most significant bit where y j 6= z j and y it is easy to show that i is the most significant bit where x i 6= z i , and x thus x ? z . Similarly, if j ? i , then j is the most significant bit where x j 6= z j and x reasoning also implies the second statement in the lemma. ut Lemma 26 For every x; z , there are TC 0 -Frege proofs of x+z - x; and also z ? Proof of Lemma 26 If z = 0 , then it is clear that x show inductively for decreasing k that Then when Assuming that z ? 0 , let z i 0 be the most significant bit such that z i . The base case of the induction will be to show that x Because z and applying Lemma 12, it suffices to show that x There are two cases. If x is equal to x n x The other case is when x be the most significant bit position greater than i 0 such that x One clearly exists because x higher bits are equal, and thus x For the inductive step, we assume that x want to show that x Using the same argument as in the base case, one can prove that (a) x . By the inductive hypothesis, (b) x Applying Lemma 25 to (a) and (b), we obtain as desired. ut Lemma 27 For every x; y; z there are Proof of Lemma 27 If Lemma 26. Then by Lemma Lemma 28 For every x; y; z there are Proof of Lemma 28 The first equality follows from Lemma 8, and the second from Lemma 26, and the fact that Lemma 29 For every x; y , there are TC 0 -Frege proofs of y Proof of Lemma 29 x by definition. Also, since y ? 0 there is a bit of y that is 1 , and suppose that it is y l . Then x ut 7.3 Some basic properties of modular arithmetic Recall that the formulas for [z] p and div p (z) take as inputs not only the variables p and z , but also variables k . The formulas give the right output if n). So the following theorems will all have the hypothesis that the values for the variables k i , r i and p i are correct, and that there are short We will state this hypothesis for the first lemma, and omit it afterwards for simplicity. For simplicity, we will also use the notations k The lemmas will be used with either is the number used for the DH formula, or with fixed hardwired value (e.g., q j is one of the primes used for the iterated product formula, and Q is the product of all these primes). If hardwired q then k can also be hardwired. Hence, there values are correct and it is straightforward to check (i.e. to prove) that the non-variable are all correct. If are inputs for the DH formula itself, and the requirements 2 are part of the requirements in the DH formula. Lemma z and p be n-bit numbers. Then there are TC 0 -Frege proofs of Proof of Lemma ut Lemma 31 For every z and p, there are TC 0 -Frege proofs of Also, the following uniqueness property has a then Proof of Lemma 31 From the previous lemma, we can express z as SUM i [(r Let l be the same as in the definition of the modulo formulas. Then The first equality follows from the definitions of the formulas [z] p and div p (z) . The remaining equalities follow from the following Lemmas: 20, 6, 8, 21, 13, 15, 14 and 30. Let us now prove the uniqueness part. Suppose then we are done. But if div p (z) ? y , then by the claim bellow which is a contradiction. (And a similar argument holds when div Proof of the claim Since v ? y , by Lemma 8, and Lemma Lemma 9 we get that Therefore by Lemmas 29 and 26, and by Lemma 25 we get p - x . ut Lemma 33 For every z; k and p, there are TC 0 -Frege proofs of Proof of Lemma 33 Let (by Lemma 31). Therefore, By the uniqueness part of Lemma 31 applied to x , [z] Lemma 34 For every x; y; z and p, there are -Frege proofs of and also of and Proof of Lemma 34 By Lemma 31, and by Lemma 33, A similar argument shows that [x Lemma 35 For every z 1 ; :::; z m and p, there are TC 0 -Frege proofs of Proof of Lemma 35 The lemma follows easily from Lemma 13 and Lemma 34. ut Lemma 36 For every x; y and p, there are TC 0 -Frege proofs of Proof of Lemma 36 The first equality follows from Lemma 34; the next equality follows from the assumption that and the third equality follows from Lemma 34. ut Lemma 37 For every x; y; z and p, there are -Frege proofs of Proof of Lemma 37 Assuming that [x it follows from the above Lemma 36 that [x . The left side of the equation is equal to: The first equality follows from Lemma 34; the second equality follows from Lemmas 5, 6 and 7; the third equality follows from Lemma 33. Similarly, it can be shown that [y and thus the lemma follows. ut Lemma 38 For every x; y and p, there are TC 0 -Frege proofs of Proof of Lemma 38 where the last equality follows from Lemma 33. ut Lemma fixed numbers such that A = BC . Then for every z , there are -Frege proofs of This lemma will be used in situations where . Recall that the numbers Q; are hardwired and also their corresponding k i , r i and the variables for the 's. Hence, we think of A; B; C as hardwired. Proof of Lemma 39 Using Lemmas 31,33,22, we get ut 7.4 Some basic properties of iterative product Lemma 40 For every z 1 ; :::; z m , and every fixed permutation ff , there are TC 0 -Frege proofs of (That is, the iterated product is symmetric.) Proof of Lemma 40 This lemma is immediate from the symmetric definition of PROD . ut Lemma 41 For every z 1 ; :::; z m , and every 1 - k - m, there are TC 0 -Frege proofs of Proof of Lemma 41 Recall that we have hard-coded the numbers u j , such that u j mod q and for all i . For all primes q j dividing Q , and for all m , we can verify the following statements: that these statements are variable-free and hence they can be easily proven by doing a formula evaluation.) Recall that for any k , the iterated product of the numbers z k ; :::; z m is calculated as follows: where r [k;:::;m] j is computed like r j as defined in Section 3.4, but using r ij only for i such that In the same way, where r [1;::;k\Gamma1;[k;::;m]] j is calculated as before by the following steps: 1. For , and also calculate r 2. For calculate a i;j such that (g a i;j also a ;j such that (g a ;j 3. Calculate c 0 4. Calculate r [1;::;k\Gamma1;[k;::;m]] j such that g c 0 by table-lookup. Therefore, all we have to do is to show that Hence, all we need to do to prove Lemma 41 is to show the following claim: 42 For every j , there are TC 0 -Frege proofs of r [1;:::;k\Gamma1;[k;:::;m]] . The first step is to prove the following claim: 43 There are . 43 is proven as follows. [r [k;::;m] The second equality follows by Lemma 39; the third equality follows by Lemma 35, and Lemma 11. To prove the fourth equality, we need to use the fact that [u j \Delta r [k;:::;m] and also for all i These facts can be easily proved just by checking all possibilities for r [k;:::;m] (proving the statement for each possibility is easy, because these statements are variable-free and hence they can be easily proven by doing a formula evaluation). In order to prove the fourth equality formally, we can show that SUM i6=j [u i \Delta r [k;:::;m] equals zero by induction on the number of terms in the sum. ut We can now turn to the proof of Claim 42. The quantity r [1;:::;m] j is obtained by doing a table lookup to find the value equal to g c j \Gamma1) . Similarly, the quantity r [1;:::;k\Gamma1;[k;:::;m]] j is obtained by doing a table lookup to find the value equal to Hence, it is enough to prove that c . Using previous lemmas Thus, it suffices to show that Recall that a ;j is the value obtained by table-lookup such that (g a ;j by Claim 43, we have that r j , in turn, is the value obtained by table-lookup to equal (g d Now it is easy to verify that our table-lookup is one-to-one. That is, for every x; . Using this property (with 7.5 The Chinese Remainder Theorem and other properties of iterative prod- uct The heart of our proof is a proof for the following lemma, which gives the hard direction of the Chinese Remainder Theorem (a proof for the other direction is simpler). Lemma 44 Let R; S be two integers, such that for every j , [R] q j : Then there are -Frege proofs of (where are the fixed primes used for the PROD formula (i.e., the first t primes), and Q is their product.) Proof of Lemma 44 Without loss of generality, we can assume that 0 - R; S - prove that R = S . Otherwise, define R use Lemma 39 to show that for every j , [R 0 . Since 0 - R 0 conclude that For every k , let Q k denote Note that the numbers Q k can be hardwired, and that one can easily prove the following statements. (These statements are variable-free and hence they can be easily proven by doing a formula evaluation.) For every i , The proof of the lemma is by induction on t (the number of q j 's). For the lemma is trivial. Assume therefore by the induction hypothesis that and hence Denote, [R] , and D [S] . Then by Lemma 31, and and since we know that [R] q t ; we have and by [R] Q , and Lemma 37 Since R; S are both lower than Q , it follows that DR ; D S are both lower than q t . Hence, by Claim Therefore, we can conclude that For every i, there are then Proof Since d 1 are only O(log n) possibilities for d 1 ; d 2 . Therefore, one can just check all the possibilities for d 1 ; d 2 . Proving the statement for each possibility is easy, because these statements are variable-free and hence they can be easily proven by doing a formula evaluation. Alternatively, one can define the function , in the domain f0; :::; q i g , and prove that f(x) is onto the range f0; :::; q i g . Then, by applying the propositional pigeonhole principle, which is efficiently provable in TC 0 -Frege, it follows that f is one to one. ut We are now able to prove the following lemmas. Lemma 46 For every z , there are TC 0 -Frege proofs of Proof of Lemma 46 Recall that PROD[z] is calculated as follows: where r j is computed by r By Claim 43, for every i , PROD[z] q i thus have for every i , PROD[z] q i The proof of the lemma now follows by Lemma 44. ut Lemma 47 For every z 1 ; z 2 , there are TC 0 -Frege proofs of Proof of Lemma us prove that for every i , The proof of the lemma then follows by Lemma 44. By two applications of Lemma 38 it is enough to prove for every i , Recall that PROD[z 1 ; z 2 ] is calculated as follows: where r [1;2] j is computed like r j as defined in Section 3.4. By Claim 43, for every i , Recall that [z 1 r 2;i . Therefore, all we have to prove is that for every r [1;2] By the definitions: r Also, r [1;2] Therefore, one can just check all the possibilities for a 1;i ; a 2;i . ut Using the previous lemmas, we are now able to prove the following: Lemma 48 For every z every p, there are TC 0 -Frege proofs of (as before, given that 2 Proof of Lemma 48 The lemmas used for each equality in turn are: Lemmas 41,47, 38,47,41, and 41. ut We are now ready to prove Lemma 4: For every z 1;1 ; :::; z m;m 0 and p , there are TC 0 -Frege proofs of (given that 2 Proof of Lemma 4 By an iterative application of the previous lemma. ut Acknowledgments We are very grateful to Omer Reingold and Moni Naor for collaboration at early stages of this work, and in particular for suggesting the use of the Diffie-Hellman cryptographic scheme. We also would like to thank Uri Feige for conversations and for his insight about extending this result to bounded-depth Frege. Part of this work was done at Dagstuhl, during the Complexity of Boolean Functions workshop (1997). --R "The Monotone Circuit Complexity of Boolean Functions" "Minimal propositional proof length is NP-hard to linearly approximate" "Generalized Diffie-Hellman modulo a composite is not weaker than factoring" " Log depth circuits for division and related problems," "Non-automatizability of bounded-depth Frege proofs" "Lower bounds for Cutting Planes proofs with small coefficients," "The Undecidability of k-Provability" "On Godel's theorems on lengths of proofs II: Lower bounds for recognizing k symbol provability," "Cutting planes, connectivity and threshold logic," "An Exponential Lower Bound for the Size of Monotone Real Circuits," "Constant depth reducibility," "New directions in cryptography," "Upper and lower bounds for tree-like Cutting Planes proofs," "Interpolation theorems, lower bounds for proof systems and independence results for bounded arithmetic" "Discretely ordered modules as a first-order extension of the cutting planes proof system," "Threshold circuits of small majority depth," "A key distribution system equivalent to factoring," "A lower bound for the complexity of Craig's interpolants in sentential logic," "Lower bounds for resolution and cutting planes proofs and monotone computations," "Algebraic models of computation and interpolation for algebraic proof systems," "Lower Bounds for the Monotone Complexity of some Boolean Func- tions" "Unprovability of lower bounds on the circuit size in certain fragments of bounded arithmetic," "Composite Diffie-Hellman public-key generating systems are hard to break," --TR --CTR Alexander Razborov, Propositional proof complexity, Journal of the ACM (JACM), v.50 n.1, p.80-82, January Pavel Pudlk, On reducibility and symmetry of disjoint NP pairs, Theoretical Computer Science, v.295 n.1-3, p.323-339, 24 February Olaf Beyersdorff, Classes of representable disjoint NP-pairs, Theoretical Computer Science, v.377 n.1-3, p.93-109, May, 2007 Albert Atserias, Conjunctive query evaluation by search-tree revisited, Theoretical Computer Science, v.371 n.3, p.155-168, March, 2007 Samuel R. Buss, Polynomial-size Frege and resolution proofs of st-connectivity and Hex tautologies, Theoretical Computer Science, v.357 n.1, p.35-52, 25 July 2006 Paolo Liberatore, Complexity results on DPLL and resolution, ACM Transactions on Computational Logic (TOCL), v.7 n.1, p.84-107, January 2006 Maria Luisa Bonet , Nicola Galesi, Optimality of size-width tradeoffs for resolution, Computational Complexity, v.10 n.4, p.261-276, May 2002 Albert Atserias , Mara Luisa Bonet, On the automatizability of resolution and related propositional proof systems, Information and Computation, v.189 n.2, p.182-201, March 15, 2004 Juan Luis Esteban , Nicola Galesi , Jochen Messner, On the complexity of resolution with bounded conjunctions, Theoretical Computer Science, v.321 n.2-3, p.347-370, August 2004

diffie-hellman;threshold circuits;propositional proof systems;frege proof systems

356491

Constructing Planar Cuttings in Theory and Practice.

We present several variants of a new randomized incremental algorithm for computing a cutting in an arrangement of n lines in the plane. The algorithms produce cuttings whose expected size is O(r2), and the expected running time of the algorithms is O(nr). Both bounds are asymptotically optimal for nondegenerate arrangements. The algorithms are also simple to implement, and we present empirical results showing that they perform well in practice. We also present another efficient algorithm (with slightly worse time bound) that generates small cuttings whose size is guaranteed to be close to the best known upper bound of J. Matou{s}ek [Discrete Comput. Geom., 20 (1998), pp. 427--448].

Introduction A natural approach for solving various problems in computational geometry is the divide-and- conquer paradigm. A typical application of this paradigm to problems involving a set L of n lines in the plane, is to fix a parameter r ? 0, and to partition the plane into regions R (those regions are usually vertical trapezoids, or triangles), such that the number of lines of L that intersect the interior of R i is at most n=r, for any m. This allows us to split the problem at hand into subproblems, each involving the subset of lines intersecting a region R i . Such a partition is known as a (1=r)-cutting of the plane. See [Aga91] for a survey of algorithms that use cuttings. For further work related to cuttings, see [AM95]. The first (though not optimal) construction of cuttings, is due to Clarkson [Cla87]. Chazelle and Friedman [CF90] showed the existence of (1=r)-cuttings with bound that is worst-case tight). They also showed that such cuttings, consisting of vertical trapezoids, can be computed in O(nr) time. Although this construction is asymptotically optimal, it does not seem to produce a practically small number of regions. Coming up with the smallest possible number of regions (i.e., reducing the constant of proportionality) is important for the efficiency of (recursive) data structures that use cuttings. Currently, the best lower bound on the number of vertical trapezoids in a (1=r)-cutting in an arrangement of lines, is 2:54(1 \Gamma o(1))r 2 , and the A preliminary version of the paper appeared in the 14th ACM Symposium of Computational Geometry, 1998. This work has been supported by a grant from the U.S.-Israeli Binational Science Foundation. This work is part of the author's Ph.D. thesis, prepared at Tel-Aviv University under the supervision of Prof. Micha Sharir. y School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel; sariel@math.tau.ac.il; http://www.math.tau.ac.il/ ~ sariel/ optimal cutting has at most 8r 2 +6r+4 trapezoids, see [Mat98]. Improving the upper and lower bounds on the size of cuttings is still open, indicating that our understanding of cuttings is still far from being satisfactory. In Section 3, we outline Matousek construction for achieving the upper bound and show a slightly improved construction (see below for details). In spite of the theoretical importance of cuttings (in the plane and in higher dimensions), we are not aware of any implementation of efficient algorithms for constructing cuttings. In this paper we propose a new and simple randomized incremental algorithm for constructing cuttings, and prove the expected worst-case tight performance bounds of the new algorithm, as stated in the abstract. We also present empirical results on several algorithms/heuristics for computing cuttings that we have implemented. They are mostly variants of our new algorithm, and they all perform well in practice. An O(r 2 ) bound on the expected size of the cuttings for some of those variants can be proved, while for the others no formal proof of performance is currently available. We leave this as an open question for further research. Matousek [Mat98] gave an alternative construction for cuttings, showing that there exists a (1=r)-cutting with at most (roughly) 8r 2 vertical trapezoids. Unfortunately, this construction relies on computing the whole arrangement, and its computation thus takes O(n 2 ) time. We present a new randomized algorithm that is based on Matousek's construction; it generates a (1=r)-cutting of size (1+ ")8r 2 , in O log expected time, where prescribed constant. In Section 2, we present the new algorithm, and analyze its expected running time and the expected number of trapezoids that it produces. Specifically, the expected running time is O(nr) and the expected size of the output cutting is O(r 2 ). In Section 3 we present our variant of Matousek's construction. In Section 4 we present our empirical results, comparing the new algorithm with several other algorithms/heuristics for constructing cuttings. These algorithms are mostly variants of our main algorithm, but they also include a variant of the older algorithm of Chazelle and Friedman. The cuttings generated by the new algorithm and its variants are of size, roughly, 14r 2 . (The algorithms generate smaller cutting when r is small. For example, for the constant is about 9.) In contrast, the Chazelle-Friedman algorithm generates cuttings of size roughly 70r 2 . Some variants of our algorithm are based on cuttings by convex polygons with a small number of edges. These perform even better in practice, and we have a proof of optimality only for one of the methods PolyVertical, which can be interpreted as an extension of CutRandomInc. We conclude in Section 5 by mentioning a few open problems. Incremental Randomized Construction of Cuttings Given a set " S of n lines in the plane, let A( " S) denote the arrangement of " namely, the partition of the plane into faces, edges, and vertices as induced by the lines of " S. Let AVD S) denote the partition of the plane into vertical trapezoid, obtained by erecting two vertical segments up and down from each vertex of A( " S), and extending each of them until it either reaches a line of " or all the way to infinity. Computing the decomposed arrangement AVD S) can be done as follows. Pick a random permutation S =! s S. incrementally the decomposed arrangements AVD (S i ), for inserting the i-th line s i of S into AVD (S To do so, we compute the zone Z i of s i in AVD (S i\Gamma1 ), which is the set of all trapezoids in AVD (S i\Gamma1 ) that intersect s i . We split each trapezoid of Z i into at most 4 trapezoids, such that no trapezoid intersects s i in its interior, as in [SA95]. Finally, we perform a pass over all the newly created trapezoids, merging vertical trapezoids that are adjacent, and have identical top and bottom lines. The merging step guarantees that the resulting decomposition is independently of the insertion order of elements in S i ; see [dBvKOS97]. However, if we decide to skip the merging step, the resulting structure, denoted as A j (S i ), depends on the order in which the lines are inserted into the arrangement. In fact, A j (S i ) is additional superfluous vertical walls. Each such vertical wall is a fragment of a vertical wall that was created at an earlier stage and got split during a later insertion step. S be a set of n lines in the plane, and let c ? 0 be a constant. A c-cutting of " S is a partition of the plane into regions R , such that, for each m, the number of lines of " S that intersect the interior of R i is at most cn. A region C in the plane is c-active, if the number of lines of " S that intersect the interior of C is larger than cn. A (1=r)-cutting is thus a partition of the plane into m regions such that none of them is (1=r)-active. Chazelle and Friedman [CF90] showed that one can compute, in O(nr) time, a (1=r)-cutting that consists of O(r 2 ) vertical trapezoids. We propose a new algorithm for computing a cutting that works by incrementally computing the arrangements A j (S i ), using a random insertion order of the lines. The new idea in the algorithm, is that any "light" trapezoid (i.e., a trapezoid that is not (1=r)-active) constructed by the algorithm is immediately added to the final cutting, and the algorithm does not maintain the arrangement inside such a trapezoid from this point on. In this sense, one can think of the algorithm as being greedy; that is, it adds a trapezoid to the cutting as soon as one is constructed, until the whole plane is covered. The algorithm, called CutRandomInc, is depicted in Figure 1. If CutRandomInc outputs C k , for some k ! n, then C k has no (1=r)-active trapezoids, and it is thus a (1=r)-cutting. Otherwise, if C n is output, then again it has no (1=r)-active trapezoids, because any such trapezoid must have been processed and split earlier (when one of the lines crossing the trapezoid is inserted). Thus C n is a (1=r)-cutting. This implies the correctness of CutRandomInc. The covering C i of the plane maintained by CutRandomInc depends heavily on the order in which the lines are inserted into the arrangement. To bound the expected running time of CutRandomInc, and the expected size of the cutting that it computes, we adapt the analysis of Agarwal et al. [AMS94] to our case. The following elegant argument, due to Agarwal et al. [AEG98], shows that the expected complexity of A j (S i ) remains quadratic (i.e. we use CutRandomInc to compute a 0-cutting.): Lemma 2.2 Let " S be a set of n lines, and let S be a random permutation of " S. Then the expected complexity of A j (S) is O(n 2 ), where A j (S) is the decomposed arrangement computed by the incremental algorithm outlined at the beginning of the section, without performing merging. Proof: Let V be the set of all intersection points of pairs of lines of " (the vertices of A( " S)). For be an indicator variable, such that D(v; i) (resp. U(v; i)) is 1 if the vertical wall emanating from v still exists in A j (S) as we go downward (resp. upward) Algorithm Input: A set " S of n lines, a positive integer r Output: A (1=r)-cutting of " S by vertical trapezoids begin Choose a random permutation S =! s S. while there are (1=r)-active trapezoids in C i do Zone i / The set of (1=r)-active trapezoids in C i\Gamma1 that intersect s i . Zone 0 is the operation of splitting a vertical trapezoid T crossed by a line s into at most four vertical trapezoids, as in [dBvKOS97], such that the new trapezoids cover T , and they do not intersect s in their interior. while return C i Figure 1: Algorithm for constructing a (1=r)-cutting of an arrangement of lines from v after crossing i lines. Clearly, the complexity of A j (S) is proportional to However, if D(v; then the two lines defining v must appear in S before the first i lines that intersect the downward ray emanating from v. Thus, P r , and the same inequality holds for U(v; i). Therefore, the expected complexity of A j (S) is O O The analysis is applied in the following abstract framework. Let " S be a set of n objects (in our case the objects are lines in IR 2 ). A selection of " S is an ordered sequence of distinct elements of " S. Let oe( " S) denote the set of all selections of " S. For a permutation S of " S, let S i denote the subsequence consisting of the first i elements of S, for n. For each R 2 S), we define a collection CT (R) of 'regions' (in our case, a region will be either a trapezoid or a segment), each defined by a small subset of R. Let T denote the set of all possible regions. We associate two subsets D(\Delta); K (\Delta) ' S with each region \Delta 2 T , where D(\Delta) is the defining set of \Delta, in the sense that \Delta 2 CT (R) only if D(\Delta) ' R. The size of D(\Delta) is assumed to be bounded by a fixed constant. The set K (\Delta) is the killing set of \Delta; namely, if K (\Delta) "R 6= ;, then denote the weight of \Delta. Let CT (R; denote the set of all regions of CT (R) having weight at least k, where k is a positive integer. For a region \Delta in the plane, we denote by CT (R; k; \Delta) the set of all regions of CT (R; k) that are contained in \Delta. S be a set of objects such that, for any sequence R 2 S), the following axioms hold: (A) For any \Delta 2 CT (R), we have D(\Delta) ' R, and K (\Delta) (R), then for any subsequence R 0 of R, such that D(\Delta) ' R 0 , we have (R 0 ). The above is a natural extension of the settings of Agarwal et al. [AMS94], where the insertion order of objects in the sample is important. For any natural number k, define The following key lemma asserts that the expected number of heavy regions decreases exponentially as a function of their weight. Lemma 2.3 Given a set " S of n objects, let R be a random sequence of r n distinct elements of S, where each such sequence (of size r) is chosen with equal probability, and let t be a parameter, Assuming that " S satisfies Axioms (A) and (B), we have (R 0 )j where R 0 is a random subsequence of " S, as above, of size r Proof: The proof is a straightforward adaptation of the proof of Lemma 2.2 of [AMS94] to our ordered sampling. Intuitively, as the execution of CutRandomInc progresses, the number of trapezoids with heavy weight becomes smaller. Unfortunately, Lemma 2.3 can not be applied directly to analyze the distribution of weights of the active trapezoids in C i . Since, the axiom (B) does not hold for the active trapezoids in C i . See Remark 2.10 below. To analyze CutRandomInc, we prove a weaker version of Lemma 2.3, by relying on the fact that C i "lies" between two structures for which Lemma 2.3 hold. In the following, " S denote a set of n lines in the plane. We denote by R a selection of " S of length r n. Definition 2.4 A vertical segment which serves as a left or right side of a trapezoid, is called a splitter. Let CT W (R) denote the set of splitters of the trapezoids of A j (R), A splitter in CT W (R) is uniquely defined by 4 lines. Let S) CT W (R) denote the set of all splitters that might appear in A j (R). We denote by T the set of vertical trapezoids having a top and bottom line from " S. Similarly, let CT V D (R) denote the set of trapezoids of AVD (R). A trapezoid in CT V D (R) is defined by 4 lines. Figure 2: A vertical trapezoid is of weight k, either because one of its vertical sides intersects at least k=4 lines, or at least k=2 of the lines pass from the bottom to the top of the trapezoid. Lemma 2.5 Let " S be a set of n lines in the plane, and let R be a selection of " S. Axioms (A) and (B) holds for CT W (R) , and CT V D (R). Proof: Let s be a splitter in CT W (R). The segment s is a part of a vertical ray emanating from an intersection point p of two lines l 1 ; l 2 of R. Additionally, there are two additional lines that define the bottom and top intersection points of s. Clearly, under general position assumption, if s 2 CT W (R) then line that intersects s can appear in R. As for axiom (B), let R 0 be a subsequence of R that contains D(s). Let i be the minimal index, such that l 1 . Clearly, at this point A j (R 0 contains a vertical wall that contains s, since the vertex p is in A(R 0 either the downward or upward ray emanating from p must include s, otherwise s can not appear in A j (R). Thus, after inserting l 3 ; l 4 into the arrangement (R 0 i ), the splitter s will appear in the resulting arrangement. Implying that s 2 CT W (R 0 ). As for the second part of the lemma, this is well known [dBvKOS97]. Lemma 2.6 Let \Delta be a trapezoid of weight k in AVD (R). Let l g be a set of disjoint trapezoids contained inside \Delta, such that they have the same bottom and top lines as \Delta, their splitters belong to CT W (R), and their weight is at least k 0 . Then, the number of trapezoids of V is Proof: If either vertical sides (i.e. splitters) of \Delta i intersects k 0 =4 lines, then we charge this trapezoid to the relevant splitter of CT W (R; k 0 =4; \Delta). Otherwise, there are at least k lines of S that intersects only the top and bottom parts of namely, those lines intersection with \Delta lie inside \Delta i . See Figure 2. Thus, the number of such trapezoids is k=(k 0 =2). Definition 2.7 Let k be a positive integer number, and let U be a set of disjoint trapezoids. The set U is (k; R)-compliant if each trapezoid \Delta of U is of weight at least k, ffi is contained in a single trapezoid of AVD (R), and \Delta is a union of trapezoids of A j (R). Note, that the set of (1=r)-active trapezoids of C i (the covering of the plane computed after the i-th iteration of CutRandomInc) are (n=r; S i )-compliant. Moreover, this remains true even if CutRandomInc performs merging. Lemma 2.8 Let R be a selection of " S of r n distinct lines of " S, where each such selection (of size r) is chosen with equal probability, let U be a (n=r; R)-compliant set of trapezoids, and let t be a parameter, 1 t r=6. We have O dn=re, and T S). Proof: Since U is (n=r; R)-compliant, we have by Lemma 2.6: r ew(\Delta)(i+1)d n r e O r By Lemma 2.3, we have: O O O Theorem 2.9 The expected size of the (1=r)-cutting generated by CutRandomInc is O(r 2 ), and the expected running time is O(nr), for any integer 1 r n. Proof: Let CS(n; r) denote the maximum expected size of the (1=r)-cutting generated by r), where the maximum is taken over all sets " S of n lines in the plane. Suppose we execute CutRandomInc until cr lines are inserted, where c is a constant to be specified shortly. At this stage, the expected size of the covering cr computed by CutRandomInc is O((cr) 2 ). Indeed, each trapezoid in C is a union of one or more trapezoids of A j (S cr ). Hence, by Lemma 2.2, we have E[jCj] E[j A j (S cr Hence, if the algorithm terminates before cr lines are inserted, the expected size of the covering is O(r 2 ). For each CS(\Delta) to be the expected number of vertical trapezoids that are contained in \Delta and belong to the final covering computed by the algorithm, if we resume the execution of CutRandomInc until it terminates. However, CS(\Delta) CS(w(\Delta); dw(\Delta)r=ne), since we can interpret the execution of CutRandomInc within \Delta, as executing CutRandomInc from fresh on K (\Delta), in order to compute a w(\Delta)r -cutting inside \Delta. Indeed, if we set C 0 , in the algorithm, to be a given trapezoid \Delta, then CutRandomInc will compute a cutting inside \Delta, and then only the lines in K (\Delta) will be relevant to the behavior of the algorithm; see Figure 1. Moreover, the analysis of the performance of CutRandomInc does not depend on the shape C 0 is initialized to. Thus, we have CS(n; r) O \Delta2C;w(\Delta)?n=r CS w(\Delta)r O \Delta2C; cr cr CS c Applying Lemma 2.2 and Lemma 2.8, we have that the expected number of vertical trapezoids in C with weight tn=(cr) is O(2 \Gammat=4 (cr) 2 =t), where 1 t cr=6. Thus, cr=6 t=c O CS cr c O t=c CS cr c If we choose c to be a sufficiently large constant, the solution to this recurrence is O(r 2 ), as is easy to verify by induction. We next analyze the expected running time of the algorithm. We implement CutRandomInc using a conflict graph; namely, for each trapezoid \Delta of C i , we maintain a list of the elements of K (\Delta), and similarly, for each line of " S, we maintain a list of active trapezoids of C i that it intersects (the "zone" of the line in C i ). Let W i denote the expected work in the i-th iteration of the algorithm. It is easy to verify that say. For i ? 10, we analyze the expected value of W i , by applying Lemma 2.2 and Lemma 2.8, to CutRandomInc after iterations were performed (i.e., our random sample is of size i \Gamma 1). Then, the probability of a trapezoid to be processed at the i-th iteration of the algorithm is w(\Delta)=(n trapezoids are inactive). Moreover, if \Delta is being processed at the i-th stage, then the work associated with \Delta (at this stage) is O(w(\Delta)). We thus have, \Delta2C r w(\Delta) \Delta2C O Hence, ii \Delta2C O O by Lemma 2.2 and Lemma 2.8. Thus, the expected running time of the algorithm in the first cr iterations, is O(cnr). Let T (n; r) be the maximum expected running time of CutRandomInc( where the maximum is taken over all sets " S of n lines in the plane. Consider a set " S over which T (n; r) is maximized. Arguing as above, we have \Delta2C;w(\Delta)?n=r \Delta2C; cr w(\Delta)(t+1) n cr c t=c O cr c cr=6 O (cnr) +O t=c cr c cr2 \Gammacr=24 using Lemma 2.8. For c sufficiently large, the solution to this recurrence is easily verified, using induction, to be O(nr). This completes the proof of the theorem. Remark 2.10 We can not apply Lemma 2.3 directly on the (1=r)-active regions of C i , because the axiom (B) does not hold for this case. See Figure 3 for an example that shows that axiom may be violated if we use merging. Even if we do not use merging in CutRandomInc, axiom (B) is still violated. See Figure 4 The algorithm CutRandomIncworks also for planar arrangements of segments and x-monotone curves (such that the number of intersection of each pair of curves is bounded by a constant). l 3 l 2 l 1 l 4 l 6 l 5 a b c l 3 l 2 l 1 l 4 l 6 l 5 a c l 3 l 2 l 1 l 6 l 5 a b c (i) (ii) (iii) Figure 3: Axiom (B) fails if merging is used: The thick lines represent two sets of 100 parallel lines, and we want to compute a (1=10)-cutting. We execute CutRandomInc with the first 6 lines l 6 in this order. Note that any trapezoid that intersects a thick line is active. The first trapezoid \Delta 0 inside 4abc that becomes inactive is created when the line l 5 is being inserted; see parts (i) and (ii). However, if we skip the insertion of the line l 4 (as in part (iii)), the corresponding inactive trapezoid \Delta 00 will extend downwards and intersect \Delta. Since \Delta 00 is inactive, the decomposition of the plane inside \Delta 00 is no longer maintained. In particular, this implies that the trapezoid \Delta will not be created, since it is being blocked by \Delta 00 , and no merging involving areas inside \Delta 00 will take place. This is a contradiction to axiom (B), since l 4 does not belong to the killing or defining sets of \Delta. l 3 l 2 l 4 l 1 l 5 a c l 6 l 7 l 3 l 2 l 4 l 5 a c l 6 l 7 Figure 4: Axiom (B) fails if even if merging is not being used by CutRandomInc. Indeed, if CutRandomInc insert the lines in the order l 1 then the trapezoid \Delta is being created. See (i). However, if we skip the insertion of the line l 1 , then the trapezoid \Delta is not being created, because the ray emanating downward from l 2 " l 3 intersects it. This follows immediately by observing that Lemma 2.2 and Lemma 2.8 can be extended for those cases, and that Axioms (A) and (B) hold for the vertical decomposition of such arrangements, and for the set of splitters of such arrangements. Lemma 2.11 Let \Gamma be a set of x-monotone curves such that each pair intersects in at most a constant number of points. Then the expected size of the (1=r)-cutting generated by CutRandomInc for \Gamma is O(r 2 ), and the expected running time is O(nr), for any integer 1 r n. However, the arrangement of a set of n segments or curves might have subquadratic complexity (since the number of intersection points might be subquadratic). This raises the question whether CutRandomInc generates smaller cuttings for such sparse arrangements. Definition 2.12 Let \Gamma be a set of curves in the plane. We denote by (\Gamma) the number of intersection points between pairs of curves of \Gamma. Lemma 2.13 Let " \Gamma be a set of n curves, so that each pair of curves from \Gamma have at most intersection points, and let \Gamma be a random permutation of " \Gamma. Then the expected complexity of A j (\Gamma) is O(n log n \Gamma)), where A j (\Gamma) is the decomposed arrangement computed by the incremental insertion algorithm, without performing merging. Proof: Note that any intersection point of a pair of curves of " \Gamma, induces an upward and downward vertical "walls", and the expected number of vertical walls in A j (\Gamma) is O(( " \Gamma)), arguing as in the proof of Lemma 2.2. Additionally, there are vertical walls defined by the endpoints of the curves of \Gamma. Let p be an endpoint of a curve \Gamma. The probability that the vertical upward ray v p emanating from will introduce i vertical walls in A j (\Gamma), is the probability that fl will be chosen before the first i curves of \Gamma that this vertical ray intersects. Thus, the expected number of superfluous vertical walls introduced by v p is Thus, the total number of vertical walls in A j (\Gamma) introduced by the endpoints of arcs in " \Gamma is O(n log n), and the Lemma readily follows. Corollary 2.14 Let " \Gamma be a set of n curves, so that each pair of curves from " \Gamma have at most intersection points, and let \Gamma r be a random selection of r elements of " \Gamma. Then the expected complexity of A j (\Gamma r ) is O(r log r Proof: We note that the probability of an intersection point of A( " \Gamma) to appear in is r(r\Gamma1) . Hence, the expected number of intersection points of arcs of " in A is O The lemma now readily follows by applying Lemma 2.13 to A j (\Gamma r ). Theorem 2.15 Let " \Gamma be a set of n curves, such that each pair of curves of " intersect in at most a constant number of points. Then the expected size of the (1=r)-cutting generated by CutRandomInc, when applied to \Gamma, is O and the expected running time is O(n log \Gamma). Proof: The proof is a tedious extension of the proof of Theorem 2.9. We derive similar recurrences to the ones used in the proof of Theorem 2.9. In deriving and solving those recurrences, we repeatedly apply the bounds stated in Lemma 2.11. We omit the details. Remark 2.16 An interesting question is whether CutRandomInc can be extended to higher di- mensions. If we execute CutRandomInc in higher dimensions, we need to use a more complicated technique in decomposing each of our "vertical trapezoids" whenever it intersects a newly inserted hyperplane. Chazelle and Friedman's algorithm uses bottom vertex triangulation for this decomposition. However, in our case, it is easy to verify that CutRandomInc might generate simplices that their defining set is no longer a constant number of hyperplanes. This implies that Lemma 2.3 can no longer be applied to CutRandomInc in higher dimension. We leave the problem of extending CutRandomInc to higher dimensions as an open problem for further research. Generating Small Cuttings In this section, we present an efficient algorithm that generates cuttings of guaranteed small size. The algorithm is based on Matousek's construction of small cuttings [Mat98]. We first review his construction, and then show how to modify it for building small cuttings efficiently. Definition 3.1 ([Mat98]) Let L be a set of n lines in the plane in general position, i.e., every pair of lines intersect in exactly one point, no three have a common point, no line is vertical or horizontal, and the x-coordinate of all intersections are pairwise distinct. The level of a point in the plane is the number of lines of L lying strictly below it. Consider the set E k of all edges of the arrangement of L having level k (where 0 k ! n). These edges form an x-monotone connected polygonal line, which is called the level k of the arrangement of L. Definition 3.2 ([Mat98]) Let E k be the level k in the arrangement A(L) with edges (from left to right), and let p i be a point in the interior of the edge e i , for t. The q-simplification of the level k, for an integer parameter 1 q t, is defined as the x- monotone polygonal line containing the part of e 0 to the left of the point p 0 , the segments p 0 p q , and the part of e t to the right of p t . Let simp q polygonal line. Let L be a set of n lines, and let E i;q denote the union of the levels denote the set of edges of the q-simplifications of the levels of E i;q . Matousek showed that the vertical decomposition of the plane induced by simp q (E i;q ), where assume that n is divisible by 2r), is a (1=r)-cutting of the plane, for any 1. Moreover, the following holds: Theorem 3.3 ([Mat98]) Let L be a set of n lines in general position, and let r be a positive integer, and let n=(2r). Then the subdivision of the plane defined by the vertical decomposition of simp q (E m;q ) is a (1=r)-cutting of A(L), where is the index i which jE i;q j is minimized. Moreover, the cutting generated has at most 8r 2 Remark 3.4 (i) Matousek's construction can be slightly improved, by noting that the leftmost and rightmost points in a q-simplification of a level can be placed at "infinity"; that is, we replace the first and second edges in the q-simplification by a ray emanating from p q which is parallel to e 0 . We can do the same thing to the two last edges of the simplified level. We denote this improved simplification by simp 0 q . It is easy to prove that using this improved simplification results in a (1=r)-cutting of A(L) with at most 8r 2 trapezoids. (ii) Inspecting Matousek's construction, we see that if we can only find an i such that prescribed constant, then the vertical decomposition induced by simp q (E i;q ) is a (1=r)-cutting having c(8r trapezoids. construction is carried out by computing the picking the minimal number n i , which is guaranteed to be no larger than the average n 2 =q. Unfortunately, implementing this scheme directly, requires computing the whole arrangement A(L), so the resulting running time is O(n 2 ). Let us assume for the moment that one can compute any of the numbers n i quickly. Then, as the following lemma testifies, one can compute a number n i which is (1 computing all the n i s. Lemma 3.5 Let n positive integers, whose sum known in ad- vance, and let " ? 0 be a prescribed constant. One can compute an index 0 k ! q, such that ")m=qe, by repeatedly picking uniformly and independently a random index q, and by checking whether n i ")m=qe. The expected number of iterations required is Proof: Let Y i be the random variable which is the value of n i picked in the i-th iteration. Using Markov's inequality 1 , one obtain: we have that the probability for failure in the i-th iteration is 1 The inequality asserts that P r t , for a random variable Y that assumes only nonnegative values. Let X denote the number of iterations required by the algorithm. Then E[X] is bounded by the expected number of trials to the first success in a geometric distribution with probability 1+" . Thus, the expected number of iterations is bounded by 1+" To apply Lemma 3.5 in our setting, we need to supply an efficient algorithm for computing the level of an arrangement of lines in the plane. Lemma 3.6 Let L be a set of n lines in the plane. Then one can compute, in O((n time, the level k of A(L), where is the complexity of the level. Proof: The technique presented here is well known (see [BDH97] for a recent example):- we include it for the sake of completeness of exposition. Let e t be the edges of the level k from left to right (where e are rays). Let e be an edge of the level k. Clearly, there exists a face f of A(L) having e on its boundary such that f lies above e. In particular, all the edges on the bottom part of @f belong to the level k. r be the faces of A(L) having the level k as their "floor", from left to right. The ray e 0 can be computed in O(n) time since it lies on line l k of L, with the k-th largest slope. Moreover, by intersecting l k with the other lines of L, one can compute e 0 in linear time. Any face of A(L) is uniquely defined as an intersection of half-planes induced by the lines of L. For the faces f 1 , we can compute the half-planes and their intersection, that corresponds to f 1 , in O(n log n) time, see [dBvKOS97]. To carry out the computation of bottom parts of dynamically maintain the intersection representing f i as we traverse the level k from left to right. To do so, we will use the data-structure of Overmars and Van Leeuwen [OvL81] that maintains such an intersection, with O(log 2 n) time for an update operation. As we move from f i to f i+1 through a vertex v, we have to "flip" the two half-planes associated with the two lines passing through v. Thus such operation will cost us O(log 2 n) time. Similarly, if we are given an edge e on the boundary of f i we can compute the next edge in O(log 2 n) time. Thus, we can compute the level k of A(L) in O((n Combining Lemma 3.5 and Lemma 3.6, we have: Theorem 3.7 Let L be a set of n lines in the plane, and let be a prescribed constant. Then one can compute a (1=r)-cutting of A(L), having at most (1 trapezoids. The expected running time of the algorithm is O nr log 2 n Proof: By the above discussion, it is enough to find an index 0 i q \Gamma 1, such that dn=(2r)e. By Remark 3.4 (ii), the vertical decomposition of simp 0 (1=r)-cutting of the required size. Picking i randomly, we have to check whether jE i;q j M . We can compute E i;q , by computing the levels in an output sensitive manner, using Lemma 3.6. Note that if jE i;q j ? M , we can abort as soon as the number of edges we computed exceeds M . Thus, checking if jE i;q j M takes O((1 time. By Lemma 3.6, the expected number of iterations the algorithm performs until the inequality jE i;q j M will be satisfied is Thus, the expected running time of the algorithm is O nr log 2 n since the vertical decomposition of simp 0 (which is the resulting cutting) can be computed in additional O((1 In fact, one can also compute, in O((1 for each trapezoid in the cutting, the lines of L that intersect it. 4 Empirical Results In this section, we present the empirical results we got for computing cuttings in the plane using CutRandomInc and various related heuristics that we have tested. The test program with a GUI of the alogrithm presented in this paper, is avaliable on the web in source form. It can be downloaded from: http://www.math.tau.ac.il/ ~ sariel/CG/cutting/cuttings.html 4.1 The Implemented Algorithms - Using Vertical Trapezoids We have implemented the algorithm CutRandomInc presented in Section 2, as well as several other algorithms for constructing cuttings. We have also experimented with the algorithm of Section 3. In this section, we report on the experimental results that we obtained. Most of the algorithms we have implemented are variants of CutRandomInc. The algorithms implemented are the following: Classical: This is a variant of the algorithm of Chazelle and Friedman [CF90] for constructing a cutting. We pick a sample R S of r lines, and compute its arrangement For each active trapezoid \Delta 2 A, we pick a random sample R \Delta ' K (\Delta) of size 6k log k, where (\Delta)j=ne, and compute the arrangement of AVD (R \Delta ) inside \Delta. If AVD (R \Delta ) is not a (1=r)-cutting, then the classical algorithm performs resampling inside \Delta until it reaches a cutting. Our implementation is more naive, and it simply continues recursively into the active subtrapezoids of AVD (R \Delta ). Cut Randomized Incremental: This is CutRandomInc without merging, as described in Figure 1. The following four heuristics, for which we currently do not have a proof of any concrete bound on the expected size of the cutting that they generate, also perform well in practice. Parallel Incremental: Let C i be the covering generated in the i-th iteration of the algorithm. For each active trapezoid pick a random line from K (\Delta), and insert it into \Delta (i.e., splitting \Delta). Continue until there are no active trapezoids. Note that unlike CutRandomInc the insertion operations are performed locally inside each trapezoid, and the line chosen for insertion in each trapezoid is independent of the lines chosen for other trapezoids. Randomized Incremental: This is CutRandomInc with merging. Greedy Trapezoid: This is a variant of CutRandomInc where we try to be "smarter" about the line inserted into the partition in each iteration. Let V i be the set of trapezoids of C i with maximal weight. We pick randomly a trapezoid \Delta out of the trapezoids of V i , and pick randomly a line s from K (\Delta). We then insert s into C i . Greedy Line: Similar to Greedy Trapezoid, but here we compute the set U of lines of " for which w 0 (s) is maximal, where w 0 (s) is the number of active trapezoids in C i that intersect the line s. We pick randomly a line from U and insert it into the current partition of the plane. Greedy Weighted Line: Similar to Greedy Line, but our weight function is: w(\Delta) \Xi n 3r namely, we give a higher priority to lines that intersect heavier (1=r)-active trapezoids. 4.2 Polygonal Cuttings In judging the quality of cuttings, the size of the cutting is of major concern. However, other factors might also be important. For example we want the regions defining the cutting to be as simple as possible. Furthermore, there are applications where we are not interested directly in the size of the cutting, but rather in the overall number of vertices defining the cutting regions. This is useful when applying cuttings in the dual plane, and transforming the vertices of the cutting back to the primal plane, as done in the computation of partition trees [Mat92]. A natural question is the following: Can one compute better cuttings, if one is willing to use cutting regions which are different from vertical trapezoids? For example, if one is willing to cut using non-convex regions having a non-constant description complexity, the size of the cutting can be improved to ????????. However, if one wishes to cut a collection of lines by triangles, instead of trapezoids, the situation becomes somewhat dis- appointing, because the smallest cuttings currently known for this case, are generated by taking the cutting of Remark 3.4, and by splitting each trapezoid into two triangles. This results in cuttings having (roughly) 16r 2 triangles. In this section, we present a slightly different approach for computing cuttings, suggested to us by Jiri how to write name??? Matousek, that works extremely well in practice. The new approach is based on cuttings by using small polygonal convex regions, instead of vertical trapezoids. Namely, we apply CutRandomInc, where each region is a convex polygon (of constant complexity). Whenever we insert a new line into an active region, we split the polygon into two new polygons. Of course, it might be that the number of vertices of a new polygon is too large. If so, we split each such polygon into two subpolygons ensuring that the number of vertices of the new polygons are below our threshold. Intuitively, the benefit in this approach is that the number of superfluous entities (i.e. vertical walls in the case of vertical trapezoids) participating in the definition of the cutting regions is a Figure 5: In the PolyTree algorithm, each time a polygon is being split by a line, we might have to further split it because a split region might have too many vertices. much smaller. Moreover, since the cutting regions are less restrictive, the algorithm can be more flexible in its maintenance of the active regions. Here are the different methods we tried: PolyTree: We use CutRandomInc where each region is a convex polygon having at most k- sides. When inserting a new line, we first split each of the active regions that intersect it into two subpolygons. If a split region R has more than k sides, we further split it using the diagonal of R that achieves the best balanced partition of R; namely, it is the pair of vertices a; b realizing the following minimum: min ab ab (R) is the set of vertices of R, and w(R) is the number of lines intersecting R, and H ab ab ) is the closed halfplane lying to the right (resp. left) of ab. See Figure 5. PolyTriangle: Modified PolyTree for generating cuttings by triangles. In each stage, we check whether a newly created region R can be triangulated into a set of inactive triangles. If do so, by applying an arbitrary triangulation to A region R and check if all the triangles generated in our (arbitrary) triangulation of R are inactive. If so, we replace R in our cuttings by its Modified PolyTree for generating cuttings by triangles. Whenever a region is being created we check whether it has a leaf triangle (a triangle defined by three consecutive vertices of the region) that is inactive. If we find such an inactive triangle, we add it immediately to the final cutting. We repeat this process until the region can not be further shrunk. 2 Computing the "best" (i.e. the weight of the heaviest triangle is minimized) triangulation is relatively compli- cated, and requires dynamic programming. It is not clear that it is going to perform better than PolyDeadLeaf, described below. We use PolyTree, but instead of splitting along a diagonal, we split along a vertical ray emanating from one of the vertices of the region. The algorithm also tries to remove dead regions from the left and right side of the region. Intuitively, each region is now an extended vertical trapezoid having a convex ceiling and floor, with at most two additional vertical walls. Theorem 4.1 The expected size of the (1=r)-cutting generated by PolyVertical is O(r 2 ), and the expected running time is O(nr), for any integer 1 r n. Proof: We only sketch the proof. First, note that the number of regions maintained by PolyVertical in the i-th iteration is O(i 2 ), since each region maintained by PolyVertical is a union of trapezoids of A j (S i ). And the total complexity of A j (S i ) is O(i 2 ) (Lemma 2.2. Let be the maximal number of vertical in a region maintained curing the execution of PolyVertical (This is a parameter of the algorithm). We know that if a region P is (1=r)-active after the i-th iteration of the algorithm, then P must contain at least one vertical trapezoid of 2.8, the expected number of such trapezoids, having weight larger than t dn=re is O Thus, we have an exponential decay bound on the distribution of heavy trapezoids, during the execution of the algorithm. We now derive similar recurrences to the recurrences used in Theorem 2.9 to bound the running time, and size of the cutting generated by PolyVertical. Remark 4.2 Note, that for all the polygonal cutting methods, except PolyVertical, it is not even clear that the number of regions they maintain, in the i-th iteration, is O(i 2 ). Thus, the proof of Theorem 2.9 does not work for those methods. 4.3 Implementation Details As an underlying data-structure for our testing, we implemented the history-graph data-structure [Sei91]. Our random arrangements were constructed by choosing n points uniformly and independently on the left side of the unit square, and similarly on the right side of the unit square. We sorted the points, and connected them by lines in a transposed manner. This yields a random arrangement with all the intersections inside the unit square. We had implemented our algorithm in C++. We had encountered problems with floating point robustness at an early stage of the implementation, and decided to use exact arithmetic instead, using LEDA rational numbers [MN95]. While this solved the robustness problems, we had to deal with a few other issues: ffl Speed: Using exact arithmetic instead of floating point arithmetic resulted in a slow down by a factor 20-40. The time to perform an operation in exact arithmetic is proportional to the bit-sizes of the numbers involved. To minimize the size of the numbers used in the computations, we normalized the line equations so that the coefficients are integer numbers (in reduced form). ffl Memory consumption: A LEDA rational is represented by a block of memory dynamically allocated on the heap. In order to save, both in the memory consumed and the time used by the dynamic memory allocator, we observe that in a representation of vertical decomposition the same number appears in several places (i.e., an x-coordinate of an intersection point appears in 6 different vertical trapezoids). We reduce memory consumption, by storing such a number only once. To do so, we use a repository of rational numbers generated so far by the algorithm. Whenever we compute a new x-coordinate, we search it in the repository, and if it does not exist, then we insert it. In particular, each vertical trapezoid is represented by two pointers to its x lef t and x right coordinates, and pointers to its top and bottom lines. The repository is implemented using Treaps [SA96]. 4.4 Results The empirical results we got for the algorithms/heuristics of Subsection 4.1, are depicted in Tables For each value of r, and each value of n, we computed a random arrangement of lines inside the unit square, as described above. For each such arrangement, we performed 10 tests for each algorithm/heuristic. The tables present the size of the minimal cutting computed in those tests. Each entry is the size of the output cutting divided by r 2 . In addition, each table caption presents a range containing the size of the cutting that can be obtained by Matousek's algorithm [Mat98]. As noted in Remark 2.10, it is an interesting question whether or not using merging results in smaller cuttings generated by CutRandomInc. We tested this empirically, and the results are presented in Table 6. As can be seen in Table 6, using merging does generate smaller cuttings, but the improvement in the cutting size is rather small. The difference in the size of the cuttings generated seems to be less than 2r 2 . 4.5 Implementing Matousek's Construction In Table 7, we present the empirical results for Matousek's construction, comparing it with the slight improvement described in Remark 3.4. For small values of r the improved version yields considerably smaller cuttings than Matousek's construction, making it the best method we are aware of for constructing small cuttings. We had implemented Matousek's algorithm naively, using quadratic space and time. Cur- rently, this implementation can not be used for larger inputs because it runs out of memory. Implementing the more efficient algorithm described in Theorem 3.7 is non-trivial since it requires the implementation of the data-structure of Overmars and van Leeuwen [OvL81]. However, if it is critical to reduce the size of a cutting for large inputs, the algorithm of Theorem 3.7 seems to be the best available option. To look into papers of cuttings. Is it possible that just picking r lines and computing the arrangement might be enough, when using the distribution lemma? didn't this idea appear in Matousek paper? Can partition trees work with lazy cuttings? If not, what is the penalty? If lazy cuttings are indeed useful, to indicate in the introduction why cuttings are interesting. To add reference to Matousek deterministic construction of cuttings. 4.6 Polygonal Cuttings The results for polygonal cuttings are presented in Tables 8, 9. As seen in the tables the polygonal cutting methods perform well in practice. In particular, the PolyTree method generated cuttings of average size (roughly) 7:5r 2 , beating all the cutting methods that use vertical trapezoids. As for triangles, the situation is even better: PolyDeadLeaf generates cuttings by triangles of size 12r 2 . (That is better by an additive factor of about 4r 2 than the best theoretical bound). To summarize, polygonal cutting methods seems to be the clear winner in practice. The generate quickly cuttings of small size, with small number of vertices, and small number of triangles. Conclusions In this paper, we presented a new approach, different from that of [CF90], for constructing cuttings. The new algorithm is rather simple and easy to implement. We proved the correctness and bounded the expected running time of the new algorithm, while demonstrating that the new algorithm performs much better in practice than the algorithm of [CF90]. We believe that the results in this paper shows that planar cuttings are practical, and might be useful in practice when constructing data-structures for range-searching. Moreover, the empirical results show that the size of the cutting constructed by the new algorithm is not considerably larger (and in some cases better) than the cuttings that can be computed by the currently best theoretical algorithm (too slow to be useful in practice due to its running time) of Matousek [Mat98]. The empirical constants that we obtain are generally between 10 and 13 (for vertical trapezoids). For polygonal cuttings we get a constant of 7 by cutting by convex polygons (using PolyTree) having at most 6 vertices. Moreover, the various variants of CutRandomInc seem to produce constants that are rather close to each other. As noted above, the method described in Remark 3.4 generates the smallest cuttings by vertical trapezoids (but is rather slow because of our naive implementation). Tables 9 present the running time we got for the various cutting algorithms. This information should be taken with reservation, since no serious effort had gone into optimizing the code for speed, and those measurements tend to change from execution to execution. (Recall also that we use exact arithmetic, which slows down the running time significantly.) However, it does provide a general comparison between the running times of the various methods in practice. Given this results, in we recommend for use in practice one of the polygonal-cutting methods. They perform well in practice, and they should be used whenever possible. If we are restricted to vertical trapezoid, CutRandomInc seems like a reasonable algorithm to be used in practice (without merging, as this is the only "non-trivial" part in the implementation of the algorithm). There are several interesting open problems for further research: ffl Can one obtain a provable bound on the expected size of the cutting generated by the PolyTree methods? ffl Can one prove the existence of a cutting smaller than the one guaranteed by the algorithm in Remark 3.4 for specific values of r? For example, Table 1 suggests a smaller cutting should exist for 2. In particular, the test results hint that a cutting made out of vertical trapezoid should exist, while the cutting size guaranteed by Matousek's algorithm [Mat98] is 48. ffl Can one generate smaller cuttings by modifying CutRandomInc to be smarter in its decision when to merge trapezoids? ffl Is there a simple and practical algorithm for computing cuttings in three and higher di- mensions? The current algorithms seems to be far from practical. Acknowledgments The author wishes to thank Pankaj Agarwal, Boris Aronov, Herve Bronnimann, Bernard Chazelle , Jiri Matousek, and Joe Mitchell for helpful discussions concerning the problems studied in this paper and related problems. I wish to thank Micha Sharir for his help and guidance in preparing the paper. --R Kinetic binary space partitions for triangles. Geometric partitioning and its applications. The area bisectors of a polygon and force equilibria in programmable vector fields. A deterministic view of random sampling and its use in geometry. New applications of random sampling in computational geometry. Computational Geometry: Algorithms and Applications. LEDA: a platform for combinatorial and geometric computing. Maintenance of configurations in the plane. Randomized search trees. A simple and fast incremental randomized algorithm for computing trapezoidal decompositions and for triangulating polygons. --TR --CTR Micha Sharir , Emo Welzl, Point-line incidences in space, Proceedings of the eighteenth annual symposium on Computational geometry, p.107-115, June 05-07, 2002, Barcelona, Spain Micha Sharir , Emo Welzl, PointLine Incidences in Space, Combinatorics, Probability and Computing, v.13 n.2, p.203-220, March 2004 Siu-Wing Cheng , Antoine Vigneron, Motorcycle graphs and straight skeletons, Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms, p.156-165, January 06-08, 2002, San Francisco, California Sariel Har-Peled , Micha Sharir, Online point location in planar arrangements and its applications, Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms, p.57-66, January 07-09, 2001, Washington, D.C., United States Hayim Shaul , Dan Halperin, Improved construction of vertical decompositions of three-dimensional arrangements, Proceedings of the eighteenth annual symposium on Computational geometry, p.283-292, June 05-07, 2002, Barcelona, Spain Pankaj K. Agarwal , Micha Sharir, Pseudo-line arrangements: duality, algorithms, and applications, Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms, p.800-809, January 06-08, 2002, San Francisco, California

range-searching;cuttings;computational geometry

356493

Gadgets, Approximation, and Linear Programming.

We present a linear programming-based method for finding "gadgets," i.e., combinatorial structures reducing constraints of one optimization problem to constraints of another. A key step in this method is a simple observation which limits the search space to a finite one. Using this new method we present a number of new, computer-constructed gadgets for several different reductions. This method also answers a question posed by Bellare, Goldreich, and Sudan [SIAM J. Comput., 27 (1998), pp. 804--915] of how to prove the optimality of gadgets: linear programming duality gives such proofs.The new gadgets, when combined with recent results of H stad [ Proceedings of the 29th ACM Symposium on Theory of Computing, 1997, pp. 1--10], improve the known inapproximability results for MAX CUT and MAX DICUT, showing that approximating these problems to within factors of $16/17 + \epsilon$ and $12/13+ \epsilon,$ respectively, is NP-hard for every $\epsilon > 0$. Prior to this work, the best-known inapproximability thresholds for both problems were 71/72 (M. Bellare, O. Goldreich, and M. Sudan [ SIAM J. Comput., 27 (1998), pp. 804--915]). Without using the gadgets from this paper, the best possible hardness that would follow from Bellare, Goldreich, and Sudan and H{s}tad is $18/19$. We also use the gadgets to obtain an improved approximation algorithm for MAX3 SAT which guarantees an approximation ratio of .801. This improves upon the previous best bound (implicit from M. X. Goemans and D. P. Williamson [ J. ACM, 42 (1995), pp. 1115--1145]; U. Feige and M. X. Goemans [ Proceedings of the Third Israel Symposium on Theory of Computing and Systems, 1995, pp. 182--189]) of .7704.

Introduction . A \gadget" is a nite combinatorial structure which translates a given constraint of one optimization problem into a set of constraints of a second optimization problem. A classical example is in the reduction from 3SAT to MAX 2SAT, due to Garey, Johnson and Stockmeyer [6]. Given an instance of 3SAT on variables X and with clauses C the reduction creates an instance of MAX 2SAT on the original or \primary" variables along with new or \auxiliary" variables Y . The clauses of the MAX 2SAT instance are obtained by replacing each clause of length 3 in the 3SAT instance with a \gadget", in this case a collection of ten 2SAT clauses. For example the clause C would be replaced with the following ten clauses on the variables new auxiliary variable Y The property satised by this gadget is that for any assignment to the primary vari- ables, if clause C k is satised, then 7 of the 10 new clauses can be satised by setting only 6 of the 10 are satisable. (Notice that the gadget An extended abstract of this paper appears in the Proceedings of the 37th IEEE Symposium on Foundations of Computer Science, pages 617-626, Burlington, Vermont, 14-16 October 1996. y Columbia University, Department of Computer Science, 1214 Amsterdam Avenue, New York, NY 10027, USA. luca@cs.columbia.edu. Part of this work was done while the author was at the University of Rome \La Sapienza" and visiting IBM Research. z IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heights NY 10598. fsorkin, dpwg@watson.ibm.com. x MIT, Laboratory for Computer Science, 545 Technology Square, Cambridge, MA 02139, USA. madhu@mit.edu. Work supported in part by an Alfred P. Sloan Foundation fellowship. Part of this work was done while the author was at the IBM Thomas J. Watson Research Center. L. TREVISAN, G. B. SORKIN, M. SUDAN, AND D. P. WILLIAMSON associated with each clause C k uses its own auxiliary variable Y k , and thus Y k may be set independently of the values of variables not appearing in C k 's gadget.) Using this simple property of the gadget it is easy to see that the maximum number of clauses satised in the MAX 2SAT instance by any assignment is 7m if and only if the instance of 3SAT is satisable. This was used by [6] to prove the NP-hardness of solving MAX 2SAT. We will revisit the 3SAT-to-2SAT reduction in Lemma 6.5. Starting with the work of Karp [12], gadgets have played a fundamental role in showing the hardness of optimization problems. They are the core of any reduction between combinatorial problems, and they retain this role in the spate of new results on the non-approximability of optimization problems. Despite their importance, the construction of gadgets has always been a \black art", with no general methods of construction known. In fact, until recently no one had even proposed a concrete denition of a gadget; Bellare, Goldreich and Sudan [2] nally did so, with a view to quantifying the role of gadgets in non-approximability results. Their denition is accompanied by a seemingly natural \cost" measure for a gadget. The more \costly" the gadget, the weaker the reduction. However, rstly, nding a gadget for a given reduction remained an ad hoc task. Secondly, it remained hard to prove that a gadget's cost was optimal. This paper addresses these two issues. We show that for a large class of reductions, the space of potential gadgets that need to be considered is actually nite. This is not entirely trivial, and the proof depends on properties of the problem that is being reduced to. However, the method is very general, and encompasses a large number of problems. An immediate consequence of the niteness of the space is the existence of a search procedure to nd an optimal gadget. But a naive search would be impracticably slow, and search-based proofs of the optimality (or the non-existence) of a gadget would be monstrously large. Instead, we show how to express the search for a gadget as a linear program (LP) whose constraints guarantee that the potential gadget is indeed valid, and whose objective function is the cost of the gadget. Central to this step is the idea of working with weighted versions of optimization problems rather than unweighted ones. (Weighted versions result in LPs, while unweighted versions would result in integer programs, IPs.) This seemingly helps only in showing hardness of weighted optimization problems, but a result due to Crescenzi, Silvestri and Trevisan [3] shows that for a large class of optimization problems (including all the ones considered in this paper), the weighted versions are exactly as hard with respect to approximation as the unweighted ones. Therefore, working with a weighted version is as good as working with an unweighted one. The LP representation has many benets. First, we are able to search for much more complicated gadgets than is feasible manually. Second, we can use the theory of LP duality to present short(er) proofs of optimality of gadgets and non-existence of gadgets. Last, we can solve relaxed or constrained versions of the LP to obtain upper and lower bounds on the cost of a gadget, which can be signicantly quicker than solving the actual LP. Being careful in the relaxing/constraining process (and with a bit of luck) we can often get the bounds to match, thereby producing optimal gadgets with even greater e-ciency! Armed with this tool for nding gadgets (and an RS/6000, OSL, and often GADGETS, APPROXIMATION, AND LINEAR PROGRAMMING 3 some of the known gadgets and construct many new ones. (In what follows we often talk of \gadgets reducing problem X to problem Y" when we mean \gadgets used to construct a reduction from problem X to problem Y".) Bellare et al. [2] presented gadgets reducing the computation of a \verier" for a PCP (probabilistically checkable proof system) to several problems, including MAX 3SAT, MAX 2SAT, and MAX CUT. We examine these in turn and show that the gadgets in [2] for MAX 3SAT and MAX 2SAT are optimal, but their MAX CUT gadget is not. We improve on the e-ciency of the last, thereby improving on the factor to which approximating MAX CUT can be shown to be NP-hard. We also construct a new gadget for the MAX DICUT problem, thereby strengthening the known bound on its hardness. Plugging our gadget into the reduction (specically Lemma 4.15) of [2], shows that approximating MAX CUT to within a factor of 60=61 is NP-hard, as is approximating MAX DICUT to within a factor of 44=45. 2 For both problems, the hardness factor proved in [2] was 71=72. The PCP machinery of [2] has since been improved by Hastad [9]. Our gadgets and Hastad's result show that, for every > 0, approximating MAX CUT to within a factor of 16=17 + is NP-hard, as is approximating MAX DICUT to within a factor of 12=13+ . Using Hastad's result in combination with the gadgets of [2] would have given a hardness factor of for both problems, for every > 0. Obtaining better reductions between problems can also yield improved approximation algorithms (if the reduction goes the right way!). We illustrate this point by constructing a gadget reducing MAX 3SAT to MAX 2SAT. Using this new reduction in combination with a technique of Goemans and Williamson [7, 8] and the state-of- the-art :931-approximation algorithm for MAX 2SAT due to Feige and Goemans [5] (which improves upon the previous :878-approximation algorithm of [8]), we obtain a :801-approximation algorithm for MAX 3SAT. The best result that could be obtained previously, by combining the technique of [7, 8] and the bound of [5], was :7704. (The best previously published result is a :769-approximation algorithm, due to Ono, Hirata, and Asano [14].) Finally, our reductions have implications for probabilistically checkable proof sys- tems. Let PCP c;s [log; q] be the class of languages that admit membership proofs that can be checked by a probabilistic verier that uses a logarithmic number of random bits, reads at most q bits of the proof, accepts correct proofs of strings in the language with probability at least c, and accepts purported proofs of strings not in the language with probability at most s. We show: rst, for any > 0, there exist constants c and s, c=s > 10=9 , such that NP PCP c;s [log; 2]; and second, for all c; s with c=s > 2:7214, PCP c;s [log; 3] P. The best bound for the former result obtainable from [2, 9] is 22=21 ; the best previous bound for the latter was 4 [16]. All the gadgets we use are computer-constructed. In the nal section, we present an example of a lower bound on the performance of a gadget. The bound is not computer constructed and cannot be, by the nature of the problem. The bound still relies on dening an LP that describes the optimal gadget, and extracting the lower 1 Respectively, an IBM RiscSystem/6000 workstation, the IBM Optimization Subroutine Library, which includes a linear programming package, and (not that we are partisan) IBM's APL2 programming language. Approximation ratios in this paper for maximization problems are less than 1, and represent the weight of the solution achievable by a polynomial time algorithm, divided by the weight of the optimal solution. This matches the convention used in [18, 7, 8, 5] and is the reciprocal of the measure used in [2]. 4 L. TREVISAN, G. B. SORKIN, M. SUDAN, AND D. P. WILLIAMSON bound from the LP's dual. Subsequent work. Subsequent to the original presentation of this work [17], the approximability results presented in this paper have been superseded. Karlo and Zwick [10] present a 7/8-approximation algorithm for MAX 3SAT. This result is tight unless NP=P [9]. The containment result PCP c;s [log; 3] P has also been improved by Zwick [19] and shown to hold for any c=s 2. This result is also tight, again by [9]. Finally, the gadget construction methods of this paper have found at least two more applications. Hastad [9] and Zwick [19] use gadgets constructed by these techniques to show hardness results for two problems they consider, MAX 2LIN and MAX NAE3SAT respectively. Version. An extended abstract of this paper appeared as [17]. This version corrects some errors, pointed out by Karlo and Zwick [11], from the extended abstract. This version also presents inapproximability results resting on the improved PCP constructions of Hastad [9], while mentioning the results that could be obtained otherwise Organization of this paper. The next section introduces precise denitions which formalize the preceding outline. Section 3 presents the niteness proof and the LP-based search strategy. Section 4 contains negative (non-approximability) results and the gadgets used to derive them. Section 5 brie y describes our computer system for generating gadgets. Section 6 presents the positive result for approximating MAX 3SAT. Section 7 presents proofs of optimality of the gadgets for some problems and lower bounds on the costs of others. It includes a mix of computer-generated and hand-generated lower bounds. 2. Denitions. We begin with some denitions we will need before giving the denition of a gadget from [2]. In what follows, for any positive integer n, let [n] denote the set ng. Definition 2.1. A (k-ary) constraint function is a boolean function f : 1g. We refer to k as the arity of a k-ary constraint function f . When it is applied to variables X (see the following denitions) the function f is thought of as imposing the constraint f(X Definition 2.2. A constraint family F is a collection of constraint functions. The arity of F is the maximum of the arity of the constraint functions in F . Definition 2.3. A constraint C over a variable set is a constraint function and are distinct members of [n]. The constraint C is said to be satised by an assignment an to X We say that constraint C is from F if f 2 F . Constraint functions, constraint families and constraints are of interest due to their dening role in a variety of NP optimization problems. Definition 2.4. For a nitely specied constraint family F , MAX F is the optimization problem given by: Input: An instance consisting of m constraints C non-negative real weights w instance is thus a triple w).) Goal: Find an assignment ~ b to the variables ~ which maximizes the weight of satised constraints. Constraint functions, families and the class fMAX F j Fg allow descriptions of GADGETS, APPROXIMATION, AND LINEAR PROGRAMMING 5 optimization problems and reductions in a uniform manner. For example, if 2SAT is the constraint family consisting of all constraint functions of arity at most 2 that can be expressed as the disjunction of up to 2 literals, then MAX 2SAT is the corresponding MAX F problem. Similarly MAX 3SAT is the MAX F problem dened using the constraint family consisting of all constraint functions of arity up to 3 that can be expressed as the disjunction of up to 3 literals. One of the motivations for this work is to understand the \approximability" of many central optimization problems that can be expressed as MAX F problems, including MAX 2SAT and MAX 3SAT. For 2 [0; 1], an algorithm A is said to be a -approximation algorithm for the MAX F problem, if on every instance w) of MAX F with n variables and m constraints, A outputs an assignment ~a s.t. b)g. We say that the problem MAX F is -approximable if there exists a polynomial time-bounded algorithm A that is a -approximation algorithm for MAX F . We say that MAX F is hard to approximate to within a factor (-inapproximable), if the existence of a polynomial time -approximation algorithm for MAX F implies NP=P. Recent research has yielded a number of new approximability results for several MAX F problems (cf. [7, 8]) and a number of new results yielding hardness of approximations (cf. [2, 9]). One of our goals is to construct e-cient reductions between MAX F problems that allow us to translate \approximability" and \inapproximabil- ity" results. As we saw in the opening example such reductions may be constructed by constructing \gadgets" reducing one constraint family to another. More specically, the example shows how a reduction from 3SAT to 2SAT results from the availability, for every constraint function f in the family 3SAT, of a gadget reducing f to the family 2SAT. This notion of a gadget reducing a constraint function f to a constraint family F is formalized in the following denition. Definition 2.5 (Gadget [2]). For f0; 1g, and a constraint family F : an -gadget (or \gadget with performance ") reducing f to F is a set of variables Y collection of real weights associated constraints C j from F over primary variables and auxiliary variables Y with the property that, for boolean assignments ~a to X the following are satised: The gadget is strict if, in addition, We use the shorthand w) to denote the gadget described above. It is straightforward to verify that the introductory example yields a strict 7- gadget reducing the constraint function to the family 2SAT. 6 L. TREVISAN, G. B. SORKIN, M. SUDAN, AND D. P. WILLIAMSON Observe that an w) can be converted into an 0 > gadget by \rescaling", i.e., multiplying every entry of the weight vector ~ w by strictness is not preserved). This indicates that a \strong" gadget is one with a small ; in the extreme, a 1-gadget would be the \optimal" gadget. This intuition will be conrmed in the role played by gadgets in the construction of reductions. Before describing this, we rst list the constraints and constraint families that are of interest to us. For convenience we now give a comprehensive list of all the constraints and constraint families used in this paper. Definition 2.6. Parity check (PC) is the constraint family fPC f0; 1g, PC i is dened as follows: Henceforth we will simply use terms such as MAX PC to denote the optimization problem MAX F where (referred to as MAX 3LIN in [9]) is the source of all our inapproximability results. For any k 1, Exactly-k-SAT (EkSAT) is the constraint family ff : that is, the set of k-ary disjunctive constraints. For any k 1, kSAT is the constraint family SAT is the constraint family l1 ElSAT. The problems MAX 3SAT, MAX 2SAT, and MAX SAT are by now classical optimization problems. They were considered originally in [6]; subsequently their central role in approximation was highlighted in [15]; and recently, novel approximation algorithms were developed in [7, 8, 5]. The associated families are typically the targets of gadget constructions in this paper. Shortly, we will describe a lemma which connects the inapproximability of MAX F to the existence of gadgets reducing PC 0 and PC 1 to F . This method has so far yielded in several cases tight, and in other cases the best known, inapproximability results for MAX F problems. In addition to 3SAT's use as a target, its members are also used as sources; gadgets reducing members of MAX 3SAT to MAX 2SAT help give an improved MAX 3SAT approximation algorithm. 3-Conjunctive SAT (3ConjSAT) is the constraint family ff 000 where: 1. f 000 (a; b; c) 2. f 001 (a; b; c) 3. f 011 (a; b; c) 4. f 111 (a; b; c) Members of 3ConjSAT are sources in gadgets reducing them to 2SAT. These gadgets enable a better approximation algorithm for the MAX 3ConjSAT problem, which in turn sheds light on the the class PCP c;s [log; 3]. is the constraint function given by CUT(a; a b. CUT/0 is the family of constraints fCUT;Tg, where CUT/1 is the family of constraints fCUT;Fg, where GADGETS, APPROXIMATION, AND LINEAR PROGRAMMING 7 MAX CUT is again a classical optimization problem. It has attracted attention due to the recent result of Goemans and Williamson [8] providing a .878-approximation algorithm. An observation from Bellare et al. [2] shows that the approximability of MAX CUT/0, MAX CUT/1, and MAX CUT are all identical; this is also formalized in Proposition 4.1 below. Hence MAX CUT/0 becomes the target of gadget constructions in this paper, allowing us to get inapproximability results for these three problems. is the constraint function given by DICUT(a; MAX DICUT is another optimization problem to which the algorithmic results of [8, 5] apply. Gadgets whose target is DICUT will enable us to get inapproximability results for MAX DICUT. 2CSP is the constraint family consisting of all binary functions, i.e. MAX 2CSP was considered in [5], which gives a .859-approximation algorithm; here we provide inapproximability results. Respect of monomial basis check (RMBC) is the constraint family may be thought of as the test (c; d)[a] ? as the test as the test (:c; d)[a] ? as the test b, where the notation (v refers to the i 1'st coordinate of the vector (v Our original interest in RMBC came from the work of Bellare et al. [2] which derived hardness results for MAX F using gadgets reducing every constraint function in PC and RMBC to F . This work has been eectively superseded by Hastad's [9] which only requires gadgets reducing members of PC to F . However we retain some of the discussion regarding gadgets with RMBC functions as a source, since these constructions were signicantly more challenging, and some of the techniques applied to overcome the challenges may be applicable in other gadget constructions. A summary of all the gadgets we found, with their performances and lower bounds, is given in Table 1. We now put forth a theorem, essentially from [2] (and obtainable as a generalization of its Lemmas 4.7 and 4.15), that relates the existence of gadgets with F as target, to the hardness of approximating MAX F . Since we will not be using this theorem, except as a motivation for studying the family RMBC, we do not prove it here. Theorem 2.7. For any family F , if there exists an 1 -gadget reducing every function in PC to F and an 2 -gadget reducing every function in RMBC to F , then for any > 0, MAX F is hard to approximate to within 1 :15 . In this paper we will use the following, stronger, result by Hastad. Theorem 2.8. [9] For any family F , if there exists an 0 -gadget reducing PC 0 to F and an 1 -gadget reducing PC 1 to F , then for any > 0, MAX F is hard to approximate to within 1 1 8 L. TREVISAN, G. B. SORKIN, M. SUDAN, AND D. P. WILLIAMSON source previous our lower bound Table All gadgets described are provably optimal, and strict. The sole exception (y) is the best possible strict gadget; there is a non-strict 3-gadget. All \previous" results quoted are interpretations of the results in [2], except the gadget reducing 3SAT to 2SAT, which is due to [6], and the gadget reducing PC to 3SAT, which is folklore. Thus, using CUT=0, DICUT, 2CSP, EkSAT and kSAT as the target of gadget constructions from PC 0 and PC 1 , we can show the hardness of MAX CUT, MAX DICUT, MAX 2CSP, MAX EkSAT and MAX kSAT respectively. Furthermore, minimizing the value of in the gadgets gives better hardness results. 3. The Basic Procedure. The key aspect of making the gadget search spaces nite is to limit the number of auxiliary variables, by showing that duplicates (in a sense to be claried) can be eliminated by means of proper substitutions. In general, this is possible if the target of the reduction is a \hereditary" family as dened below. Definition 3.1. A constraint family F is hereditary if for any f 2 F of arity k, and any two indices [k], the function f when restricted to X i X j and considered as a function of k 1 variables, is identical (up to the order of the arguments) to some other function f 0 2 F [f0; 1g (where 0 and 1 denote the constant functions). Definition 3.2. A family F is complementation-closed if it is hereditary and, for any f 2 F of arity k, and any index i 2 [k], the function f 0 given by contained in F . Definition 3.3 (Partial Gadget). For and a constraint family F : an S-partial -gadget (or \S-partial gadget with performance ") reducing f to F is a nite collection of constraints Cm from F over primary variables and nitely many aux- GADGETS, APPROXIMATION, AND LINEAR PROGRAMMING 9 iliary variables Y a collection of non-negative real weights w with the property that, for boolean assignments ~a to X the following are satised: We use the shorthand w) to denote the partial gadget. The following proposition follows immediately from the denitions of a gadget and a partial gadget. Proposition 3.4. For a constraint function . Then for every family 1. An S 1 -partial -gadget reducing f to F is an -gadget reducing f to F . 2. An S 2 -partial -gadget reducing f to F is a strict -gadget reducing f to F . Definition 3.5. For 1 and S f0; 1g k , w) be an S-partial -gadget reducing a constraint f : f0; 1g k ! f0; 1g to a constraint family F . We say that the function b is a witness for the partial gadget, witnessing the set S, if b(~a) satises equations (3.2) and (3.4). Specically: The witness function can also be represented as an jSj rows are the vectors (~a; b(~a)). Notice that the columns of the matrix correspond to the variables of the gadget, with the rst k columns corresponding to primary variables, and the last n corresponding to auxiliary variables. In what follows we shall often prefer the matrix notation. Definition 3.6. For a set S f0; 1g k let MS be the matrix whose rows are the vectors ~a 2 S, let k 0 S be the number of distinct columns in MS , and let k 00 S be the number of columns in MS distinct up to complementation. Given a constraint f of arity k and a hereditary constraint family F that is not complementation-closed, an (S; f; F)-canonical witness matrix (for an S-partial gadget reducing f to F) is the jSj whose rst k columns correspond to the k primary variables and whose remaining columns are all possible column vectors that L. TREVISAN, G. B. SORKIN, M. SUDAN, AND D. P. WILLIAMSON are distinct from one another and from the columns corresponding to the primary variables. If F is complementation-closed, then a canonical witness matrix is the whose rst k columns correspond to the k primary variables and whose remaining columns are all possible column vectors that are distinct up to complementation from one another and from the columns corresponding to the primary variables. The following lemma is the crux of this paper and establishes that the optimal gadget reducing a constraint function f to a hereditary family F is nite. To motivate the lemma, we rst present an example, due to Karlo and Zwick [11], showing that this need not hold if the family F is not hereditary. Their counterexample has g. Using k auxiliary variables, Y may construct a gadget for the constraint X , using the constraints X Y with each constraint having the same weight. For an appropriate choice of this weight it may be veried that this yields a (2 2=k)-gadget for even k; thus the performance tends to 2 in the limit. On the other hand it can be shown that any gadget with k auxiliary variables has performance at most 2 thus no nite gadget achieves the limit. It is clear that for this example the lack of hereditariness is critical: any hereditary family containing PC 1 would also contain f , providing a trivial 1-gadget. To see why the hereditary property helps in general, consider an -gadget reducing f to F , and let W be a witness matrix for . Suppose two columns of W , corresponding to auxiliary variables Y 1 and Y 2 of , are identical. Then we claim that does not really need the variable Y 2 . In every constraint containing Y 2 , replace it with Y 1 , to yield a new collection of weighted constraints. By the hereditary property of F , all the resulting constraints are from F . And, the resulting instance satises all the properties of an -gadget. (The universal properties follow trivially, while the existential properties follow from the fact that in the witness matrix Y 1 and Y 2 have the same assignment.) Thus this collection of constraints forms a gadget with fewer variables and performance at least as good. The niteness follows from the fact a witness matrix with distinct columns has a bounded number of columns. The following lemma formalizes this argument. In addition it also describes the canonical witness matrix for an optimal gadget | something that will be of use later. Lemma 3.7. For 1, set S f0; 1g k , constraint hereditary constraint family F, if there exists an S-partial -gadget reducing f to F , with witness matrix W , then for any (S; f; F)-canonical witness matrix W 0 , and some 0 , there exists an 0 -gadget 0 reducing f to F , with W 0 as a witness matrix. Proof. We rst consider the case where F is not complementation-closed. Let w) be an S-partial -gadget reducing f to F and let W be a witness matrix for . We create a gadget 0 with auxiliary variables Y 0 one associated with each column of the matrix W 0 other than the rst k. With each variable Y i of we associate a variable Z such that the column corresponding to Y i in W is the same as the column corresponding to Z in W 0 . Notice that Z may be one of the primary variables or one of the auxiliary variables By denition of a canonical witness, such a column and hence variable Z does exist. Now for every constraint C j on variables Y i 1 in with weight w j , we introduce the constraint C j on variables Y 0 in 0 with weight w j where Y 0 is the variable associated with Y i l . Notice that in this process the variables involved GADGETS, APPROXIMATION, AND LINEAR PROGRAMMING 11 with a constraint do not necessarily remain distinct. This is where the hereditary property of F is used to ensure that a constraint C j 2 F , when applied to a tuple of non-distinct variables, remains a constraint in F . In the process we may arrive at some constraints which are either always satised or never satised. For the time being, we assume that the constraints 0 and 1 are contained in F , so this occurrence does not cause a problem. Later we show how this assumption is removed. This completes the description of 0 . To verify that 0 is indeed an S-partial -gadget, we notice that the universal constraints (conditions (3.1) and (3.3) in Definition are trivially satised, since 0 is obtained from by renaming some variables and possibly identifying some others. To see that the existential constraints (conditions (3.2) and (3.4) in Denition 3.3) are satised, notice that the assignments to the variables ~ Y that witness these conditions in are allowable assignments to the corresponding variables in ~ Y 0 and in fact this is what dictated our association of variables in ~ Y to the variables in ~ Y 0 . Thus 0 is indeed an S-partial -gadget reducing f to F , and, by construction, has W 0 as a witness matrix. Last, we remove the assumption that 0 must include constraints 0 and 1. Any constraints 0 can be safely thrown out of the gadget without changing any of the pa- rameters, since such constraints are never satised. On the other hand, constraints 1 do aect . If we throw away a 1 constraint of weight w j , this reduces the total weight of satised clauses in every assignment by w j . Throwing away all such constraints reduces by the total weight of the 1 constraints, producing a gadget of (improved) performance 0 . Finally, we describe the modications required to handle the case where F is complementation-closed (in which case the denition of a canonical witness changes). Here, for each variable Y i and its associated column of W , either there is an equal column in W 0 , in which case we replace Y i with the column's associated variable or there is a complementary column in W 0 , in which case we replace Y i with the negation of the column's associated variable, :Y 0 The rest of the construction proceeds as above, and the proof of correctness is the same. It is an immediate consequence of Lemma 3.7 that an optimum gadget reducing a constraint function to a hereditary family does not need to use more than an explicitly bounded number of auxiliary variable. Corollary 3.8. Let f be a constraint function of arity k with s satisfying assignments. Let F be a constraint family and 1 be such that there exists an -gadget reducing f to F . 1. If F is hereditary then there exists an 0 -gadget with at most 2 s k 0 auxiliary variables reducing f to F , where 0 , and k 0 is the number of distinct variables among the satisfying assignments of f . 2. If F is complementation-closed then there exists an 0 -gadget with at most auxiliary variables reducing f to F , for some 0 , where k 00 is the number of distinct variables, up to complementation, among the satisfying assignments of f . Corollary 3.9. Let f be a constraint function of arity k. Let F be a constraint family and 1 be such that there exists a strict -gadget reducing f to F . 1. If F is hereditary then there exists a strict 0 -gadget with at most 2 2 k auxiliary variables reducing f to F , for some 0 . 2. If F is complementation-closed then there exists a strict 0 -gadget with at most auxiliary variables reducing f to F , for some 0 . L. TREVISAN, G. B. SORKIN, M. SUDAN, AND D. P. WILLIAMSON We will now show how to cast the search for an optimum gadget as a linear programming problem. Definition 3.10. For a constraint function f of arity k, constraint family F , M) is a linear program dened as follows: Cm be all the possible distinct constraints that arise from applying a constraint function from F to a set of n variables. Thus for every j, 1g. The LP variables are w corresponds to the weight of the constraint C j . Additionally the LP has one more variable . Let S f0; 1g k and b be such that (i.e., M is the witness matrix corresponding to the witness function b for the set S). The LP inequalities correspond to the denition of an S-partial gadget. Finally the LP has the inequalities w j 0. The objective of the LP is to minimize . Proposition 3.11. For any constraint function f of arity k, constraint family F , and s witnessing the set S f0; 1g k , if there exists an S-partial gadget reducing f to F with witness matrix M , then LP(f; F ; M) nds such a gadget with the minimum possible . Proof. The LP-generated gadget consists of k primary variables corresponding to the rst k columns of M ; n auxiliary variables Y corresponding to the remaining n columns of M ; constraints C as dened in Denition 3.10; and weights w returned by LP(f; F ; M ). By construction the LP solution returns the minimum possible for which an S-partial -gadget reducing f to F with witness M exists. Theorem 3.12 (Main). Let f be a constraint function of arity k with s satisfying assignments. Let k 0 be the number of distinct variables of f and k 00 be the number of distinct variables up to complementation. Let F be a hereditary constraint family with functions of arity at most l. Then: If there exists an -gadget reducing f to F , then there exists such a gadget with at most v auxiliary variables, where closed and If there exists a strict -gadget reducing f to F then there exists such a gadget with at most v auxiliary variables, where complementation-closed and Furthermore such a gadget with smallest performance can be found by solving a linear program with at most jF j (v variables and 2 v+k constraints. GADGETS, APPROXIMATION, AND LINEAR PROGRAMMING 13 Remark: The sizes given above are upper bounds. In specic instances, the sizes may be much smaller. In particular, if the constraints of F exhibit symmetries, or are not all of the same arity, then the number of variables of the linear program will be much smaller. Proof. By Proposition 3.11 and Lemma 3.7, we have that LP(f; F ; WS ) yields an optimal S-partial gadget if one exists. By Proposition 3.4 the setting gadget, and the setting strict gadget. Corollaries 3.8 and 3.9 give the required bound on the number of auxiliary variables; and the size of the LP then follows from the denition. To conclude this section, we mention some (obvious) facts that become relevant when searching for large gadgets. First, if S 0 S, then the performance of an S 0 - partial gadget reducing f to F is also a lower bound on the performance of an S-partial gadget reducing f to F . The advantage here is that the search for an S 0 -partial gadget may be much faster. Similarly, to get upper bounds on the performance of an S-partial gadget, one may use other witness matrices for S (rather than the canonical one); in particular ones with (many) fewer columns. This corresponds to making a choice of auxiliary variables not to be used in such a gadget. 4. Improved Negative Results. 4.1. MAX CUT. We begin by showing an improved hardness result for the MAX CUT problem. It is not di-cult to see that no gadget per Denition 2.5 can reduce any member of PC to CUT: for any setting of the variables which satises equation (2.2), the complementary setting has the opposite parity (so that it must be subject to inequality (2.3)), but the values of all the CUT constraints are unchanged, so that the gadget's value is still , violating (2.3). Following [2], we use instead the fact that MAX CUT and MAX CUT/0 are equivalent with respect to approximation as shown below. Proposition 4.1. MAX CUT is equivalent to MAX CUT/0. Specically, given an instance I of either problem, we can create an an instance I 0 of the other with the same optimum and with the feature that an assignment satisfying constraints of total weight W to the latter can be transformed into an assignment satisfying constraints of the same total weight in I. Proof. The reduction from MAX CUT to MAX CUT/0 is trivial, since the family CUT/0 contains CUT; and thus the identity map provides the required reduction. In the reverse direction, given an instance ( ~ w) of MAX CUT/0 with n variables and m clauses, we create an instance ( ~ w) of MAX CUT with variables and m clauses. The variables are simply the variables ~ X with one additional variable called 0. The constraints of ~ C are transformed as follows. If the constraint is a CUT constraint on variables X i and X j it is retained as is. If the constraint is replaced with the constraint CUT(X Given a assignment ~a to the vector ~ notice that its complement also satises the same number of constraints in I 0 . We pick the one among the two that sets the variable 0 to 0, and then observe that the induced assignment to ~ X satises the corresponding clauses of I. Thus we can look for reductions to CUT/0. Notice that the CUT=0 constraint family is hereditary, since identifying the two variables in a CUT constraint yields the constant function 0. Thus by Theorem 3.12, if there is an -gadget reducing PC 0 to CUT=0, then there is an -gadget with at most 13 auxiliary variables (16 variables in all). Only are possible on 16 variables. Since we only 14 L. TREVISAN, G. B. SORKIN, M. SUDAN, AND D. P. WILLIAMSONx 1 Fig. 4.1. 8-gadget reducing PC 0 to CUT. Every edge has weight .5. The auxiliary variable which is always 0 is labelled 0. need to consider the cases when Y 0, we can construct a linear program as above with constraints to nd the optimal -gadget reducing PC 0 to CUT=0. A linear program of the same size can similarly be constructed to nd a gadget reducing PC 1 to CUT=0. Lemma 4.2. There exists an 8-gadget reducing PC 0 to CUT=0, and it is optimal and strict. We show the resulting gadget in Figure 4.1 as a graph. The primary variables are labelled is a special variable. The unlabelled vertices are auxiliary variables. Each constraint of non-zero weight is shown as an edge. An edge between the vertex 0 and some vertex x corresponds to the constraint T (x). Any other edge between x and y represents the constraint CUT(x; y). Note that some of the 13 possible auxiliary variables do not appear in any positive weight constraint and thus are omitted from the graph. All non-zero weight constraints have weight .5. By the same methodology, we can prove the following. Lemma 4.3. There exists a 9-gadget reducing PC 1 to CUT=0, and it is optimal and strict. The gadget is similar to the previous one, but the old vertex 0 is renamed Z, and a new vertex labelled 0 is joined to Z by an edge of weight 1. The two lemmas along with Proposition 4.1 above imply the following theorem. Theorem 4.4. For every > 0, MAX CUT is hard to approximate to within Proof. Combining Theorem 2.8 with Lemmas 4.2 and 4.3 we nd that MAX CUT/0 is hard to approximate to within 16=17 . The theorem then follows from Proposition 4.1. gadgets. Finding RMBC gadgets was more di-cult. We discuss this point since it leads to ideas that can be applied in general when nding large gad- gets. Indeed, it turned out that we couldn't exactly apply the technique above to nd an optimal gadget reducing, say, RMBC 00 to CUT=0. (Recall that the is the function (a 3 ; a 4 .) Since there are 8 satisfying assignments to the 4 variables of the RMBC 00 constraint, by Theorem 3.12, we would need to consider 2 auxiliary variables, leading to a linear program with which is somewhat beyond the capacity of current computing GADGETS, APPROXIMATION, AND LINEAR PROGRAMMING 15 machines. To overcome this di-culty, we observed that for the RMBC 00 function, the value of a 4 is irrelevant when a and the value of a 3 is irrelevant when a led us to try only restricted witness functions for which ~ b(0; a 2 ; a 3 ; and ~ b(1; a (dropping from the witness matrix columns violating the above conditions), even though it is not evident a priori that a gadget with a witness function of this form exists. The number of distinct variable columns that such a witness matrix can have is at most 16. Excluding auxiliary variables identical to a 1 or a 2 , we considered gadgets with at most 14 auxiliary variables. We then created a linear program with constraints. The result of the linear program was that there exists an 8-gadget with constant 0 reducing RMBC 00 to CUT, and that it is strict. Since we used a restricted witness function, the linear program does not prove that this gadget is optimal. However, lower bounds can be established through construction of optimal S- partial gadgets. If S is a subset of the set of satisfying assignments of RMBC 00 , then its dening equalities and inequalities (see Denition 3.3) are a subset of those for a gadget, and thus the performance of the partial gadget is a lower bound for that of a true gadget. In fact, we have always been lucky with the latter technique, in that some choice of the set S has always yielded a lower bound and a matching gadget. In particular, for reductions from RMBC to CUT, we have the following result. Theorem 4.5. There is an 8-gadget reducing RMBC 00 to CUT=0, and it is optimal and strict; there is an 8-gadget reducing RMBC 01 to CUT=0, and it is optimal and strict; there is a 9-gadget reducing RMBC 10 to CUT=0, and it is optimal and strict; and there is a 9-gadget reducing RMBC 11 to CUT=0, and it is optimal and strict. Proof. In each case, for some set S of satisfying assignments, an optimal S- partial gadget also happens to be a true gadget, and strict. In the same notation as in Denition 2.6, the appropriate sets S of 4-tuples (a; b; c; d) are: for RMBC 00 , 4.2. MAX DICUT. As in the previous subsection, we observe that if there exists an -gadget reducing an element of PC to DICUT, there exists an -gadget with auxiliary variables. This leads to linear programs with 1615 variables (one for each possible DICUT constraint, corresponding to a directed edge) and 2 linear constraints. The solution to the linear programs gives the following. Lemma 4.6. There exist 6:5-gadgets reducing PC 0 and PC 1 to DICUT, and they are optimal and strict. The PC 0 gadget is shown in Figure 4.2. Again x 1 , x 2 and x 3 refer to the primary variable and an edge from x to y represents the constraint :x^b. The PC 1 gadget is similar, but has all edges reversed. Theorem 4.7. For every > 0, MAX DICUT is hard to approximate to within gadgets. As with the reductions to CUT=0, reductions from the RMBC family members to DICUT can be done by constructing optimal S-partial gadgets, and again (with fortuitous choices of S) these turn out to be true gadgets, and strict. Theorem 4.8. There is a 6-gadget reducing RMBC 00 to DICUT, and it is optimal and strict; there is a 6.5-gadget reducing RMBC 01 to DICUT, and it is optimal and strict; there is a 6.5-gadget reducing RMBC 10 to DICUT, and it is optimal and L. TREVISAN, G. B. SORKIN, M. SUDAN, AND D. P. WILLIAMSON3 Fig. 4.2. 8-gadget reducing PC 0 to DICUT. Edges have weight 1 except when marked otherwise. strict; and there is a 7-gadget reducing RMBC 11 to DICUT, and it is optimal and strict. Proof. Using, case by case, the same sets S as in the proof of Theorem 4.5, again yields in each case an optimal S-partial gadget that also happens to be a true, strict gadget. 4.3. MAX 2-CSP. For reducing an element of PC to the 2CSP family we need consider only 4 auxiliary variables, for a total of 7 variables. There are two non-constant functions on a single variable, and twelve non-constant functions on pairs of variables, so that there are 2 7 functions to consider overall. We can again set up a linear program with a variable per function and 2 7 linear constraints. We obtain the following. Lemma 4.9. There exist 5-gadgets reducing PC 0 and PC 1 to 2CSP, and they are optimal and strict. The gadget reducing PC 0 to 2CSP is the following: The gadget reducing PC 1 to 2CSP can be obtained from this one by complementing all the occurrences of X 1 . Theorem 4.10. For every > 0, MAX 2CSP is hard to approximate to within MAX 2CSP can be approximated within :859 [5]. The above theorem has implications for probabilistically checkable proofs. Reversing the well-known reduction from constraint satisfaction problems to probabilistically checkable proofs (cf. [1]) 3 , Theorem 4.10 yields the following theorem. Theorem 4.11. For any > 0, constants c and s exist such that NP c;s [log; 2] and c=s > 10=9 . The previously known gap between the completeness and soundness achievable reading two bits was 74=73 [2]. It would be 22=21 using Hastad's result [9] in combination 3 The reverse connection is by now a folklore result and may be proved along the lines of [2, Proposition 10.3, Part (3)]. GADGETS, APPROXIMATION, AND LINEAR PROGRAMMING 17 with the argument of [2]. Actually the reduction from constraint satisfaction problems to probabilistically checkable proofs is reversible, and this will be important in Section 7. RMBC gadgets. Theorem 4.12. For each element of RMBC, there is a 5-gadget reducing it to 2CSP, and it is optimal and strict. Proof. Using the same selected assignments as in Theorems 4.5 and 4.8 again yields lower bounds and matching strict gadgets. 5. Interlude: Methodology. Despite their seeming variety, all the gadgets in this paper were computed using a single program (in the language APL2) to generate an LP, and call upon OSL (the IBM Optimization Subroutine Library) to solve it. This \gadget-generating" program takes several parameters. The source function f is specied explicitly, by a small program that computes f . The target family F is described by a single function, implemented as a small program, applied to all possible clauses of specied lengths and symmetries. The symmetries are chosen from among: whether clauses are unordered or ordered; whether their variables may be complemented; and whether they may include the constants 0 or 1. For example, a reduction to MAX CUT=0 would take as F the function x 1 x 2 , applied over unordered binomial clauses, in which complementation is not allowed but the constant 0 is allowed. This means of describing F is relatively intuitive and has never restricted us, even though it is not completely general. Finally, we specify an arbitrary set S of selected assignments, which allows us to search for S-partial gadgets (recall Denition 3.3). From equations (3.2) and (3.4), each selected assignment ~a generates a constraint that Selecting all satisfying assignments of f reproduces the set of constraints (2.2) for an -gadget, while selecting all assignments reproduces the set of constraints (2.2) and (2.4) for a strict -gadget. Selected assignments are specied explicitly; by default, to produce an ordinary gadget, they are the satisfying assignments of f . The canonical witness for the selected set of assignments is generated by our program as governed by Denition 3.6. Notice that the denition of the witness depends on whether F is complementation-closed or not, and this is determined by the explicitly specied symmetries. To facilitate the generation of restricted witness matrices, we have also made use of a \don't-care" state (in lieu of 0 or 1) to reduce the number of selected assign- ments. For example in reductions from RMBC 00 we have used selected assignments of (00 1). The various LP constraints must be satised for both values of any don't-care, while the witness function must not depend on the don't-care values. So in this example, use of a don't-care reduces the number of selected assignments from 8 to 4, reduces the number of auxiliary variables from about 2 8 to 2 4 (ignoring duplications of the 4 primary variables, or any symmetries), and reduces the number of constraints in the LP from 2 2 8 (a more reasonable 65,536). Use of don't-cares provides a technique complementary to selecting a subset of all satisfying assignments, in that if the LP is feasible it provides an upper bound and a gadget, but the gadget may not be optimal. In practice, selecting a subset of satisfying assignments has been by far the more useful of the two techniques; so far we have always been able to choose a subset which produces a lower bound and a gadget to match. L. TREVISAN, G. B. SORKIN, M. SUDAN, AND D. P. WILLIAMSON After constructing and solving an LP, the gadget-generating program uses brute force to make an independent verication of the gadget's validity, performance, and strictness. The hardest computations were those for gadgets reducing from RMBC; on an IBM Risc System/6000 model 43P-240 workstation, running at 233MHz, these took up to half an hour and used 500MB or so of memory. However, the strength of [9] makes PC virtually the sole source function of contemporary interest, and all the reductions from PC are easy; they use very little memory, and run in seconds on an ordinary 233MHz Pentium processor. 6. Improved Positive Results. In this section we show that we can use gadgets to improve approximation algorithms. In particular, we look at MAX 3SAT, and a variation, MAX 3ConjSAT, in which each clause is a conjunction (rather than a disjunction) of three literals. An improved approximation algorithm for the latter problem leads to improved results for probabilistically checkable proofs in which the verier examines only 3 bits. Both of the improved approximation algorithms rely on strict gadgets reducing the problem to MAX 2SAT. We begin with some notation. Definition 6.1. A )-approximation algorithm for MAX 2SAT is an algorithm which receives as input an instance with unary clauses of total weight m 1 and binary clauses of total weight m 2 , and two reals produces reals s 1 u 1 and s 2 u 2 and an assignment satisfying clauses of total weight at least . If there exists an optimum solution that satises unary clauses of weight no more than u 1 and binary clauses of weight no more than u 2 , then there is a guarantee that no assignment satises clauses of total weight more than s 1 +s 2 . That is, supplied with a pair of \upper bounds" )-approximation algorithm produces a single upper bound of s 1 +s 2 , along with an assignment respecting a lower bound of 1 Lemma 6.2. [5] There exists a polynomial-time (:976; :931)-approximation algorithm for MAX 2SAT. 6.1. MAX 3SAT. In this section we show how to derive an improved approximation algorithm for MAX 3SAT. By restricting techniques in [8] from MAX SAT to MAX 3SAT and using a :931-approximation algorithm for MAX 2SAT due to Feige and Goemans [5], one can obtain a :7704-approximation algorithm for MAX 3SAT. The basic idea of [8] is to reduce each clause of length 3 to the three possible subclauses of length 2, give each new length-2 clause one-third the original weight, and then apply an approximation algorithm for MAX 2SAT. This approximation algorithm is then \balanced" with another approximation algorithm for MAX 3SAT to obtain the result. Here we show that by using a strict gadget to reduce 3SAT to MAX 2SAT, a good )-approximation algorithm for MAX 2SAT leads to a :801- approximation algorithm for MAX 3SAT. Lemma 6.3. If for every f 2 E3SAT there exists a strict -gadget reducing f to 2SAT, there exists a )-approximation algorithm for MAX 2SAT, and there exists a -approximation algorithm for MAX 3SAT with Proof. Let be an instance of MAX 3SAT with length-1 clauses of total weight clauses of total weight m 2 , and length-3 clauses of total weight m 3 . GADGETS, APPROXIMATION, AND LINEAR PROGRAMMING 19 We use the two algorithms listed below, getting the corresponding upper and lower bounds on number of satisable clauses: Random: We set each variable to 1 with probability 1=2. This gives a solution of weight at least m 1 Semidenite programming: We use the strict -gadget to reduce every length- 3 clause to length-2 clauses. This gives an instance of MAX 2SAT. We apply the )-approximation algorithm with parameters to nd an approximate solution to this problem. The approximation algorithm gives an upper bound s 1 on the weight of any solution to the MAX 2SAT instance and an assignment of weight 1 translated back to the MAX 3SAT instance, the assignment has weight at least and the maximum weight satisable in the MAX 3SAT instance is at most The performance guarantee of the algorithm which takes the better of the two solutions is at least We now dene a sequence of simplications which will help prove the bound. To nish the proof of the lemma, we claim that To see this, notice that the rst inequality follows from the substitution of variables . The second follows from the fact that setting m 1 to t 1 only reduces the numerator. The third inequality follows from setting . The fourth is obtained by substituting a convex combination of the arguments instead of max and then simplifying. The convex combination takes a 1 fraction of the rst argument, 2 of the second and 3 of the third, where L. TREVISAN, G. B. SORKIN, M. SUDAN, AND D. P. WILLIAMSON 3: Observe that 1 and that the condition on guarantees that 2 0. Remark 6.4. The analysis given in the proof of the above lemma is tight. In particular for an instance with m clauses such that is easy to see that The following lemma gives the strict gadget reducing functions in E3SAT to 2SAT. Notice that nding strict gadgets is almost as forbidding as nding gadgets for RMBC, since there are 8 existential constraints in the specication of a gadget. This time we relied instead on luck. We looked for an S-partial gadget for the set found an S-partial 3:5-gadget that turned out to be a gadget! Our choice of S was made judiciously, but we could have aorded to run through all 8 sets S of size 4 in the hope that one would work. Lemma 6.5. For every function f 2 E3SAT, there exists a strict (and optimal) 3:5-gadget reducing f to 2SAT. Proof. Since 2SAT is complementation-closed, it is su-cient to present a 3:5- gadget reducing (X 1 _X 2 _X 3 ) to 2SAT. The gadget is every clause except the last has weight 1=2, and the last clause has weight 1. Combining Lemmas 6.2, 6.3 and 6.5 we get a :801-approximation algorithm. Theorem 6.6. MAX 3SAT has a polynomial-time :801-approximation algorithm 6.2. MAX 3-CONJ SAT. We now turn to the MAX 3ConjSAT problem. The analysis is similar to that of Lemma 6.3. Lemma 6.7. If for every f 2 3ConjSAT there exists a strict reducing f to 2SAT composed of 1 length-1 clauses and 2 length-2 clauses, and there exists a )-approximation algorithm for MAX 2SAT, then there exists a -approximation algorithm for MAX 3ConjSAT with =8 Proof. Let be an instance of MAX 3ConjSAT with constraints of total weight m. As in the MAX 3SAT case, we use two algorithms and take the better of the two solutions: Random: We set every variable to 1 with probability half. The total weight of satised constraints is at least m=8. GADGETS, APPROXIMATION, AND LINEAR PROGRAMMING 21 Semidenite programming: We use the strict -gadget to reduce any constraint to 2SAT clauses. This gives an instance of MAX 2SAT and we use the )-approximation algorithm with parameters The algorithm returns an upper bound s 1 on the total weight of satisable constraints in the MAX 2SAT instance, and an assignment of measure at least 1 translated back to the MAX 3ConjSAT instance, the measure of the assignment is at least 1 thermore, s 1 1 m, s 2 2 m, and the total weight of satisable constraints in the MAX 3ConjSAT instance is at most s Thus we get that the performance ratio of the algorithm which takes the better of the two solutions above is at least We now dene a sequence of simplications which will help prove the bound. tmt In order to prove the lemma, we claim that To see this, observe that the rst inequality follows from the substitution of variables 1)m. The second follows from setting The third inequality follows from the fact that setting t 2 to (1 1 )m only reduces the numerator. The fourth is obtained by substituting a convex combination of the arguments instead of max and then simplifying. The following gadget was found by looking for an S-partial gadget for 011g. Lemma 6.8. For any f 2 3ConjSAT there exists a strict (and optimal) 4-gadget reducing f to 2SAT. The gadget is composed of one length-1 clause and three length-2 clauses. Proof. Recall that 2SAT is complementation-closed, and thus it is su-cient to exhibit a gadget reducing f(a 1 ; a 2 ; a 3 to 2SAT. Such gadget is Y , clauses have weight 1. The variables are primary variables and Y is an auxiliary variable. 22 L. TREVISAN, G. B. SORKIN, M. SUDAN, AND D. P. WILLIAMSON Theorem 6.9. MAX 3ConjSAT has a polynomial-time .367-approximation algorithm It is shown by Trevisan [16, Theorem 18] that the above theorem has consequences for PCP c;s [log; 3]. This is because the computation of the verier in such a proof system can be described by a decision tree of depth 3, for every choice of random string. Further, there is a 1-gadget reducing every function which can be computed by a decision tree of depth k to kConjSAT. Corollary 6.10. PCP c;s [log; 3] P provided that c=s > 2:7214. The previous best trade-o between completeness and soundness for polynomial-time PCP classes was c=s > 4 [16]. 7. Lower Bounds for Gadget Constructions. In this section we shall show that some of the gadget constructions mentioned in this paper and in [2] are optimal, and we shall prove lower bounds for some other gadget constructions. The following result is useful to prove lower bounds for the RMBC family. Lemma 7.1. If there exists an -gadget reducing an element of RMBC to a complementation-closed constraint family F , then there exists an -gadget reducing all elements of PC to F . Proof. If a family F is complementation-closed, then an -gadget reducing an element of PC (respectively RMBC) to F can be modied (using complementations) to yield -gadgets reducing all elements of PC (respectively RMBC) to F . For this reason, we will restrict our analysis to PC 0 and RMBC 00 gadgets. Note that, for any be an gadget over primary variables x auxiliary variables y reducing RMBC to 2SAT. Let 0 be the gadget obtained from by imposing x 4 x it is immediate to verify that 0 is an -gadget reducing PC 0 to F . 7.1. Reducing PC and RMBC to 2SAT. Theorem 7.2. If is an -gadget reducing an element of PC to 2SAT, then 11. Proof. It su-ces to consider PC 0 . We prove that the optimum of (LP1) is at least 11. To this end, consider the dual program of (LP1). We have a variable y ~a; ~ b for any ~a 2 f0; 1g 3 and any ~ b 2 f0; 1g 4 , plus additional variables ^ y ~a; ~ b opt (~a) for any opt is the \optimal" witness function dened in Section 3. The formulation is maximize subject to P y ~a; ~ b opt (~a) (DUAL1) There exists a feasible solution for (DUAL1) whose cost is 11. Corollary 7.3. If is an -gadget reducing an element of RMBC to 2SAT, then 11. 7.2. Reducing PC and RMBC to SAT. Theorem 7.4. If is an -gadget reducing an element of PC to SAT, then 4. GADGETS, APPROXIMATION, AND LINEAR PROGRAMMING 23 Proof. As in the proof of Theorem 7.2 we give a feasible solution to the dual to obtain the lower bound. The linear program that nds the best gadget reducing PC 0 to SAT is similar to (LP1), the only dierence being that a larger number N of clauses are considered, namely, . The dual program is then maximize subject to P y ~a; ~ b opt (~a) (DUAL2) Consider now the following assignment of values to the variables of (DUAL2) (the unspecied values have to be set to zero): 1where d is the Hamming distance between binary sequences. It is possible to show that this is a feasible solution for (DUAL2) and it is immediate to verify that its cost is 4. Corollary 7.5. If is an -gadget reducing an element RMBC to SAT, then 7.3. Reducing kSAT to lSAT. Let k and l be any integers k > l 3. The standard reduction from EkSAT to lSAT can be seen as a d(k 2)=(l 2)e-gadget. In this section we shall show that this is asymptotically the best possible. Note that since lSAT is complementation-closed we can restrict ourselves to considering just one constraint function of EkSAT, say f(a Theorem 7.6. For any k > l > 2, if is an -gadget reducing f to lSAT then k=l. Proof. We can write a linear program whose optimum gives the smallest such that an -gadget exists reducing f to lSAT. Let b be the witness function used to formulate this linear program. We can assume that b is -ary and we let Also let N be the total number of constraints from lSAT that can be dened over k +K variables. Assume some enumeration C of such constraints. The dual LP is maximize subject to P y ~a; ~ b kSAT lSAT (~a) y ~a; ~ b kSAT lSAT (~a) y ~a; ~ b kSAT lSAT (~a) The witness function ~ b kSAT lSAT is an \optimal" witness function for gadgets reducing kSAT to lSAT. L. TREVISAN, G. B. SORKIN, M. SUDAN, AND D. P. WILLIAMSON Let A k f0; 1g k be the set of binary k-ary strings with exactly one non-zero component (note that jA k be the k-ary string all whose components are equal to 0 (respectively, 1). The following is a feasible solution for (DUAL3) whose cost is k=l. We only specify non-zero values. y ~a; ~ b kSAT lSAT (~a) In view of the above lower bound, a gadget cannot provide an approximation- preserving reduction from MAX SAT to MAX kSAT. More generally, there cannot be an approximation-preserving gadget reduction from MAX SAT to, say, MAX (log n)SAT. In partial contrast with this lower bound, Khanna et al. [13] have given an approximation- preserving reduction from MAX SAT to MAX 3SAT and Crescenzi and Trevisan [4] have provided a tight reduction between MAX SAT and MAX (log n)SAT, showing that the two problems have the same approximation threshold. Acknowledgments . We thank Pierluigi Crescenzi and Oded Goldreich for several helpful suggestions and remarks. We are grateful to John Forrest and David Jensen for their assistance in e-ciently solving large linear programs. We thank Howard Karlo and Uri Zwick for pointing out the error in the earlier version of this paper, and the counterexample to our earlier claim. We thank the anonymous referees for their numerous comments and suggestions leading to the restructuring of Section 3. --R Proof veri Free bits To weight or not to weight: Where is the question? Approximating the value of two prover proof systems Some simpli New 3/4-approximation algorithms for the maximum satis ability problem Improved approximation algorithms for maximum cut and satis Reducibility among combinatorial problems. On syntactic versus computational views of approximability. Approximation algorithms for the maximum satis Parallel approximation algorithms using positive linear programming. On the approximation of maximum satis Approximation algorithms for constraint satisfaction problems involving at most three variables per constraint. --TR --CTR Eran Halperin , Dror Livnat , Uri Zwick, MAX CUT in cubic graphs, Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms, p.506-513, January 06-08, 2002, San Francisco, California Eran Halperin , Dror Livnat , Uri Zwick, MAX CUT in cubic graphs, Journal of Algorithms, v.53 n.2, p.169-185, November 2004 Gunnar Andersson , Lars Engebretsen, Property testers for dense constraint satisfaction programs on finite domains, Random Structures & Algorithms, v.21 n.1, p.14-32, August 2002 Takao Asano , David P. Williamson, Improved approximation algorithms for MAX SAT, Journal of Algorithms, v.42 n.1, p.173-202, January 2002 Manthey, Non-approximability of weighted multiple sequence alignment, Theoretical Computer Science, v.296 n.1, p.179-192, 4 March Don Coppersmith , David Gamarnik , Mohammad Hajiaghayi , Gregory B. Sorkin, Random MAX SAT, random MAX CUT, and their phase transitions, Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms, January 12-14, 2003, Baltimore, Maryland Lane A. Hemaspaandra, SIGACT news complexity theory column 34, ACM SIGACT News, v.32 n.4, December 2001 Philippe Chapdelaine , Nadia Creignou, The Complexity of Boolean Constraint Satisfaction Local Search Problems, Annals of Mathematics and Artificial Intelligence, v.43 n.1-4, p.51-63, January 2005 Alexander D. Scott , Gregory B. Sorkin, Solving Sparse Random Instances of Max Cut and Max 2-CSP in Linear Expected Time, Combinatorics, Probability and Computing, v.15 n.1-2, p.281-315, January 2006 Johan Hstad, Some optimal inapproximability results, Journal of the ACM (JACM), v.48 n.4, p.798-859, July 2001 Amin Coja-Oghlan , Cristopher Moore , Vishal Sanwalani, MAX k-CUT and approximating the chromatic number of random graphs, Random Structures & Algorithms, v.28 n.3, p.289-322, May 2006

reductions;intractability;combinatorial optimization;approximation algorithms;NP-completeness;probabilistic proof systems

357383

An Evaluation of Statistical Approaches to Text Categorization.

This paper focuses on a comparative evaluation of a wide-range of text categorization methods, including previously published results on the Reuters corpus and new results of additional experiments. A controlled study using three classifiers, kNN, LLSF and WORD, was conducted to examine the impact of configuration variations in five versions of Reuters on the observed performance of classifiers. Analysis and empirical evidence suggest that the evaluation results on some versions of Reuters were significantly affected by the inclusion of a large portion of unlabelled documents, mading those results difficult to interpret and leading to considerable confusions in the literature. Using the results evaluated on the other versions of Reuters which exclude the unlabelled documents, the performance of twelve methods are compared directly or indirectly. For indirect compararions, kNN, LLSF and WORD were used as baselines, since they were evaluated on all versions of Reuters that exclude the unlabelled documents. As a global observation, kNN, LLSF and a neural network method had the best performance&semi; except for a Naive Bayes approach, the other learning algorithms also performed relatively well.

Introduction Text categorization is the problem of assigning predefined categories to free text documents. A growing number of statistical learning methods have been applied to this problem in recent years, including regression models[5, 18], nearest neighbor classifiers[3, 19], Bayes belief networks [14, 9], decision trees[5, 9, 11], rule learning algorithms[1, 15, 12], neural networks[15] and inductive learning techniques[2, 8]. With more and more methods available, cross- method evaluation becomes increasingly important. However, without an unified methodology of empirical validation, an objective comparison is difficult. The most serious problem is the lack of standard data collections. Even when a shared collection is chosen, there are still many ways to introduce inconsistency. For example, the commonly used Reuters newswire corpus[6] has at least four different versions, depending on how the training/test sets were divided, and what categories are included or excluded in the evaluation. Lewis and Ringuette used this corpus to evaluate a decision tree approach and a naive Bayes classifier, where they included a large portion of unlabelled documents (47% in the training set, and 58% in the test set) [9]. It is not clear whether these unlabelled documents are all negative instances of the categories in consideration, or that they are unlabelled simply as an oversight. Apte et al. run a rule learning algorithm, SWAP-1, on the same set of documents after removing the unlabelled documents[1]. They observed an 12-14% improvement of SWAP-1 over the results in Lewis&Ringuette's experiments, and concluded that SWAP-1 can often substantially improve results over decision trees, and that "text classification has a number of characteristics that make optimized rule induction particularly suitable." This would be a significant finding if the same data were used in the two experiments. However, given that 58% of the test documents were removed from the original set, it is questionable whether the observed difference came from the change in the data, or from the difference in the methods. An analysis later in Sections 3 and 5 will further clarify the point: the inclusion or exclusion of unlabelled documents could have a significant impact to the results; ignoring this issue makes an evaluation problematic. It would be ideal if a universal test collection were shared by all the text categorization researchers, or if a controlled evaluation of a wide range of categorization methods were conducted, similar to the Text Retrieval Conference for document retrieval[4]. The reality, however, is still far from the ideal. Cross-method comparisons have often been attempted but only for two or three methods. The small scale of these experiments could lead to overly general statements based on insufficient observations at one extreme, or the inability to state significant differences at the other extreme. A solution for these problems is to integrate the available results of categorization methods into a global evaluation, by carefully analyzing the test conditions in different experiments, and by establishing a common basis for cross-collection and cross-experiment integration. This paper reports on an effort in this direction. Section 2 outlines the fourteen methods being investigated. Section 3 analyzes the collection differences in commonly used corpora, using three classifiers to examine to what degree a difference in conditions effects the evaluation of a classifier. Section 4 defines a variety of performance measures in use and addresses the equivalence and comparability between them. Section 5 reports on new evaluations, and compares them with previously published results. The performance of a baseline classifier on multiple data collections is used as a reference point for a cross- collection observation. Section 6 concludes the findings. Categorization Methods The intention here is to integrate available results from individual experiments into a global evaluation. Two commonly used corpora, the Reuters news story collection[9] and the OHSUMED bibliographical document collection[7] are chosen for this purpose. Fourteen categorization methods are investigated, including eleven methods which were previously evaluated using these corpora, and three methods which were newly evaluated by this author. Not all of the results are directly comparable because different versions or subsets of these corpora were used. These methods are outlined below; the data sets and the result comparability will be analyzed in the next section. 1. CONSTRUE, an expert system consisting of manuallydeveloped categorization rules for Reuters news stories[6]. 2. Decision tree (DTree) algorithms for classification[9, 11]. 3. A naive Bayes model (NaiveBayes) for classification where word independence is assumed in category prediction[9, Table 1. Data collections examination using WORD, kNN and LLSF in category ranking Corpus Set UniqCate TrainDoc TestDos (labelled) WORD kNN LLSF CONSTRUE* CONSTRUE.2 Reuters Lewis* 113 14,704 6,746 (42%) .10 .84 - Apte 93 7,789 3,309 (100%) .21 .93 .92 full range 14,321 183,229 50,216 (100%) .16 .52 - OHSUMED HD big* HD small* 28 183,229 50,216 (100%) - * Unlabelled documents are included. * Heart Diseases (a sub-domain) Categories only, with a training-set category frequency of at least 75. * Heart Diseases (a sub-domain) Categories only, with a training-set category frequency between 15 to 74. 4. SWAP-1, an inductive learning algorithm for classification using rules in Disjunctive Normal Form (DNF)[1]. 5. A neural network approach (NNets) to classification[15]. 6. CHARADE, a DNF rule learning system for classification by I. Moulinier[12]. 7. RIPPER, a DNF rule learning system for classification by W. Cohen[2]. 8. Rocchio, a vector space model for classification where a training set of documents are used to construct a prototype vector for each category, and category ranking given a document is based on a similarity comparison between the document vector and the category vectors [8]. 9. An exponentiated gradient (EG) inductive learning algorithm which approximates a least squares fit [8]. 10. The Widrow-Hoff (WH) inductive learning algorithm which approximates a least squares fit[8]. 11. Sleeping Experts (EXPERTS), an inductive learning system using n-gram phrases in classification [2]. 12. LLSF, a linear least squares fit (LLSF) approach to classification [18]. A single regression model is used for ranking multiple categories given a test document. The input variables in the model are unique terms (words or phrases) in the training documents, and the output variables are unique categories of the training documents. 13. kNN, a k-nearest neighbor classifier[16]. Given an arbitrary input document, the system ranks its nearest neighbors among training documents, and uses the categories of the k top-ranking neighbors to predict the categories of the input document. The similarity score of each neighbor document is used as the weight of its categories, and the sum of category weights over the k nearest neighbors are used for category ranking. 14. A simple, non-learning method which ranks categories for a document based on word matching (WORD) between the document and category names. The conventional Vector Space Model is used for representing documents and category names (each name is treated as a bag of words), and the SMART system [13] is used as the search engine. 3 Collection Analysis 3.1 Two corpora The Reuters corpus, a collection of newswire stories from 1987 to 1991, is commonly used for text categorization research, starting from an early evaluation of the CONSTRUE expert system [6, 9, 1, 15, 12, 2] 1 . This collection is newly refined version named Reuters-21578 is available through Lewis' home page http://www.research.att.com/ ~ lewis. split into training and test sets when used to evaluate various learning systems. However, the split is not the same in different studies. Also, various choices were made for the inclusion and exclusion of some categories in an evaluation, as described in the next section. The OHSUMED corpus, developed by William Hersh and colleagues at the Oregon Health Sciences University, is a subset of the documents in the MEDLINE database 2 . It consists of 348,566 references from 270 medical journals from the years 1987 to 1991. All of the references have titles, but only 233,445 of them have abstracts. We refer to the title plus abstract as a document. The documents were manually indexed using subject categories (Medical Subject Headings, or MeSH; about 18,000 categories defined) in the National Library of Medicine. The OHSUMED collection has been used with the full range of categories (14,321 MeSH categories actually occurred) in some experiments[17], or with a subset of categories in the heart disease sub-domain (HD, 119 categories) in other experiments[8]. 3.2 Different versions Table 1 lists the different versions or subsets of Reuters and OHSUMED. Each is referred as a "set" or "collection", and labelled for reference. To examine the collection differences from a text categorization point of view, three classifiers (WORD, kNN and LLSF) were applied to these collections. The assumption is that if two collections are statistically homogeneous, then the results of a classifier on these collections should not differ too much. Inversely, if a dramatic performance change is observed between collections, then this would indicate a need for further analysis. Since the behavior of a single classifier may lead to biased conclusions, the multiple and fundamentally different classifiers were used instead. All the systems produces a ranked list of candidate categories given a document. The conventional 11-point average precision[13] was used to measure the goodness of category ranking. WORD and kNN were tested on all the collections, while LLSF was only tested on the smaller collections due to computational limitations. The HD sets were examined together with the OHSUMED superset instead of being examined separately. Several observations emerge from Table 1: Homogeneous collections. The Apte set, the PARC set and the Lewis.2 of the Reuters documents are relatively homogeneous, evident from the similar performance of WORD, kNN and LLSF on these sets. The Lewis.2 is derived (by this author) from the original Lewis set by removing the unlabelled documents. The Apte set is obtained by further restricting the categories to have a training set frequency of at least two. In both sets, a continuous chunk of documents (the early ones) are used for training, and the remaining chunk of documents (the later ones) are used for testing. The PARC set is drawn from the CONSTRUE set by eliminating the unlabelled documents and some rare categories[15]. Instead of taking continuous chunks of documents for training and testing, it uses a different partition. The collection is sliced into many subsets using non-overlapping time windows. The odd subsets are used for training, and the even subsets are used for testing. The differences between the PARC set, the Apte set and the Lewis.2 set do not seem to have a significant impact on the performance of the classifiers. An outlier collection. The CONSTRUE collection has an unusual test set. The training set contains all the documents in the Lewis set, Apte set or PARC set, and therefore should be statistically similar. The test set contains only 723 documents which are not included in the other sets. The performance of WORD and kNN on this set are clearly in favor of word matching over statistical learning. Comparing the Apte set to the CONSTRUE set, the relative improvement in WORD is 33% (changing from 21% to 28% in average precision), while the performance change in kNN is \Gamma13% (from 92% to 80%). Although we do not know what criteria were used in selecting the test documents, it is clear that using this set for evaluation would lead to inconsistent results, compared to using the other sets. The small size of this test set also makes its results statistically less reliable for evaluation. collection. The categorization task in OHSUMED seems to be more difficult than in Reuters, as evidenced from the significant performance decrease in both WORD and kNN. The category space is two magnitudes larger than Reuters. The number of categories per document is also larger, about 12 to 13 categories on average in OHSUMED while about 1.2 categories in Reuters. This means that the word/category correspondences are more "fuzzy" in OHSUMED. Consequently, the categorization is more difficult to learn. The collections named "HD big" (containing common categories) or "HD small" (containing 28 secondarily common categories) are sub-domains of the heart diseases sub-domain. Since they contains only about 0.2-.3% of the full range of the categories, performance of a classifier on these sets may not be sufficiently representative of its performance over the full domain. This does 2 OHSUMED is anonymously ftp-able from medir.ohsu.edu in the directory /pub/ohsumed not invalidate the use of the HD data sets, but it should be taken into consideration in a cross-collection comparison of categorization methods. collection. The Lewis set of the Reuters corpus seems to be problematic given the large portion of suspiciously unlabelled documents. Note that 58% of the test documents are unlabelled. According to D. Lewis, "it may (or may not) have been a deliberate decision by the indexer" 3 . It is observed by this author that on randomly selected test documents, the categories assigned by kNN appeared to be correct in many cases, but they were counted as failures because these documents were given as unlabelled. This raises a serious question as to whether or not these unlabelled documents should be included in the test set, and treated as negative instances of all categories, as they were handled in the previous experiments[9, 2]. The following analysis addresses this question. Assume the test set has 58% unlabelled documents, and suppose that all of the unlabelled documents should be assigned categories but are erroneously unlabelled. Let us further assume A to be a perfect classier which assigns a category to a document if and only if they match, and B a trivial classifier which never assigns any category to a document. Now if we use the errorful test set as the gold standard to evaluate the two systems, system A will have an assessed error rate of 58% instead of the true rate of zero percent. System B will have an assessed error rate of 42% instead of the true rate of 100%. Clearly, conclusions based on such a test set can be extremely misleading. In other words, it can make a better method look worse, and a worse method look better. Of course we do not know precisely how many documents in the Lewis set should be labelled with categories, so the argument above is only indicative. Nevertheless, to avoid unnecessary confusion, it would be more sensible to remove the unlabelled documents, or use the Apte set or PARC set instead. This point will be further addressed in Section 5, with a discussion on the problems with the experimental results on the Lewis set. Performance Measures Classifiers either produce scores, and hence ranked lists of potential category labels, or make binary decisions to assign categories. A classifier that produces a score can be made into a binary classifier by thresholding the score. The inverse process is considerably more difficult. An evaluation method applicable to a scoring classifier may not apply to a binary method. We present evaluations suitable to the two cases and indicate in the following which are used for comparison. 4.1 Evaluation of category ranking The recall and precision of a category ranking is similar to the corresponding measures used in text retrieval. Given a document as the input to a classifier, and a ranked list of categories as the output, the recall and precision at a particular threshold on this ranked list are defined to be: categories found and correct total categories correct categories found and correct total categories found where "categories found" means that the categories are above the threshold. For a collection of test documents, the category ranking for each document is evaluated first, then the performance scores are averaged across documents. The conventional 11-point average precision is used to measure the performance of a classifier on a collection of documents[13]. 4.2 Evaluation of binary classification Performance measures in binary classification can be defined using a two-way contingency table (Table 2). The table contains four cells: ffl a counts the assigned and correct cases, 3 Refer to the documentation of the newly refined Reuters-21578 collection. counts the assigned and incorrect cases, ffl c counts the not assigned but incorrect cases, and ffl d counts the not assigned and correct cases. Table 2. A contingency table YES is correct No is correct Assigned YES a b Assigned NO c d The recall (r), precision (p), error (e) and fallout (f) are defined to be: c) if a Given a classifier, the values of often depend on internal parameter tuning; there is a trade-off between recall and precision in general. A commonly used measure in method comparison [9, 1, 15, 12] is the break-even point (BrkEvn) of recall and precision, i.e., when r and p are tuned to be equal. Another common is called the F -measure, defined to be: where fi is the parameter allowing differential weighting of p and r. When the value of fi is set to one (denoted as precision is weighted equally: When the value of F 1 (r; p) is equivalent to the break-even point. Often the break-even point is close to the optimal score of F 1 (r; p), but they are not necessarily equivalent. In other words, the optimal score of F 1 (r; p) given a system can be higher-valued than the break-even point of this system. Therefore, the break-even point of one system should not be compared directly with the optimal F 1 value of another system. 4.3 Global averaging There are two ways to measure the average performance of a binary classifier over multiple categories, namely, the macro-average and the the micro-average. In macro-averaging, one contingency table per category is used, and the local measures are computed first and then averaged over categories. In micro-averaging, the contingency tables of individual categories are merged into a single table where each cell of a, b, c and d is the sum of the corresponding cells in the local tables. The global performance then is computed using the merged table. Macro-averaging gives an equal weight to the performance on every category, regardless how rare or how common a category is. Micro- averaging, on the other hand, gives an equal weight to the performance on every document (category instance), thus favoring the performance on common categories. The micro-average is used in the following evaluation section. Table 3 summarizes the results of all the categorization methods investigated in this study. The results of kNN, LLSF and WORD are newly obtained. The results of the other methods are either directly from previous publications. Table 3. Results of different methods in category assignments Reuters Reuters OHSUMED OHSUMED Reuters Reuters Apte PARC full range HD big Lewis CONSTRUE BrkEvn BrkEvn F NNets (N) - .82* - DTree NaiveBayes (L) .71 (\Gamma16%) - .65 - "L" indicates a linear model, and "N" indicates a non-linear model; "*" marks the local optimal on a fixed collection; "(.)" includes the performance improvement relative to kNN; "[.]" includes a F(1) score; the corresponding break-even point should be the same or slightly lower. 5.1 The new experiments The KNN, LLSF and WORD experiments used the SMART system for unified preprocessing, including stop word removal, stemming and word weighting. A phrasing option is also available in SMART but not used in these experiments. Several term weighting options (labelled as "ltc", "atc", "lnc" , "bnn" etc. in SMART's notation) were tried, which combine the term frequency (TF) measure and the Inverted Document Frequency (IDF) measure in a variety of ways. The best results (with "ltc" in most cases) are reported in the Table 3. In kNN and LLSF, aggressive vocabulary reduction based on corpus statistics was also applied as another step of the preprocessing. This is necessary for LLSF which would otherwise be too computationally expensive to apply to large training collections. Computational tractability is not an issue for kNN but vocabulary reduction is still desirable since it improves categorization accuracy. About 1-2% improvements in average precision and break-even point were observed in both kNN and LLSF when an 85% vocabulary reduction was applied. Several word selection criteria were tested, including information gain, mutual information, a - 2 statistic and document frequency[20]. The best results (using the - 2 statistic) were included in Table 3. Aggressive vocabulary reduction was not used in WORD because it would reduce the chance of word-based matching between documents and category names. KNN, LLSF and WORD produces a ranked list of categories first when a test document is given. A threshold on category scores then is applied to obtain binary category assignments to the document. The thresholding on category scores was optimized on training sets (for individual categories) first, and then applied to the test sets. Other parameters in these systems include: ffl k in kNN indicates the number of nearest neighbors used for category prediction, and ffl p in LLSF indicates the number of principal components (or singular vectors) used in computing the linear regression. The performance of kNN is relatively stable for a large range of k, so three values (30, 45 and 65) were tried, and the best results are included in the result table. A satisfactory performance of LLSF depends on whether p is sufficiently large. In the experiments of LLSF on the Reuters sets, the optimal or nearly optimal results were obtained when using about 800 to 1000 singular vectors. A Sun SPARC Ultra-2 Server was used for the experiments. LLSF has not yet applied to the full set of OHSUMED training documents due to computational limitations. 5.2 Cross-experiment comparison A row-wise comparison in Table 3 allows observation of the performance variance of a method across collections. Unfortunately, most of the rows are sparse except for kNN and WORD. A column-wise comparison allows observation of different methods on a fixed collection. A star marks the best result for each collection. KNN is chosen to provide the baseline performance on each collection. Several characteristics of this method make it preferable, i.e., efficient to test, easy to scale up, and relatively robust as a learning method. LLSF is equally effective, based on the empirical results obtained so far; however, its training is computationally intensive, and thus has not yet been applied to the full range of the OHSUMED collection. WORD is chosen to provide an secondary reference point in addition to kNN, to enable a quantitative comparison between learning approaches to a simple method that requires no knowledge or training. The Reuters Apte set has the densest column where the results of eight systems are available. Although the document counts reported by different researchers are somewhat inconsistent[1, 2] 4 , the differences are relatively small compared to the size of the corpus (i.e., at most 21 miscounted out of over ten thousands training documents, and at most 7 miscounted out of over three thousands of test documents), so the impact of such differences on the evaluation results for this set maybe be considered negligible. The results on the Lewis set, on the other hand, are more problematic. That is, the inclusion of the 58% "mysteriously" unlabelled documents in the test set makes the results difficult to interpret. For example, most of the methods (kNN, RIPPER, Rocchio and WORD) which were evaluated on both the Apte set and the Lewis set show a significant decrease in their performance scores on the Lewis set, but the scores of EXPERTS are almost insensitive to the inclusion or exclusion of the large amounts of unlabelled documents in the test set. Moreover, EXPERTS has a score near the lower end among all the learning methods evaluated on the Apte set, but the highest score on the Lewis set. Cohen concluded EXPERTS the best performer ever reported on the Lewis set without an explanation on its mysterious insensitivity to the large change in test documents[2]. This is suspicious because the inclusion of a large amounts of incorrectly labelled documents in the test set should decrease the performance of a good classifier, as analyzed in Section 3. Another example of potential difficulties is the misleading comparison by Apte et al. between SWAP-1 (or rule learning), NaiveBayes and DTree methods (Section 1). They claim an advantage for SWAP-1 based on a score on the Apte set versus scores for the other methods on the Lewis set. To see the perils in such an inference, kNN has a score of 85% on the Apte set, versus the SWAP-1 score of 79% on the same set. On the Lewis set, however, the kNN score is 69%, i.e., 10% lower than Apte SWAP-1 score. Should we then conclude that SWAP-1 is better than kNN, or the opposite? More interestingly, a recent result using a DTree algorithm (via C4.5) due to Moulinier scores 79% on the Apte set[11], which is exactly the same as the SWAP-1 result. How should this be interpreted? To make the point clear, the Lewis set should not be used for text categorization evaluation unless the status of the unlabelled documents is resolved. Results obtained on this set can be seriously misleading, and therefore should not be used for a comparison or to draw any conclusions. Inferences based on the CONSTRUE set should also be questioned because the test set is much smaller than the other sets, contains 20% mysteriously unlabelled documents, and may possibly be a biased selection (Section 3). Finally, it may worth mentioning that the cross-method comparisons here are not necessarily precise, because some experimental parameters might contribute to a difference in the results but are not available. For instance, different choices could be made in stemming, term selection, term weighting, sampling strategies for training data, thresholding for binary decisions, and so on. Without detailed information, we cannot be sure that a one or two percent difference in break-even point or F-measure is an indication of the theoretical strength or weakness of a learning method. It is also unclear how a significance test should be designed, given that the performance of a method is compressed into a single number, e.g., to the break-even point of averaged recall and precision. A variance analysis would be difficult given that the necessary input data is not generally published. Further research is needed on this issue. Nonetheless, missing detailed information should not prohibit the good use of available information. As long as the related issues are carefully addressed, as shown above, an integrated view across methods and experiments is possible, especially for significant variations in results on a fully-labelled common test set. 4 Inconsistent numbers about the documents in the Apte set were found in previous papers and the corpus documentation, presumably due to counting errors or processing errors by the individuals. The numbers included in Table 1 are those agreed by at least two research sites. Details are available through yiming@cs.cmu.edu. 6 Discussions Despite the imperfectness of the comparison across collections and experiments, the integrated results are clearly informative, enabling a global observation which is not possible otherwise. Several points in the results appear to be interesting regarding the analysis of classification models. The impressive performance of kNN is rather surprising given that the method is quite simple and computationally efficient. It has the best performance, together with LLSF, on the Apte set, and is equally effective as NNets on the PARC set. On the OHSUMED set, it is the only learning method evaluated on the full domain, i.e., a category space which is more than one hundred times larger than those used in the evaluations of most learning algorithms. When extending the target space from the sub-domain of 49 "HD big" categories to the full domain of 14,321 categories, the performance decline of kNN is only 5% in absolute value, or a 9% decrease relative. In contrast, the performance of WORD declined from 44% to 27%, or a 39% decrease relatively. This suggests that kNN is more powerful than WORD in making fine distinctions between categories. Or, it "failed" more gracefully when the category space grows by several orders of magnitude. The good performance of WH on "HD big" calls for deeper analysis. WH is an incremental learning algorithm trained based on an least squares fit criterion. Its optimal performance therefore should be bounded by or close to a least squares fit solution obtained in a batch-mode training, such as LLSF. It would be interesting in future research to compare the empirical results of LLSF with WH. It is also worth asking whether there is something else, beyond the core theory, which contributed to the good performance. In the WH experiment on "HD big", Lewis used a "pocketing" strategy to select a subset of training instances from a large pool[8]. This is similar or equivalent to a sampling strategy which divides available training instances into small chunks, examines one chunk at a time using a validation set, and adds a new chunk to the selected ones only if it improves the performance on the validation set. This strategy would be particularly effective when the training data are highly noisy, such as OHSUMED documents. Nevertheless, the sampling strategy is not a part of the WH algorithm, and can be used in any other classifiers. It would be interesting to examine the effect of the pocketing strategy in kNN on OHSUMED in feature research, for example. Rocchio has a relatively poor performance compared to the other learning methods, and is almost as poor as WORD on the "HD big" subset, surprisingly. This suggests that Rocchio may not be a good choice (although commonly used) for the baseline in evaluating learning methods, because it is inferior to most methods and thus would be not very informative especially when the comparison includes only one or two other learning methods. In other words, Rocchio is a straw man rather than a challenging standard. KNN would be a better alternative, for instance. The mixture of the linear (L) and non-linear (N) classifiers among the top-ranking performers (WH, NNets, kNN and LLSF) suggests that no general conclusion can be fetched regarding reliable improvement of non-linear approaches over linear approaches, or vice versa. It is also hard to draw a conclusion about the advantage of a multiple-category classification model (kNN or LLSF) over unary classification models (WH, NNets, EG, RIPPER etc.) Either the category independence assumption in the latter type of methods is reasonable, or an improvement in kNN and LLSF is needed in the handling of the dependence or mutual exclusiveness among categories. Resolving this issue requires future research. The rule induction algorithms (SWAP-1, RIPPER and CHARADE) have a similar performance, but below the local optimum of kNN on the Apte set, and also below some other classifiers (WH, NNets) based on an indirect comparison across collections via kNN as the baseline. This observation raises a question with respect to a claim about the particular advantage of rule learning in text categorization. The claim was based on context-sensitivity, i.e., the power in capturing term combinations[1, 2]. It seems that the methods which do not explicitly identify term combinations but use the context implicitly (such as in WH, NNets, kNN and LLSF) performed at least as well. It may be worth mentioning that a classifier can have a degree of context-sensitivity without explicitly identifying term combinations or phrases. The classification function in LLSF, for instance, is sensitive to weighted linear combinations of words that co-occur in training documents. This does not makes it equivalent to a non-linear model, but makes a fundamental distinction from the methods based on a term independence assumption, such as naive Bayes models. This may be a reason for the impressive performance of kNN and LLSF. It would be interesting to compare them with NaiveBayes if the latter were tested on the Apte set, for example. Conclusions The following conclusions are reached from this study: 1. The performance of a classifier depends strongly on the choice of data used for evaluation. Using a seriously problematic collection[8], comparing categorization methods without analyzing collection differences[1], and drawing conclusion based on the results of flawed experiments[2] raise questions about the validity of some published evaluations. These problems need to be addressed to clarify of the confusions among researchers, and to prevent the repetition of similar mistakes. Providing information and analysis on these problems is a major effort in this study. 2. Integrating results from different evaluations into a global comparison across methods is possible, as shown in this paper, by evaluating one or more baseline classifiers on multiple collections, by normalizing the performance of other classifiers using a common baseline classifier, and by analyzing collection biases based on performance variations of several baseline classifiers. Such an integration allows insights on methods and collections which are rarely apparent in comparisons involving two or three classifiers. It also shows an evaluation methodology which is complementary to the effort to standardize collections and unify evaluations. 3. WH, kNN, NNets and LLSF are the top performers among the learning methods whose results were empirically validated in this study. Rocchio had a relatively poor performance, on the other hand. All the learning methods outperformed WORD, the non-learning method. However, the differences between some learning methods are not as large as previously claimed[1, 2]. It is not evident in the collected results that non-linear models are better than linear models, or that more sophisticated methods outperform simpler ones. Conclusive statements on the strengths and weaknesses of different models requires further research. 4. Scalability of a classifier when the problem size grows by several magnitudes, or when the category space becomes a hundred times denser, has been rarely examined in text categorization evaluations. KNN is the only learning method evaluated on the full set of the OHSUMED categories. Its robustness in scaling up and dealing with harder problems, and its computational efficiency make it the method of choice for approaching very large and noisy categorization problems. Acknowledgement I would like to thank Jan Pedersen at Verity, David Lewis and William Cohen at AT&T, and Isabelle Moulinier at University of Paris VI for providing the information of their experiments. 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comparative study;evaluation;text categorization;statistical learning algorithms

357485

Secure Execution of Java Applets Using a Remote Playground.

AbstractMobile code presents a number of threats to machines that execute it. We introduce an approach for protecting machines and the resources they hold from mobile code and describe a system based on our approach for protecting host machines from Java 1.1 applets. In our approach, each Java applet downloaded to the protected domain is rerouted to a dedicated machine (or set of machines), the playground, at which it is executed. Prior to execution, the applet is transformed to use the downloading user's web browser as a graphics terminal for its input and output and so the user has the illusion that the applet is running on her own machine. In reality, however, mobile code runs only in the sanitized environment of the playground, where user files cannot be mounted and from which only limited network connections are accepted by machines in the protected domain. Our playground thus provides a second level of defense against mobile code that circumvents language-based defenses. The paper presents the design and implementation of a playground for Java 1.1 applets and discusses extensions of it for other forms of mobile code, including Java 1.2.

Introduction Advances in mobile code, particularly Java, have considerably increased the exposure of networked computers to attackers. Due to the "push" technologies that often deliver such code, an attacker can download and execute programs on a victim's machine without the victim's knowledge or consent. The attacker's code could conceivably delete, modify, or steal data on the victim's machine, or otherwise abuse other resources available from that machine. Moreover, mobile code "sandboxes" intended to constrain mobile code have in many cases proven unsatisfactory, in that implementation errors enable mobile code to circumvent the sandbox's security mechanisms [1, 9]. One of the oldest ideas in security, computer or oth- erwise, is to physically separate the attacker from the resources of value. In this paper we present a novel approach for physically separating mobile code from those resources. The basic idea is to execute the mobile code somewhere other than the user's machine, where the resources of value to the user are not avail- able, and to force the mobile code to interact with the user only from this sanitized environment. Of course, this could be achieved by running the mobile code at the server that serves it (thereby eliminating its mo- bility). However, the challenge is to achieve this physical separation without eliminating the benefits derived from code mobility, in particular reducing load on the code's server and increasing performance by co-locating the code and the user. In order to achieve this protection at an organizational level, we propose the designation of a distinguished machine (or set of machines), a playground, on which all mobile code served to a protected domain is executed. That is, any mobile code pushed to a machine in this protected domain is automatically rerouted to and executed on the domain's play- ground. To enable the user to interact with the mobile code during its execution, the user's computer acts as a graphics terminal to which the mobile code displays its output and from which it receives its input. How- ever, at no point is any mobile code executed on the user's machine. Provided that valuable resources are not available to the playground, the mobile code can entirely corrupt the playground with no risk to the do- main's resources. Moreover, because the playground can be placed in close network proximity to the machines in the domain it serves, performance degradation experienced by users is minimal. There can even be many playgrounds serving a domain to balance load among them. In this paper we report on the design and implementation of a playground for Java 1.1 applets. As described above, our system reroutes all Java applets retrieved via the web to the domain's playground, where the applets are executed using the user's browser essentially as an I/O terminal. By disallowing the playground to mount protected file systems or open arbitrary network connections to domain machines-in the limit, locate the playground just "outside" the do- main's firewall-the domain's resources can be protected even if the playground is completely corrupted. Our system is largely transparent to users and applet developers, and in some configurations requires no changes to web browsers in use today. While there is a class of applets that are not amenable to execution on our present playground prototype, e.g., due to performance requirements or code structure, in our experience this class is a small fraction of Java applets. As described above, the playground need not be trusted for our system to work securely. Indeed, the only trusted code that is common to all configurations of our system is the browser itself and a small "graphics server", itself a Java applet, that runs in the browser. The graphics server implements interfaces that the untrusted applet, running on the play- ground, calls to interact with the user. The graphics server is a simply structured piece of code, and thus should be amenable to analysis. In one configuration of our system, trust is limited to only the graphics server and the browser, but doing so requires a minor change to browsers available today. If we are constrained to using today's browsers off-the-shelf, then a web proxy component of our system, described in Section 3, must also be trusted. The rest of this paper is structured as follows. In Section 2 we relate our work to previous efforts at protecting resources from hostile mobile code. We give an overview of our system in Section 3 and refine this description in Section 4, where we describe the implementation of the system in some detail. The security of the system is discussed in Section 5, and limitations are discussed in Section 6. Related work There are three general approaches that have been previously proposed for securing hosts from mobile code. The first to be deployed on a large scale for Java is the "sandbox" model. In this model, Java applets are executed in a restricted execution environment (the sandbox) within the user's browser; this sandbox attempts to prevent the applet from performing illegal actions. This approach has met with mixed success, in that even small implementation errors can enable applets to entirely bypass the security restrictions enforced by the sandbox [1]. The second general approach is to execute only mobile code that is trusted based on some criteria. For ex- ample, Balfanz has proposed a Java filter that allows users to specify the servers from which to accept Java applets (see http://www.cs.princeton.edu/sip/). Here the criterion by which an applet is trusted is the server that serves it. A related approach is to determine whether to trust mobile code based on its author, which can be determined, e.g., if the code is digitally signed by the author. This is the approach adopted for securing Microsoft's ActiveX content, and is also supported for applets in JDK 1.1. Combinations of this approach and the sandbox model are implemented in JDK 1.2 [3, 4] and Netscape Communicator (see [14]), which enforce access controls on an applet based on the signatures it possesses (or other properties). A third variation on this theme is proof-carrying code [11], where the mobile code is accompanied by a proof that it satisfies certain properties. However, these techniques have not yet been applied to languages as rich as Java (or Java bytecodes). Our approach is compatible with both of the approaches described above. Our playground executes applets in sandboxes (hence the name "playground"), and could easily be adapted to execute only "trusted" applets based on any of the criteria above. Our approach provides an orthogonal defense against hostile applets, and in particular, in our system a hostile applet is still physically separated from valuable resources after circumventing these other defenses. The third approach to securing hosts from mobile code is simply to not run mobile code. A course-grained approach for Java is to simply disable Java in the browser. Another approach is to filter out all applets at a firewall [10] (see also [9, Chapter 5]), which has the advantage of allowing applets served from behind the firewall to be executed. Independently of our work, a system similar to ours has recently been marketed by Digitivity, Inc., a California-based company. While there are no descriptions of their system in the scientific liter- ature, we have inferred several differences in our systems from a white paper [5], their web site (http://www.digitivity.com), and discussions with company representatives. First, elements of the Digi- tivity system-notably the protocol for communication between the user's browser and (their analog of) the playground, and the Java Virtual Machine (JVM) running on their playground-are proprietary, whereas our system is built using only public, widely-used protocols and JVMs. This may enable Digitiv- ity to more easily tune its system's performance, but our approach promotes greater confidence in the security of our system by exposing it to maximum public scrutiny and understanding. Second, our system does not require trust in certain elements of the system that, according to [5], are trusted in their architecture. We discuss the trusted elements of our architecture in Section 5.1.1. 3 Architecture The core idea in this paper is to establish a dedicated machine (or set of machines) called a playground at which mobile code is transparently executed, using users' browsers as I/O terminals. In this section we give an overview of the playground and supporting architecture that we implemented for Java, deferring many details to Section 4. To understand how our system works, it is first necessary to understand how browsers retrieve, load, and run Java applets. When a browser retrieves a web page written in Hypertext Markup Language (HTML), it takes actions based on the HTML tags in that page. One such tag is the !applet? tag, which might appear as follows: !applet code=hostile.class .? This tag instructs the browser to retrieve and run the applet named hostile.class from the server that served this page to the browser. The applet that returns is in a format called Java bytecode, suitable for running in any JVM. This bytecode is subjected to a bytecode verification process, loaded into the browser's JVM, and executed (see, e.g., [8]). In our system, when a browser requests a web page, the request is sent to a proxy (Figure 1, step 1). The proxy forwards the request to the end server (step 2) and receives the requested page (step 3). As the page is received, the proxy parses it to identify all !applet? tags on the returning page, and for each !applet? tag so identified, the proxy replaces the named applet with the name of a trusted graphics server applet stored locally to the browser. 1 The proxy then sends this modified page back to the browser (step 4), which loads the graphics server applet upon receiving the page. For each !applet? tag the proxy identified, the proxy retrieves the named applet (steps 5-6) and modifies its bytecode to use the graphics server in the requesting browser for all input and output. The proxy forwards the modified applet to the playground (step 7), where it is executed using the graphics server in the browser as an I/O terminal (step 8). To summarize, there are three important components in our architecture: the graphics server applet that is loaded into the user's browser, the proxy, and the playground. None of these need be executed on the same machine, and indeed there are benefits to executing them on different machines (this is discussed in Section 5). The graphics server and the playground are implemented in Java, and thus can run on any Java compliant environment; the proxy is a Perl script. The same proxy can be used for multiple browsers and mul- "stored locally" means stored in a directory named by the CLASSPATH environment variable. tiple playgrounds. In the case of multiple playgrounds, the proxy can distribute load among playgrounds for improved performance. In the following subsections, we describe the functions of these components in more detail. Security issues are discussed in Section 5. 3.1 The graphics server In this section we give an overview of the graphics server that is loaded into a user's browser in place of an applet provided by a web server. Because the graphics server is a Java applet, we must introduce some Java terminology to describe it. In Java, a class is a collection of data fields and functions (called methods) that operate on those fields. An object is an instance of a class; at any point in time it has a state-i.e., values assigned to its data fields-that can be manipulated by invoking the methods of that object (defined by the object's class). Classes are arranged in a hierar- chy, so that a subclass can inherit fields and methods from its superclass. A running Java applet consists of a collection of objects whose methods are invoked by a runtime system, and that in turn invoke one another's methods. For more information on Java see, e.g., [2]. 3.1.1 Remote AWT classes The Abstract Window Toolkit (AWT) is the standard API for implementing graphical user interfaces (GUI) in Java programs. The AWT contains classes for user input and output devices, including buttons, choice boxes, text fields, images, and a variety of types of windows, to name a few. Virtually every Java applet interacts with the user by instantiating AWT classes and invoking the methods of the objects so created. The intuitive goal of the graphics server is to provide versions of the AWT classes whose instances can be created and manipulated from the playground. For example, the graphics server should enable a program running on the playground to create a dialog window in the user's browser, display it to the user, and be informed when the user clicks the "ok" button. In the parlance of distributed object technology, such an object-i.e., one that can be invoked from outside the virtual machine in which it resides-is called a remote object, and the class that defines it is called a remote class. So, the graphics server, running in the user's browser, should allow other machines (the play- ground) to create and use "remote AWT objects" in the user's browser for interacting with the user. Accordingly, the graphics server is implemented as a collection of remote classes, where each remote class (with one exception that is described in Section 3.1.2) is a remote version of a corresponding AWT class. The 7. Modified applet [load graphics server] [change 4. Modified page 1. Request for page 8. I/O [change I/O] Browser Proxy 6. Applet 3. Page [load applet] WEB 5. Request for applet 2. Request for page Playground Figure 1: Playground architecture (modified) Java applet running on the playground creates a collection of graphical objects in the graphics server to implement its GUI remotely, and uses stubs to interact with them (see Section 3.3). To minimize the amount of code in the graphics server, each remote class is a subclass of its corresponding AWT class, which enables it to inherit many method implementations from the original AWT class. Other methods must be overwritten, for example those involving event monitoring: in the remote class, methods involving the remote object's events (e.g., mouse clicks on a remote button object) must be adapted to pass back the event to the stubs residing in the playground JVM. In our present implementation, the remote classes employ Remote Method Invocation (RMI) to communicate with the playground. RMI is available on all Java 1.1 platforms, including Netscape Communicator, Internet Explorer 4.0, JDK 1.1, and HotJava 1.0. 3.1.2 The remote Applet class As described in Section 3.1.1, most classes that comprise the graphics server are remote versions of AWT classes. The main exception to this is a remote version of the java.applet.Applet (or just Applet) class, which is the class that all Java applets must subclass. The main purpose of the Applet class is to provide a standard interface between applets and their envi- ronment. Thus, the remote version of this class serves to provide this interface between the applet running on the playground and the environment with which it must interact, namely the user's browser. More specifically, this class implements two types of methods. First, it provides remote interfaces to the methods of the Applet class, so that applets on the playground can invoke them to interact with the user's environment. Second, this class defines a new "con- structor" method for each remote AWT class (see Section 3.1.1). For example, there is a constructButton method for constructing a remote button in the user's browser. This constructor returns a reference to the newly-created button, so that the remote methods of the button can be invoked directly from the play- ground. Similarly, there is a constructor method for each of the other remote AWT classes. When initially started, the graphics server consists of only one object, whose class is the remote Applet class, called BrowserServer.class. The applet on the playground can then invoke the methods of this object (and objects so created) to create the graphical user interface that it desires for the user. 3.2 The proxy The proxy serves as the browser's and playground's interface to the web. It retrieves HTML pages for the browser and Java bytecodes for the playground, and transforms them to formats suitable for the browser or playground to use. When retrieving an HTML page for the browser, the proxy parses the returning HTML, identifies all !applet? tags in the page, and replaces them with references to the remote Applet class of the graphics server (see Section 3.1.2). Thus, when the browser receives the returned HTML page it loads this remote Applet class (stored locally), instead of the applet originally referenced in the page. When retrieving a Java bytecode file for the play- ground, the proxy transforms it into bytecode that interacts with the user on the browser machine while running at the playground. It does so by replacing all invocations of AWT methods with invocations of the corresponding remote AWT methods at the browser, or more precisely, with invocations of playground-side stubs for those remote AWT methods (which in turn call remote AWT methods). This involves parsing the incoming bytecode and making automatic textual substitutions to change the names of AWT classes to the names of the representative stubs of the corresponding remote AWT classes. 3.3 The playground The playground is a machine that loads modified applets from the proxy and executes them. As described above, the proxy modifies the applet's byte-codes so that playground-resident stubs for remote AWT methods are called instead of the (non-remote) AWT methods themselves. So, when a modified applet runs on the playground, a "skeleton" of its GUI containing stubs for corresponding remote graphics objects is built on the playground. The stubs contain code for remotely invoking the remote objects' methods at the user's browser and for handling events passed back from the browser. For example, in the case of a dialog window with an "ok" button, stubs for the window and for button objects are instantiated at the playground. Calls to methods having to do with displaying the window and button are passed to the remote objects at the user's browser, and "button press" events are passed back to methods provided by the button's stub to handle such events. These stubs are stored locally on the playground, but aside from this, the playground is configured as a standard JVM. A playground is a centralized resource that can be carefully administered. Moreover, investments in the playground (e.g., upgrading hardware or performing enhanced monitoring) can improve applet performance and security for all users in the protected do- main. There can even be multiple playgrounds for load-balancing. Implementation 4.1 An example In order to understand how the components described in Section 3 work together, in this section we illustrate the execution of a simple applet. This example describes how the applet is automatically transformed to interact with the browser remotely, and how the graphics server and its playground-side stubs interact during applet execution. This section necessarily involves low-level detail, but the casual reader can skip ahead to Section 5 without much loss of continuity. The applet we use for illustration is shown in Figure 2. This is a very simple (but complete) applet that prints the word "Click!" wherever the user clicks a mouse button. For the purposes of this discussion, it implements two methods that we care about. The first is an init() method that is invoked once and registers this (i.e., the applet object) as one that should receive mouse click events. This registration is achieved via a call to its own addMouseListener() method, which it inherits from its Applet superclass. The second method it implements is a mouseClicked() method that is invoked whenever a mouse click occurs. This method calls getGraphics(), again inherited from Applet, to obtain a Graphics object whose methods can be called to display graphics. In this case, the method invokes the drawString() method of the Graphics object to draw the string "Click!" where the mouse was clicked. import java.applet.*; import java.awt.*; import java.awt.event.*; public class Click extends Applet implements MouseListener - public void init() - // Tell this applet what MouseListener objects // to notify when mouse events occur. Since we // implement the MouseListener interface ourselves, // our own methods are called. // A method from the MouseListener interface. Invoked when the user clicks a mouse button. public void mouseClicked(MouseEvent e) - Graphics g.drawString("Click!", e.getX(), e.getY()); // The other, unused methods of the MouseListener // interface. public void mousePressed(MouseEvent e) - public void mouseReleased(MouseEvent e) - public void mouseEntered(MouseEvent e) - public void mouseExited(MouseEvent e) - Figure 2: An applet that draws "Click!" wherever the user clicks The Click applet is a standalone applet that is not intended to be executed using a remote graphics display for its input and output. Thus, when our proxy retrieves (the bytecode for) such an applet, the applet must be altered before being run on the playground. For one thing, the addMouseListener() method invocation must somehow be passed to the browser to indicate that this playground applet wants to receive mouse events, and the mouse click events must be passed back to the playground so that mouseClicked() is invoked. In our present implementation, passing this information is achieved using Java Remote Method Invocation (RMI). Associated with each remote class is a stub for calling it that executes in the calling JVM. The stub is invoked exactly as any other object is, and once invoked, it marshals its parameters and passes them across the network to the remote object that services the request. Below, the stubs are described by interfaces with the suffix Xface. For example, BrowserXface is the interface that is used to call the BrowserServer object of the graphics server. RMI provides the mechanism to invoke methods remotely, but how do we get the Click applet to use RMI? To achieve this, we exploit the subclass inheritance features of Java to interpose our own versions of the methods it invokes. More precisely, we alter the Click applet to subclass our own PGApplet, rather than the standard Applet class. This is a straight-forward modification of the bytecode for the Click applet. By changing what Click subclasses in this way, the addMouseListener() method called in Figure 2 is now the one in Figure 3. PGApplet, shown in Figure calls, when necessary, to the remote applet object of the graphics server described in Section 3.1.2. The addMouseListener() method simply adds its argument (the Click applet) to an array of mouse listeners and, if this is the first to register, registers itself as a mouse listener at the graphics server. This registration at the graphics server is handled by the addPGMouseListener() remote method of BrowserServer, the class of the remote applet object running on the browser (see Section 3.1.2). The relevant code of BrowserServer is shown in Figure 5. Recall that BrowserServer implements the BrowserXface interface that specifies the remote methods that can be called from the playground. The addPGMouseListener() method, which is one of those remote methods, records the fact that the playground applet wants to be informed of mouse events and then registers its own object as a mouse listener, so that its object's mouseClicked() method is invoked when the mouse is clicked. Such an invocation passes the mouse-click event-or more precisely, a reference to a BrowserEvent remote object that holds a reference to the actual mouse-click event object- back to the PGMouseClicked() remote method of the PGApplet class. The PGMouseClicked() method invokes mouseClicked() with a PGMouseEvent object, 2 For readability, in Figures 3-6 we omit import statements, error checking, exception handling (try/catch statements), etc. public class PGApplet extends Applet implements PGAppletXface - BrowserXface bx; MouseListener new MouseListener[.]; int // Adds a MouseListener. If this is the first, then // register this object at the graphics server as a // MouseListener. public synchronized void addMouseListener(MouseListener l) - if (ml-index == 1) - // Part of the PGAppletXface remote interface. // Invoked from the browser graphics server when the // mouse is clicked. public void PGMouseClicked(BrowserEventXface e) - int PGMouseEvent new PGMouseEvent(e); for // Returns an object that encapsulates the remote // graphics object of the browser applet. public Graphics getGraphics () - return new PGGraphics(bx.getBrowserGraphics()); Figure 3: Part of the PGApplet class (executes on the playground) public class PGGraphics extends Graphics - // Reference to the BrowserGraphics remote object // in the graphics server. private BrowserGraphicsXface bg; // The constructor for this object. Calls its // superclass constructor and saves the reference // to the BrowserGraphics remote object. public PGGraphics(BrowserGraphicsXface b) - // Invokes drawString in the graphics server. public void drawString(String str, int x, int y) - bg.drawString(str, x, y); Figure 4: Part of the PGGraphics class (executes on the playground) which holds the reference to the BrowserEvent object. That is, the PGMouseEvent object translates invocations of its own methods (e.g., getX() and getY() in Figure into invocations of the corresponding BrowserEvent remote methods, which in turn translates them into invocations of the actual Event object in the browser. For brevity, the BrowserEvent and PGMouseEvent classes are not shown. The call to getGraphics() in Click is also replaced with the PGApplet version. As shown in Figure 3, the getGraphics() method of PGApplet retrieves a reference to a remote BrowserGraphics ob- ject, via the getBrowserGraphics() method of Figure 5. The getGraphics() method of PGApplet then returns this BrowserGraphics reference encapsulated within a PGGraphics object for calling it. So, when Click invokes drawString(), the arguments are passed to the browser and executed (Figures 4,6). 4.2 Passing by reference In the example of the previous section, all parameters that needed to be passed across the network were serializable. Object serialization refers to the ability to write the complete state of an object to an output stream, and then recreate that object at some later time by reading its serialized state from an input stream [12, 2]. Object serialization is central to remote method invocation-and thus to communication between the graphics server and the playground stubs-because it allows for method parameters to be passed to a remote method and the return value to be passed back. In the example of Section 4.1, the remote method invocation bg.drawString(str, x, y) in the PGGraphics.drawString() method of Figure 4 causes no difficulty because each of str (a string) and x and y (integers) are serializable. However, not all classes are serializable. An example is the Image class, which represents a displayable image in a platform-dependent way. So, while the previous invocation of bg.drawString(str, x, y) suc- ceeds, a similar invocation bg.drawImage(img, x, y, .) fails because img (an instance of Image) cannot be serialized and sent to the graphics server. Even if all objects could be serialized, serializing and transmitting large or complex objects can result in substantial cost. For such reasons, an object that can be passed as a parameter to a remote method of the graphics server is generally constructed in the graphics server originally (with a corresponding stub on the playground). Then, a reference to this object in the browser is passed to graphics server routines in place of the object itself. In this way, only the object reference is ever passed over the network. public class BrowserServer extends Applet implements BrowserXface, MouseListener - // Reference to the MouseListener object on // the playground PGMouseListenerXface ml; // Part of the BrowserXface remote interface. Invoked // from the playground to add a remote MouseListener. public void addPGMouseListener(PGMouseListenerXface pml) - Invoked whenever the mouse is clicked. Passes the // event to the MouseListener on the playground. public void mouseClicked(MouseEvent event) - ml.PGMouseClicked(new BrowserEvent(event)); // Returns a remote object that encapsulates the // graphics context of this applet. public BrowserGraphicsXface getBrowserGraphics() - Graphics return new BrowserGraphics(g); Figure 5: Part of the BrowserServer class (executes in the browser) public class BrowserGraphics extends Graphics implements BrowserGraphicsXface - private Graphics // The constructor for this class. Calls the // superclass constructor, saves the pointer to the // "real" Graphics object (passed in), and exports // its interface to be callable from the playground. public BrowserGraphics(Graphics gx) - // Part of the BrowserGraphicsXface remote interface. // Invoked from the playground to draw a string. public void drawString(String str, int x, int y) - g.drawString(str, x, y); Figure Part of the BrowserGraphics class (executes in the browser) To illustrate this manner of passing objects by ref- erence, we continue with the example of an Image. When the downloaded applet calls for the creation of an Image object, e.g., via the Applet.getImage() method, our interposed PGApplet.getImage() passes the arguments (a URL and a string, both serializable) to a remote image creation method in the graphics server. This remote method constructs the image, places it in an array of objects, and returns the array index it occupies. Playground objects then pass this index to remote graphics server methods in place of the image itself. For example, the BrowserGraphics class in the graphics server implements versions of the drawImage() method that accept image indices and display the corresponding Image. Conversely, there are circumstances in which objects that need to be passed as parameters to remote methods cannot first be created in the browser. This can be due to security reasons-e.g., the object's class is a user-defined class that overwrites methods of an AWT class-or because the class of the parameter object is unknown (e.g., it is only known to implement some interface). In these circumstances, a reference is passed in the parameter's place, and method invocations intended for the object are passed back to the playground object for processing. Continuing with our Image() example, such "callbacks" can occur when the downloaded applet applies certain image filters to an image before displaying it. One such filter is an RGBImageFilter: A subclass of RGBImageFilter defines a per-pixel transformation to apply to an image by overwriting the filterRGB() method. To avoid loading untrusted code in the browser, such a filter must be executed on the playground with callbacks to its filterRGB() method. In some circumstances, the need to pass objects by reference can considerably hurt performance. Continuing with the RGBImageFilter example above, filtering an image may require that every image pixel be passed from the browser to the playground, transformed by the filterRGB() method, and passed back. This can result in considerable delay in rendering the image, though our experience is that this delay is reasonable for images whose pixel values are indices into a colormap array (i.e., for images that employ an IndexColorModel). 4.3 Addressing The previous sections described how an applet running on the playground is coerced into using the user's browser as its I/O terminal. Before any I/O can be performed at the browser, however, the applet running on the playground and the graphics server running in the user's browser must be able to find each other to communicate. This is complicated by the fact that an HTML page can contain any number of !applet? tags that, when modified by the proxy, result in multiple instances of the graphics server running in the browser. To retain the intended function of the page, it is necessary to correctly match each applet running on the playground with its corresponding instance of the graphics server in the browser. The addressing scheme that we use requires that the proxy make additional changes to the HTML page containing applet references prior to forwarding it to the browser. Specifically, if the page contains an !applet? tag of the form !applet code=hostile.class .? then the proxy not only replaces hostile.class with BrowserServer.class (as described in Section 3), but also adds a parameter tag to the HTML page, like this: !applet code=BrowserServer.class .? !param name=ContactAddress value=address? Parameter tags are tags that contain name/value pairs. This one assigns an address value, which the proxy generates to be unique, to be the value of ContactAddress. The !param? tags that appear between an !applet? tag and its terminator (!/applet?, not shown) are used to specify parameters to the applet when it is run. In this case, the BrowserServer.class object (i.e., the remote Applet object of the graphics server; see Section 3.1.2) looks for the ContactAddress field in its parameters and obtains the address assigned by the proxy. Once the BrowserServer object is initialized and prepared to service requests from the playground, it binds a remote reference to itself to the address assigned by the proxy; this binding is stored in an RMI name server [13]. The proxy remembers what address it assigned to each !applet? tag and provides this address to the playground in a similar fashion. That is, the proxy loads applets into the playground by sending to the playground an HTML page with identical ContactAddress !param? tags to what it forwarded to the browser (for simplicity, this step is not shown in Figure 3). A JVM on the playground loads the referenced applets (via the proxy) and uses the corresponding !param? tags provided with each to look up the corresponding graphics servers in the RMI name server. As discussed previously, the security goal of our system is to protect resources in the protected domain from hostile applets that are downloaded by users in that domain. We limit our attention to protecting data that users do not offer to hostile applets. Protecting data that users offer to hostile applets by, e.g., typing it into the applet's interface, must be achieved via other protections that are not our concern here (though we can utilize them on our playground if avail- able). 5.1 Requirements Achieving strong protection for the domain's private resources relies on at least the following two distinct requirements. 1. Prevent the JVM in the user's browser from loading any classes from the network. If this is achieved, then untrusted code can never be loaded into the browser (unless the browser machine's local class files are maliciously altered, a possibility that we do not consider here). Later in this section we describe how by disabling network class loading, many classes of attacks that have been successfully mounted on JVMs are prevented by our system. 2. Prevent untrusted applets running on the playground from accessing valuable resources. Because we assume that untrusted applets might circumvent the language-based protections on the playground, this requirement can be met only by relying on underlying operating system protections on the play- ground, or preferably by isolating the playground from valuable user resources. Below we describe alternatives for meeting these requirements in a playground system. 5.1.1 Preventing network class loading Preventing network class loading by the browser can be achieved in one of two ways in our system. Trusted proxy One approach is to depend on the proxy to rewrite all incoming !applet? tags on HTML pages to point to the graphics server class (stored locally to the browser), and to intercept and deny entry to any class files destined for protected machines (as in [10]). Rewriting all !applet? tags ensures that the browser is never directed by an incoming page to load anything but the trusted graphics server. 3 More- over, if the playground passes an unknown class to 3 This is not strictly true: in JavaScript-enabled browsers, an !applet? tag may be dynamically generated when the page is the graphics server (e.g., as a parameter to a remote method call) the graphics server is unable to load the class because the class' passage is denied by the proxy. This approach works with any browser "off-the-shelf": it requires no changes to the browser beyond specifying the proxy as the browser's HTTP and SSL proxy, which can typically be done using a simple preferences menu in the browser. A disadvantage, however, is that the proxy becomes part of the trusted computing base of the system. For this reason, the proxy must be written and protected carefully, and we refer to this approach as the "trusted proxy" approach. Untrusted proxy A second approach to preventing the browser from loading classes over the network is to directly disable network class loading in the browser. The main disadvantage of this approach is that it requires either configuration or source-code changes to all browsers in the protected domain. In particular, for most popular browsers today (including Netscape Communicator and Internet Explorer 4.0), a source-code change seems to be required to achieve this, but we expect such changes to become easier as browser's security policies become more configurable. An advantage of this approach is that it excludes the proxy from the trusted computing base of the system, and hence we call this the "untrusted proxy" approach. In this approach only the browser and the graphics server classes are in the trusted computing base. To more precisely show how the above approaches prevent network class loading by the browser, below we describe what causes classes to be loaded from the network and how our approaches prevent this. 1. Section 3 briefly described the process by which a browser loads an applet specified in an !applet? tag in an HTML page. As described in Sections 3 and 4.3, the proxy modifies the code attribute of each !applet? tag to reference the trusted graphics server applet that is stored locally to the browser. Thus, the browser is coerced into always loading the graphics server applet as its ap- plet. In the untrusted proxy approach, this coercion serves merely a functional purpose-i.e., running the graphics server to which the playground applet can connect-but security is not threatened if the proxy fails to rewrite the !applet? tag. In interpreted by the browser, possibly in response to user action; such a tag is not visible to the proxy and thus cannot be rewrit- ten. The retrieval of the applet must then be blocked by the proxy when it is attempted. the trusted proxy approach, this coercion is central to both security and function. 2. Once an applet is loaded and started, the applet class loader in the browser loads any classes referenced by the applet. If a class is not in the core Java library or stored locally, the applet class loader would normally retrieve the class over the network. However, in our approach, the applet class loader needs to load only local classes, because the graphics server applet refers only to other local classes. 3. As described in Section 3, the modified applet on the playground invokes remote methods in the server. Because our present implementation uses RMI to carry out these invocations, the RMI class loader can load additional classes to pass parameters and return values. As above, the RMI class loader first looks in local directories to find these classes before going to the network to retrieve them. It is possible that the playground applet passes an object whose class is not stored locally on the browser machine (particularly if the playground is corrupted). In the trusted proxy approach, the RMI class loader goes to the network, via the proxy, to retrieve the class, but the class is detected and denied by the proxy. In the untrusted proxy ap- proach, the RMI class loader returns an exception immediately upon determining that the class is not available locally. 5.1.2 Isolating untrusted applets Once untrusted applets are diverted to the play- ground, security relies on preventing those applets from accessing protected resources. A first step is for the playground's JVMs (and thus the applets) to execute under accounts different from actual users' accounts and that have few permissions associated with them. For some resources, e.g., user files available to the playground, this achieves security equivalent to that provided by the access control mechanisms of the playground's operating system. (Similarly, JVMs on the playground execute under different accounts, to reduce the threat of inter-JVM attacks; see Section 5.3.) If the complete compromise of the playground is feared, then further configuration of the network may provide additional defenses. For example, if network file servers are configured to refuse requests from the playground (and if machines' requests are authenti- cated, as with AFS [6]), then even the total corruption of the playground does not immediately lead to the compromise of user files. Similarly, if all machines in the protected domain are configured to refuse network connections from the playground, except to designated ports reserved for browsers' graphics servers to listen, then the compromise of the playground should gain little for the attacker. In the limit, an organization's playground can be placed outside its firewall, thereby giving applets no greater access than if they were run on the servers that served them. However, because most firewalls disallow connections from outside the firewall to inside, additional steps may be necessary so that communication can proceed uninhibited between the graphics server in the browser and the applet on the playground. In particular, RMI in Java 1.1 (used in our proto- opens network connections between the browser and the playground in both directions, i.e., from the browser to the playground and vice versa. One approach to enable these connections across a firewall is to multiplex them over a single connection from the graphics server to the playground (i.e., from inside to outside). This can be achieved if both the graphics server and the playground applet interpose a customized connection implementation (e.g., by changing the SocketImplFactory), but for technical reasons this does not appear to be possible with all off-the- shelf browsers (e.g., it appears to work with HotJava 1.0 but not Netscape 3.0). Another alternative is to establish reserved ports on which graphics servers listen for connections from playground applets, and then configure the firewall to admit connections from the playground to those ports. 5.2 RMI security Though the requirements discussed in Section 5.1 are necessary for security in our system, they may not be sufficient. In particular, RMI is a relatively new technology that could conceivably present new vulner- abilities. A first step toward securing RMI is to support authenticated and encrypted transport, so that a network attacker cannot alter or eavesdrop on communication between the browser and the playground. This can also be achieved by interposing encryption at the object serialization layer (see [12]). A more troubling threat is possible vulnerabilities in the object serialization routines that are used to marshal parameters to and return values from remote method invocations. In the worst case, a corrupted playground could conceivably send a stream of bytes that, when unserialized at the browser, corrupt the type system of the JVM in the browser. Here our decision to generally pass only primitive data types (e.g., integers, strings) as parameters to remote method invocations (see Section 4.2) would seem to be fortu- itous, as it greatly limits the number structurally interesting classes that the attacker has at its disposal for attempting such an attack. However, the possibility of a vulnerability here cannot yet be ruled out, and several research efforts are presently examining RMI in an effort to identify and correct such problems. This process of public scrutiny is one of the main advantages to building our system from public and widely used components. 5.3 Resistance to known attacks One way to assess the security of our approach is to examine it in the face of known attack types. In this section we review several classes of attack that applets can mount, and describe the extent to which our system defends against them. Accessing and modifying protected resources Several bugs in the type safety mechanisms of Java have provided ways for applets to bypass Java sand- boxes, including some in popular browsers [1, 9]. These penetrations typically enable the applet to perform any operation that the operating system allows, including reading and writing the user's files and opening network connections to other machines to attack them. We anticipate that in the foreseeable future, type safety errors will continue to exist, and therefore we must presume that applets running on our playground may run unconstrained by the sandbox. How- ever, they are still confined by the playground operating system's protections and, ultimately, to attack only those resources available to the playground. In Section 5.1.2 we described several approaches to limit what resources are available to applets on the play- ground. We expect that through proper network and operating system configuration, hostile applets can be effectively isolated from protected resources. Denial of service In a denial of service attack, a hostile applet might disable or significantly degrade access to system resources such as the CPU, disk, net-work and interactive devices. Ladue [7] presents several such applets, e.g., that consume CPU even after the user clicks away from the applet origin page, that monopolize system locks, or that pop up windows on the user's screen endlessly. Using our Java playground, most of these applets have no effect on the protected domain, and only affect the playground machine. However, an applet that pops up windows endlessly causes the graphics server running in the user's browser to create an infinite stream of windows. Uncontrolled, this may prevent access to the user terminal altogether and require that the user reboot her machine or otherwise shut off her browser. One approach to defend against this is to configure the graphics server and/or the playground to limit the number or rate of window creations. In another type of denial of service, an applet may deny service to other applets within the JVM, e.g., by killing off others' threads [7]. Although the sandbox mechanisms of most browsers are intended to separate applets in different web pages from one another, several ways of circumventing this separation have been shown [1, 7]. This can be prevented in our system if the applets for each page run in a separate JVM on the playground under a separate user account, and hence are unable to directly affect applets from another page (except by attacking the playground itself). Violating privacy The Java security policy in browsers is geared towards maintaining user privacy by disallowing loaded applets access to any local in- formation. In some cases, however, a Java applet can reveal a lot about a user whose browser executes it. For example, in [7], Ladue presents an applet that uses a sendmail trick to send mail on the user's behalf to a sendmail daemon running on the applet's server. When this applet is downloaded onto a Unix host (running the standard ident service), this mail identifies not only the user's IP address, but also the user's account name. In our system, the applet runs on the playground machine under an account other than the user's, and the information that it can reveal is limited only to what is available on the playground. 6 Limitations In our experience, our system is transparent to users for most applets. There are, however, classes of applets for which our playground architecture is not transparent, and indeed our system may be unable to execute certain applets at all. In particular, the remote interface supported by the graphics server supports the passage of certain classes as parameters and return values of its remote methods. If the code running on the playground attempts to pass an object parameter whose class is an unknown subclass of the expected parameter class, then the browser is required to load that subclass to unserialize that parameter. However, because class loading from the network is prevented in our system (see Section 5.1.1), the load does not complete and an exception is generated. At the time of this writing, the number of applets that cannot be supported due to this limitation does not seem significant; indeed, we have yet to encounter an applet for which this is a problem. A second limitation of our approach is that by moving the applet away from the machine on which the user's browser executes, the applet's I/O incurs the overhead of communicating over the network. Our experience indicates that for many applets, this cost is barely noticeable over typical local area network links such as a 10-Mbit/s Ethernet. However, for applets whose output involves intensive I/O operations, e.g., low-level image filtering (see Section 4.2), this overhead can be considerable. The emergence of more functional applets raises new challenges in transparently executing them on a playground. For example, one can envision a text editor applet that can be downloaded from the network and then used to compose a document (and save it to disk). Such an applet is inconsistent with the sandbox policies implemented in Netscape 3.X and Internet Explorer 3.X, because network-loaded applets are not allowed to write files. However, it may well be possible with the more flexible policies implemented in recently (or soon-to-be) released versions of these products (e.g., [14, 4]). Our primary direction of on-going work is extending our playground architecture to support such applets when they become available, while still offering protections to data not intended for the applet. 7 Conclusion This paper presented a novel approach to protecting hosts from mobile code and an implementation of this approach for Java applets. The idea behind our approach is to execute mobile code in the sanitized environment of an isolated machine (a "playground") while using the user's browser as an I/O terminal. We gave a detailed account of the technology to allow transparent execution of Java applets separately from their graphical interface at the user's browser. Using our system, users can enjoy applets downloaded from the network, while exposing only the isolated environment of the playground machine to untrusted code. Although we presented the playground approach and technology in the context of Java and applets, other mobile-code platforms may also utilize it. Acknowledgements We are grateful to Drew Dean, Ed Felten, Li Gong and the anonymous referees for helpful comments. --R Java security: From Hotjava to Netscape and beyond. Java in a Nutshell Java security: Present and near future. Going beyond the sandbox: An overview of the new security architecture in the Java TM Development Kit 1.2. Secure mobile code management: Enabling Java for the enterprise. Scale and Performance in a Distributed File System. Pushing the limits of Java security. The Java Virtual Machine Specification Java Security: Hostile Ap- plets Blocking Java applets at the firewall. Safe kernel extensions without run-time checking Extensible security architectures for Java. --TR --CTR F. M. T. Brazier , B. J. Overeinder , M. van Steen , N. J. E. 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Levy, SpyProxy: execution-based detection of malicious web content, Proceedings of 16th USENIX Security Symposium on USENIX Security Symposium, p.1-16, August 06-10, 2007, Boston, MA Pieter H. Hartel , Luc Moreau, Formalizing the safety of Java, the Java virtual machine, and Java card, ACM Computing Surveys (CSUR), v.33 n.4, p.517-558, December 2001

security;remote method invocation;mobile code

357769

An Optimal Hardware-Algorithm for Sorting Using a Fixed-Size Parallel Sorting Device.

AbstractWe present a hardware-algorithm for sorting $N$ elements using either a p-sorter or a sorting network of fixed I/O size $p$ while strictly enforcing conflict-free memory accesses. To the best of our knowledge, this is the first realistic design that achieves optimal time performance, running in $\Theta ( {\frac{N \log N}{p \log p}})$ time for all ranges of $N$. Our result completely resolves the problem of designing an implementable, time-optimal algorithm for sorting $N$ elements using a p-sorter. More importantly, however, our result shows that, in order to achieve optimal time performance, all that is needed is a sorting network of depth $O(\log^2 p)$ such as, for example, Batcher's classic bitonic sorting network.

Introduction Recent advances in VLSI have made it possible to implement algorithm-structured chips as building blocks for high-performance computing systems. Since sorting is one of the most fundamental computing problems, it makes sense to endow general-purpose computer systems with a special-purpose parallel sorting device, invoked whenever its services are needed. In this article, we address the problem of sorting N elements using a sorting device of I/O size p, where N is arbitrary and p is fixed. The sorting device used is either a p-sorter or a sorting network of fixed I/O size p. We assume that the input as well as the partial results reside in several constant-port memory modules. In addition to achieving time-optimality, it is crucial that we sort without memory access conflicts. In real-life applications, the number N of elements to be sorted is much larger than the fixed size p that a sorting device can accommodate. In such a situation, the sorting device must be used repeatedly in order to sort the input. The following natural question arises: "How should one schedule memory accesses and the calls to the sorting device in order to achieve the best possible sorting performance?" Clearly, if this question does not find an appropriate answer, the power of the sorting device will not be fully utilized. A p-sorter is a sorting device capable of sorting p elements in constant time. Computing models for a p- sorter do exist. For example, it is known that p elements can be sorted in O(1) time on a p \Theta p reconfigurable Work supported by ONR grant N00014-97-1-0526, NSF grants CCR-9522093 and ECS-9626215, and by Louisiana grant LEQSF(1996-99)-RD-A-16. y Department of Computer Science, Old Dominion University, Norfolk, VA 23529-0162, USA z Istituto di Elaborazione dell'Informazione, C.N.R., Pisa 56126, ITALY x Department of Computer Science, University of Texas at Dallas, Richardson, Richardson, mesh [3, 7, 8]. Beigel and Gill [2] showed that the task of sorting N elements, N - p, N log N log p calls to a p-sorter and presented an algorithm that achieves this bound. However, their algorithm assumes that the p inputs to the p-sorter can be fetched in unit time, irrespective of their location in memory. Since, in general, the address patterns of the operands of p-sorter operations are irregular, it appears that the algorithm of [2] cannot realistically achieve the time complexity of \Theta N log N log p , unless one can solve in constant time the addressing problem inherent in accessing the p inputs to the p-sorter and in scattering the output back into memory. In spite of this, the result of [2] poses an interesting open problem, namely that of designing an implementable \Theta N log N log p time sorting algorithm that uses a p-sorter. Consider an algorithm A that sorts N elements using a p-sorter in O(f(N; p)) time. It is not clear that algorithm A also sorts N elements using a sorting network T of I/O size p in O(f(N; p)) time. The main reason is that the task of sorting p elements using the network T requires O(D(T proportional to the depth D(T ), which is the maximum number of nodes on a path from an input to an output, of the network. Thus, if each p-sorter operation is replaced naively by an individual application of T , the time required for sorting becomes O(D(T ) \Delta f(N; p)). To eliminate this O(D(T )) slowdown factor, the network must be used in a pipelined fashion. In turn, pipelining requires that sufficient parallelism in the p-sorter operations be identified and exploited. Recently, Olariu, Pinotti and Zheng [9] introduced a simple but restrictive design - the row merge model - and showed that in this model N elements can be sorted in \Theta log N time using either a p-sorter or with a sorting network of I/O size p. To achieve better sorting performance, a new algorithm-structured architecture must be designed. This involves devising a sorting algorithm suitable for hardware implementation and, at the same time, an architecture on which the algorithm can be executed directly. Such an algorithm-architecture combination is commonly referred to as a hardware-algorithm. The major contribution of this article is to present the first realistic hardware-algorithm design for sorting an arbitrary number of input elements using a fixed-size sorting device in optimal time, while strictly enforcing conflict-free memory accesses. We introduce a parallel sorting architecture specially designed for implementing a carefully designed algorithm. The components of this architecture include a parallel sorting device, a set of random-access memory modules, a set of conventional registers, and a control unit. This architecture is very simple and feasible for VLSI realization. We show that in our architectural model N elements can be sorted in \Theta N log N log p time using either a p-sorter or a sorting network of fixed I/O size p and depth O(log 2 p). In conjunction with the theoretical work of [2], our result completely resolves the problem of designing an implementable, time optimal, algorithm for sorting N elements using a p-sorter. More importantly, however, our result shows that in order to achieve optimal sorting performance a p-sorter is not really necessary: all that is needed is a sorting network of depth O(log 2 p) such as, for example, Batcher's classic bitonic sorting network. As we see it, this is exceedingly important since any known implementation of a p-sorter requires powerful processing elements, whereas Batcher's bitonic sort network uses simple comparators. Architectural Assumptions In this section we describe the architectural framework within which we specify our optimal sorting algorithm using a fixed-size sorting device. We consider that a sequential sorting algorithm is adequate for the case . Consequently, from now on, we assume that This assumption implies that just for addressing purposes we need at least 2 log p bits. 1 For the reader's convenience, Figure 1 depicts our design for 9. To keep the figure simple, control signal lines are not shown. The basic architectural assumptions of our sorting model include: R R R R R R R R R Memory Modules AR AR AR AR AR AR AR AR AR Sorting Device Control Unit Figure 1: The proposed architecture for (i) A data memory organized into p independent, constant-port, memory modules Each word is assumed to have a length of w bits, with w - 2 log p. We assume that the N input elements are distributed evenly, but arbitrarily, among the p memory modules. The words having the same address in all memory modules are referred to as a memory row. Each memory module M i is randomly addressed by an address register AR i , associated with an adder. Register AR i can be loaded with a word read from memory module M i or by a row address broadcast from the CU (see below). (ii) A set of data registers, R i , (1 - i - p), each capable of storing a (w We refer to the word stored in register R i as a composed word, since it consists of three fields: ffl an element field of w bits for storing an element, ffl a long auxiliary field of log p bits, and ffl a short auxiliary field of 0:5 log p bits. 1 In the remainder of this article all logarithms are assumed to be base 2. These fields are arranged such that the element field is to the left of the long auxiliary field, which is to the left of the short auxiliary field. Each field of register R i can be loaded independently from memory module M i , from the i-th output of the sorting device, or by a broadcast from the CU. The output of register R i is connected to the i-th input of the sorting device, to the CU, and to memory module M i . We assume that: ffl In constant time, the p elements in the data registers can be loaded into the address registers or can be stored into the p modules addressed by the address registers. ffl The bits of any field of register R i , (1 - i - p), can be set/reset to all 0's in constant time. ffl All the fields of data register R i , (1 - i - p), can be compared with a particular value, and each of the individual fields can be set to a special value depending on the outcome of the comparison. Moreover, this parallel compare-and-set operation takes constant time. (iii) A sorting device of fixed I/O size p, in the form of a p-sorter or of a sorting network of depth O(log 2 p). We assume that the sorting device provides data paths of width w+ 1:5 log p bits from its input to its output. The sorting device can be used to sort composed words on any combination of their element or auxiliary fields. In case a sorting network is used as the sorting device, it is assumed that the sorting network can operate in pipelined fashion. (iv) A control unit (CU, for short), consisting of a control processor capable of performing simple arithmetic and logic operations and of a control memory used to store the control program as well as the control data. The CU generates control signals for the sorting device, for the registers, and for memory accesses. The CU can broadcast an address or an element to all memory modules and/or to the data registers, and can read an element from any data register. We assume that these operations take constant time. Described above are minimum hardware requirements for our architectural model. In case a sorting network is used as the sorting device, one can use a "half-pipelining" scheme: the input to the network is provided in groups of D rows. The next group is supplied only after the output of the previous group is obtained. D is the depth of the sorting network. For the sorting network to operate at full capacity, one may add an additional set of address (resp. data) registers. One set of address (resp. data) registers is used for read operations, while the order set is used for write operations; both operations are performed concurrently. Let us now estimate the VLSI area that our design uses for hardware other than data memory, the sorting device and the CU under the word model, i.e. assuming that the word length w is a constant. We exclude the area taken by the CU: this is because in a high-performance computer system, one of the processors can be assigned the task of controlling the parallel sorting subsystem. Clearly, the extra area is only that used for the address and the data registers, and this amounts to O(p) - which does not exceed the VLSI area of any implementation of a p-sorter or of a sorting network of I/O size p. We do not include the VLSI area for running the data memory address bus, which has a width of log N bits, and the control signal lines to data memory and to the sorting device, since they are needed for any architecture involving a data memory and a sorting device. It should be pointed out that for any architecture that has p memory modules involving a total of N - p 2 words, the control circuitry itself requires at least \Omega\Gamma0/1 flog p; log N area. Since the operations performed by the control processor are simple, we can assume that it takes constant area. The length of the control memory words is at least log N which is the length of data memory addresses. As will become apparent, our algorithms require O( N of data memory, and consequently, the control memory words have length O(log N ). The control program is very simple and takes constant memory. However, O( N are used for control information, which can be stored in data memory. 3 An Extended Columnsort Algorithm In this section, we present an extension of the well known Columnsort algorithm [5]. This extended Column- algorithm will be implemented in our architectural model and will be invoked repeatedly when sorting a large number of elements. There are two known versions of Columnsort [5, 6]: one involves eight steps, the other seven. We provide an extension of the 8-step Columnsort, because the 7-step version does not map well to our architecture. Columnsort was designed to sort, in column-major order, a matrix of r rows and s columns. The "classic" Columnsort contains 8 steps. The odd-numbered steps involve sorting each of the columns of the matrix independently. The even-numbered steps permute the elements of the matrix in various ways. The permutation of Step 2 picks up the elements in column-major order and lays them down in row-major order. The permutation of Step 4 is just the reverse of that in Step 2. The permutation of Step 6 amounts to a b rc shift of the elements in each column. The permutation of Step 8 is the reverse of the permutation in Step 6. The 8-step Columnsort works under the assumption that r - In [5], Leighton poses as an open problem to extend the range of applicability of Columnsort without changing the algorithm "drastically". We provide such an extension. We show that one additional sorting step is necessary and sufficient to complete the sorting in case r - s(s \Gamma 1). Our extension can be seen as trading one additional sorting step for a larger range of applicability of the algorithm. Figure 2: Step by step application of the extended Columnsort algorithm. The first eight steps correspond to the classic 8-step Columnsort. Figure 2 shows a matrix of r rows and s columns with which the condition r - is not satisfied. The first eight steps of this example correspond to the 8-step Columnsort algorithm which does not produce a sorted matrix. By adding one more step, Step 9, in which the elements in each column are sorted, we obtain an extended Columnsort algorithm. We assume a matrix M of r rows and s columns, numbered from 0 to r \Gamma 1 and from 0 to s \Gamma 1, respectively. Our arguments rely, in part, on the following well-known gem of computer science mentioned by Knuth [4]. Proposition 1 Let M be a matrix whose rows are sorted. After sorting the columns, the rows remain sorted. The following result was proved in [5]. Lemma 1 If some element x ends up in position M [i; j] at the end of Step 3, then x has rank at least The following result was mentioned without proof in [5]. Lemma 2 If element x ends up in position M [i; j] at the end of Step 3, then its rank is at most si Proof. We are interested in determining a lower bound on the number of elements known to be larger than or equal to x. For this purpose, we note that since at the end of Step 3, element x was in position are known to be larger than of equal to x. Among these, at most s are known to be smaller than of equal to s \Gamma j elements in their columns at the end of Step 1. The remaining must be smaller than or equal to s other elements in their column at the end of Step 1. Consequently, x is known to be smaller than or equal to at least rs elements of M . It follows that the rank of x is at most si + sj, as claimed. 2 For later reference, we now choose r such that Lemma 3 If some element x ends up in column c at the end of Step 4, then the correct position of x in the sorted matrix is in one of the pairs of columns 1). Proof. Consider, again, a generic element x that ended up in position M (i; j) at the end of Step 3. The permutation specific to Step 4 guarantees that x will be moved, in Step 4, to a position that corresponds, in the sorted matrix, to the element of rank si + j. In general, this is not the correct position of x. However, as we shall prove, x is "close" to its correct position in the following sense: if x is in column c at the end of Step 4, then in the sorted matrix x must be in one of the pairs of columns 1). Recall that by virtue of Lemmas 1 and 2, combined, x has rank no smaller than si no larger than si sj. Moreover, simple algebraic manipulations show that Now consider the elements y and z of ranks si respectively. The number N (y; z) of elements of the matrix M lying between y and z, in sorted order, is: and so, by (2), we have Observe that equation (4) implies that y and z must lie in adjacent columns of the sorted matrix. As we saw, at the end of Step 4, x lies in the position corresponding to the element of rank is in the sorted matrix. confirms that x lies somewhere between y and z. Assume that x lies in column c at the end of Step 4. Thus, the correct position of x is in one of the columns c \Gamma 1 or c in case z is in the same column as x, and in one of the columns c or c y is in the same column as x. 2 Lemma 4 The rows of M are sorted at the end of Step 4. Proof. Consider an arbitrary column k, (0 - k - s \Gamma 1), at the end of Step 3. The permutation specified in Step 4 guarantees that the first r s elements in column k will appear in positions k; k+s; k+2s; column 0; the next group of r s elements will appear in positions k; k+s; k+2s; on. Since the columns were sorted at the end of Step 3, it follows that all the rows k; k+s; of M are sorted at the end of Step 4. Since k was arbitrary, the conclusion follows. 2 Lemma 5 If some element x is in the bottom half of column c at the end of Step 5, then its correct position in the sorted matrix is in one of the columns c or c + 1. Proof. By Lemma 3, we know that the correct position of x is in one of the pairs of columns or 1). Thus, to prove the claim we only need to show that x cannot be in column c \Gamma 1. For this purpose, we begin by observing that by Proposition 1 and by Lemma 4, combined, the rows and columns are sorted at the end of Step 5. Now, suppose that element x ends up in row t, t - b rc, at the end of Step 5. If x belongs to column c \Gamma 1 in the sorted matrix, then all the elements of the matrix in columns and c belonging to rows 0; must belong to column c \Gamma 1 or below. By Lemma 3, all elements that are already in columns 0; must belong to columns 0; in the sorted matrix. Thus, at least additional elements must belong to column c \Gamma 1 or below, a contradiction. 2 In a perfectly similar way one can prove the following result. Lemma 6 If some element is in the top half of column c at the end of Step 5, then its correct position in the sorted matrix is in one of the columns c \Gamma 1 or c. Now, suppose that we find ourselves at the end of Step 8 of the 8-step Columnsort. Lemma 7 Every item x that is in column c at the end of Step 8 must be in column c in the sorted matrix. Proof. We begin by showing that no element in column c can be in column We proceed by induction on c. The basis is trivial: no element in column 0 can lie in the column to its left. Assume that (5) is true for all columns less than c. In other words, no element that ends up in one of the columns at the end of Step 8, can lie in the column to its left. We only need to prove that the statement also holds for column c. To see that this must be the case, consider first an element u that lies in the bottom half of column c at the end of Step 8. At the end of Step 5, u must have been either in the bottom half of column c or in the top half of column c + 1. If u belonged to the bottom half of column c then, by Lemma 6, it must belong to columns c or c + 1 in the sorted matrix. If u belonged to the top half of column c must belong to columns c or c + 1 in the sorted matrix. Therefore, in either case, u cannot belong to column c \Gamma 1. Next, consider an element v that lies in the top half of column c at the end of Step 8. If v belonged to all the elements in the bottom half of column c \Gamma 1 as well as the elements occurring above v in column c must belong to column c \Gamma 1. By the induction hypothesis, no element that lies in column at the end of Step 8 can lie in column c \Gamma 2. By Lemmas 5 and 6 combined, no element that lies in the top half of column c \Gamma 1 can belong to column c. But now, we have reached a contradiction: column must contain more than r elements. Thus, (5) must hold. What we just proved is that no element in a column can belong to the column to its left. A symmetric argument shows that no element belongs to the column immediately to its right, completing the proof. 2 By Lemma 7, one more sorting step completes the task. Thus, we have obtained a 9-step Columnsort that trades an additional sorting step for a larger range of r versus s. Theorem 1 The extended 9-step Columnsort algorithm correctly sorts an r \Theta s matrix such that r - s(s \Gamma 1). 4 The Basic Algorithm In this section we show how to sort, in row-major order, m, using our architectural model while enforcing conflict-free memory accesses. The resulting algorithm, referred to as the basic algorithm, will turn out to be the first stepping stone in the design of our time-optimal sorting algorithm. The basic algorithm is an implementation of the extended Columnsort discussed in Section 3 with Our presentation will focus on the efficient use of a generic sorting device of I/O size p. With this in mind, we shall keep track of the following two parameters that will become key ingredients in evaluating the running time of the algorithm: ffl the number of calls to the sorting device, and ffl the amount of time required by all the data movement tasks that do not involve sorting. Assume that we have to sort, in row-major order, the elements in memory rows. The case 2 is perfectly similar. We assume, without loss of generality, that the input is placed, in some order, in memory rows a for some integer a - 0. The sorted elements will be placed in memory rows b 2 such that the ranges [a do not overlap. Step 1: Sort all the rows independently. This step consists of the following loop: do read the i-th memory row and sort it in non-decreasing order using the sorting device; be the resulting sorted sequence; for all do in parallel store x j in the i-th word of memory module M j endfor endfor 2 calls to the sorting device and O(p 1 data movement not involving sorting. Step 2: Permuting rows. The permutation specific to Step 2 of Columnsort prescribes picking up the elements in each memory row and laying them down column by column. For an illustration, consider the case with the initial element distribution featured in the following matrix: At the end of Step 2, the permuted matrix reads: A careful examination of the permuted matrix reveals that consecutive elements in the same memory row will end up in the same memory module (e.g. elements 1, 2, 3 will occur in memory module M 1 ). Therefore, in order to achieve the desired permutation without memory-access conflicts, one has to devise a different way of picking up the elements in various memory rows. For this purpose, we find it convenient to view each element x stored in a memory module as an ordered triple hx; row(x); module(x)i where row(x) and module(x) stand for the identity of the memory row and of the memory module, respectively, containing x. Further, we let row(x)jmodule(x) denote the binary number obtained by concatenating the binary representations of row(x) and module(x). The details are spelled out in the following procedure. procedure begin for all do in parallel read the -th word of memory module M j endfor using the sorting device, sort the p elements in non-decreasing order of row(x)jmodule(x); be the resulting sorted sequence; for all do in parallel store x j in the -th word of memory module M j endfor endfor Clearly, this procedure involves p 1 iterations. In each iteration, p words are read, one from each memory module, sorted, and then written back into memory, one word per module, with no read and write memory access conflicts. It would seem as though each memory module requires an arithmetic unit to compute the address of the word to be accessed in each iteration. In fact, as we now point out, such arithmetic capabilities are not required. Specifically, we can use p 1 memory rows to store "offsets" used for memory access operations. For the above example, the offsets are At the beginning of Step 2 all the address registers contain a + 1. In the first iteration, the entries in the first row of the offset matrix are added to the contents of the address registers, guaranteeing that the correct word in each memory module is being accessed. As an illustration, referring to (7), we note that the offsets in the first row indicate that the words involved in the read operation will be found at address a memory module M 1 , address a memory module M 3 , and so on. The key observation for understanding what happens in all the iterations is that in any column of the offset matrix (7), once the entry in the first row is available, the subsequent elements in the same column can be generated by modulo 2 arithmetic. In our architecture, this computation can be performed by the adder associated with each address register. In turn, this observation implies that, in fact, the offset matrix need not be stored at all, as its entries can be generated on the fly. Yet another important point to note is that each ordered triple hx; row(x); module(x)i is a composed word with three fields and that the composed words are sorted using the combination of two fields, namely, row(x) and module(x). Clearly, module(x) has log p bits, but it seems that in order to represent row(x) we need log N bits. Actually, we can replace row(x) with the address offset contained in the offset matrix discussed above. Since the entries in that matrix are integers no larger than p 1 logp bits are sufficient. Therefore, the concatenation row(x)jmodule(x) involves 1:5 log p bits. From the above discussion it is clear that Step 2 requires p 1 2 calls to the sorting device and that the time spent on data movement operations not involving sorting is bounded by O(p 1 Step 3: Same as Step 1. Step 4: The permutation of Step 2 is performed in reverse; the permuted set of words are stored in rows a . Step 5: Same as Step 1. Step Shifting rows. We shall permute the elements slightly differently from the way specified by Columnsort. However, it is easy to verify that the elements supposed to end up in a given row, indeed end up in the desired row. Since Step 7 sorts the rows, the order in which the elements are placed in the row in Step 6 is immaterial. The permutation of Step 6 is best illustrated by considering a particular example. Specifically, the permutation specified by Step 6 of Columnsort involving the three rows shown in (6) is: Our permutation is a bit different: Assume that the p 1 consecutive input rows are stored in memory starting from memory row a + 1. In addition, we assume that memory row a is available to us. Some of its contents are immaterial and will be denoted by "?"s. The motivation is anchored in the observation that in Step 7 we do not have to sort memory rows a and a the elements in these rows will be sorted in Step 9. Consequently, the only rows that have to be sorted in Step 7 are rows a 1. The details follow. procedure ROW SHIFT begin for all \Sigma p\Upsilon , do in parallel read the i-th word of memory module M j and store it in the (i \Gamma 1)-th word of memory module M j endfor endfor It is important to note that, in our implementation, Step 6 does not involve sorting. However, O(p 1 time is spent on data movement operations that do not involve sorting. Step 7: Same as Step 1. Step 8: This is simply the reverse of the data movement in Step 6. Step 9: Same as Step 1. To summarize, we have proved the following result. Theorem 2 A set of p 3 elements stored in p 1 memory rows can be sorted, in row-major order, without memory-access conflicts, in at most 7p 1 2 calls to a sorting device of I/O size p and in O(p 1 data movement not involving sorting. In essentially the same way one can prove the following companion result to Theorem 2. Theorem 3 The task of sorting, in row-major order, a set of mp elements stored in m, memory rows can be performed, without memory-access conflicts, in at most 7m calls to a sorting device of I/O size p and in O(m) time for data movement operations not involving sorting. In the remainder of this section we present an important application of the basic algorithm. Suppose that we wish to merge two sorted sequences an and Our algorithm for merging A and B relies on the following technical result. Lemma 8 Assume that a d n 2 e - b b n 2 c+1 and let be the sequence obtained by merging e and a d n an . Then, no element in the sequence strictly larger than any element in the sequence Proof. We begin by showing that no a i , (1 - i - d ne), is strictly larger than any element in E. The assumption that a d n 2 e - b b n 2 c+1 guarantees that if the claim is false, then some element a i , strictly larger than some element c k , To evaluate the position of the element c k in the sorted sequence C, observe that all the b nc elements in C that come from A are known to be larger than or equal to a i and, therefore, strictly larger than c k . Consequently, if n is even, then b nc elements in C are larger than c k , implying that k - b nc, a contradiction. On the other hand, if n is odd, then d ne = b nc+1, and, by assumption, b d n 2 e is larger than a i and, therefore, strictly larger than c k . In this case, at least d ne elements in C are strictly larger than c k . It follows that contradicting that c k belongs to E. Next, we claim that no c i , larger than any element in E. Since C is sorted, if the statement is false, then c i ? b k for some k, (k - b nc+1). Notice that all elements of C that come from B are smaller than or equal to b k and, therefore, strictly smaller than c i . It follows that contradicting that c i belongs to D. This completes the proof of the lemma. 2 A mirror argument proves the following companion result to Lemma 8. Lemma 9 Assume that a d n be the sequence obtained by merging no element in the sequence D 2 c is strictly larger than any element in the sequence It is worth noting that Lemmas 8 and 9, combined, show that given two sorted sequences, each of size n, the task of merging them can be handled as follows: we begin by splitting the two sequences into two sequences of size n each, such that no element in the first one is strictly larger than any element in the second one. Once this "separation" is available, all that remains to be done is to sort the two sequences independently. The noteworthy feature of this approach is that it fits extremely well our architecture. 2 and consider a sorted sequence stored in m memory rows stored in m memory rows 1. The goal is to merge these two sequences and to store the resulting sorted sequence in memory rows r A ; r A 1. The details follow. procedure MERGE TWO GROUPS begin if a d mp use the basic algorithm to sort b 1 non-increasing order as c mp - c store the result in memory rows r else use the basic algorithm to sort a 1 ; a non-increasing order as c mp - c store the result in memory rows r do copy memory row r row r A copy memory row r A +m \Gamma i into memory row r B endfor if m is odd then copy the leftmost d pe elements in row r into the leftmost d pe positions of row r A copy the rightmost b pc elements in row r A into the rightmost b pc positions of row r endif endif if m is odd then copy the leftmost d pe elements in memory row r C into the leftmost d pe positions of row r copy the rightmost b pc elements in row r C into the rightmost b pc positions of row r A endif do copy memory row r C copy memory row r C endfor use the basic algorithm to sort memory rows r A ; r A non-decreasing use the basic algorithm to sort memory rows r non-decreasing order It is obvious that procedure MERGE TWO GROUPS can be implemented directly in our architectural model. One point is worth discussing, however. Specifically, the task of sorting a sequence in non-increasing order can be performed in our architecture as follows. The signs of all the elements to be sorted are flipped and the resulting sequence is then sorted in non-decreasing order. Finally, the signs are flipped back to their original value. The correctness of the procedure follows from Lemmas 8 and 9. Moreover, the procedure requires three calls to the basic algorithm. Consider the task of sorting a collection of 2mp memory rows, with m as above. Having partitioned the input into two subgroups of m consecutive memory rows each, we use the basic algorithm to sort each group. Once this is done, we complete the sorting using procedure MERGE TWO GROUPS. Thus, we have the following result. Theorem 4 The task of sorting 2mp, 2 , elements stored in 2m memory rows can be performed in five calls to the basic algorithm and O(m) time for data movement operations not involving sorting. 2 5 An Efficient Multiway Merge Algorithm Consider a collection A =! A 1 ; A sequences, each of size p i We assume that A is stored, top-down, in the order A 1 ; A consecutive memory rows. The multiway merge problem is to sort these sequences in row-major order. The goal of this section is to propose an efficient algorithm MULTIWAY MERGE for the multiway merge problem, and to show how it can be implemented on our architecture. procedure MULTIWAY MERGE(A;m; i); each of size p i Output: the resulting sorted sequence stored in row-major order in mp i\Gamma2 contiguous memory rows. g Step 1. Select a sample S of size mp i\Gamma2 2 from A by retaining every p-th element in each sequence A j , (1 - j - m), and move S to its own dmp i\Gamma4 discussed below; Step 2. by one call to the sorting device else if by one call to the basic algorithm else frecursively multiway merge Sg endif (i\Gamma2)=2 be the sorted version of Step 3. Partition A into p i\Gamma2 each containing at most 2mp elements, as discussed below, and move the elements of A to their buckets without memory access conflicts; Step 4. Sort all the buckets individually using the basic algorithm and procedure MERGE TWO GROUPS; Step 5. Coalesce the sorted buckets into the desired sorted sequence. 2 Notice that if the sample S will be stored in one memory row. The remainder of this section is devoted to a detailed implementation of this procedure on our architecture 5.1 Implementing Step 1 and Step 2 For convenience, we view A as a matrix of size mp i\Gamma2 2 \Theta p, with the t-th element of memory row j being denoted by A[j; t]. The element A[j; p] is termed the leader of memory row j. The goal of Step 1 is to extract a sample S of A by retaining the leader s of every memory row in A, along with the identity k of the subsequence A k to which the leader belongs. In this context, k is referred to as the sequence index of s. Two disjoint groups of dmp i\Gamma4 consecutive memory rows each are set aside to store the sample S and the corresponding set I of sequence indices. In the remainder of this subsection, we view the memory rows allocated to S and I as two matrices of size dmp i\Gamma4 p. The intention is that at the end of Step 1, S[x; y] and I[x; y] store the y)-th leader of A and its sequence index, respectively. To see how Step 1 can be implemented without memory access conflicts, notice that in each memory row the leader to be extracted is stored in memory module M p . For a generic memory row j, the CU interchanges temporarily the elements A[j; p] and A[j; d(j)], where p. (This interchange will be undone at the end of Step 1). Next, dmp i\Gamma4 parallel read operations are performed, each followed by two parallel write operations. The j-th parallel read operation picks up the k)-th word of memory module M k , (1 - k - p), and these p elements are stored in the j-th memory row allocated to S. The second parallel write operation stores the sequence indices of these p elements in the j-th memory row allocated to I. Thus, Step 1 can be implemented in O(mp i\Gamma2 data movement and no calls to the sorting device. The sampling process continues, recursively, until a level is reached where procedure MULTIWAY MERGE is invoked with either which case the corresponding sample set is stored in one memory row and will be sorted in one call to the sorting device, or with 4, in which case the sample set is stored in m memory rows, and will be sorted in one call to the basic algorithm. Since the operation of sorting one row is direct, we only discuss the way the basic algorithm operates in this context. Conceptually, the process of sorting the samples benefits from being viewed as one of sorting the concatenation sjk, where s is a sample element and k its sequence index. Recall that, as described in Section 2, our design assumes that the sorting device provides data paths of size w + 1:5 log p from its inputs to its outputs. This implies that Steps 1, 3, 5, 7, and 9 of the extended Columnsort can be executed directly. To sort a row r of S and the corresponding row r of I, the CU loads, in two parallel read operations, the element field and the short auxiliary field of data register R j , (1 - j - p), with S[r; j] and I[r; j], and the long auxiliary field with 0. Let s r;j and k r;j be the element and its sequence index stored in register R j and let s r;j jk r;j denote their concatenation. Next, the contents of the data registers are supplied as input to the sorting device. Let received by R j after sorting, with s r;j 0 and k r;j 0 stored, respectively in the element and short auxiliary field of R j . In two parallel write operations, the CU stores the element field and the short auxiliary field of each register R j , (1 - j - p), into S[r; j] and I[r; j], respectively. Steps 2, 4, 6, and 8 of the basic algorithm perform permutations. The implementation of Steps 6 and 8 does not involve sorting. In this case, the data movement involving the sample elements and that of the corresponding sequence indices will be performed in two companion phases. Specifically, viewing the sample set S and its corresponding sequence index set I as two matrices, the same permutation is performed on S and I. Steps 2 and 4 of the basic algorithm involve both data movement operations and sorting. The data movement operations in these steps are similar to those in Steps 6 and 8 and will not be detailed any further. Recall that the sorting operations in Steps 2 and 4 of the basic algorithm are performed on the concatenation of the two auxiliary fields storing the relative row number and the column number of the element. Hence, we perform two companion sorting phases, one for permuting the sample elements and the other for permuting sequence indices. Clearly, this can be implemented with the same time complexity. It is easy to confirm that at the end of Step 2 of procedure MULTIWAY MERGE the sample set S is sorted in row-major order. Furthermore, viewed as matrices, I[x; y] is the sequence index of the sample element S[x; y]. Let the sorted version of S be Equation (8) will be used in Step 3 to partition the elements of A into buckets. In order to do so, the leader of each row in A needs to learn its rank in (8). Our next goal is to associate with every memory row in A the rank in S. This task will be carried out in two stages. In the first stage, using the sequence index and the rank of s in S the CU assigns to s a row number row(s) in A. For every s in S, row(s) is either the exact row number from which s was extracted in Step 1 or, in case the leaders of several rows are equal, row(s) achieves a possible reassignment of leaders to rows. The details of the first stage are spelled out in procedure ASSIGN ROW NUMBERS presented below. For convenience, we use the matrix representation of S and I. These operations can be easily implemented using the addresses of words corresponding to S[x; y] and Initially, I contains the sequence indices of samples in S. When the procedure terminates, I[x; y] contains row(s) corresponding to procedure ASSIGN ROW NUMBERS begin do r k := the row number of the first memory row storing the sequence A k endfor 2 e do for do endfor endfor In the second stage, the CU assigns the rank with the memory row row(s) contained in I[x; y]. The operations performed on the matrix representations of S and I can be easily implemented using the addresses of words corresponding to S[x; y] and I[x; y]. Since only read/write operations are used in the procedure described, the total time spent on these operations is bounded by O(mp i\Gamma2 5.2 Implementing Step 3 and Step 4 Once the rank of each leader in A is known, we are ready to partition A into buckets. Our first objective is to construct a collection of buckets such that the following conditions are satisfied: (b1) every element of A belongs to exactly one bucket; (b2) no bucket contains more than 2mp elements; (b3) for every i and j, (1 - strictly larger than any element in B j . Before presenting our bucket partitioning scheme, we need a few definitions. Let s mp (i\Gamma2)=2 be as in (8). The memory row with leader s b is said to be regular with respect to bucket B j , (j Notice that equation (9) guarantees that every memory row in A is regular with respect to exactly one bucket and that the identity of this bucket can be determined by the CU in constant time. Conversely, with respect to each bucket there are exactly m regular memory rows. A memory row r with leader s b in some sequence A k , (1 - k - m), is termed special with respect to bucket B t if, with s a standing for the leader of the preceding memory row in A k , if any, we have a Let the memory rows with leaders s a and s b be regular with respect to buckets respectively, such that j. It is very important to note that equation (10) implies that the memory row whose leader is s b is special with respect to all the buckets Conceptually, our bucket partitioning scheme consists of two stages. In the first stage, by associating all regular and special rows with respect to a generic bucket 2 ), we obtain a set C j of candidate elements . In the second stage, we assign the elements of A to buckets in such a way that the actual elements assigned to bucket B j form a subset of the candidate set C j . Specifically, an element x of a memory row regular with respect to bucket B j is assigned to B j if one of the conditions below is satisfied: s (j \Gamma1)m or s (j \Gamma1)m - x - s jm whenever s (j The elements of A that have been assigned to a bucket by virtue of (11) or (12) are no longer eligible for being assigned to buckets in the remainder of the assignment process. Consider, further, an element x that was not assigned to the bucket with respect to which its memory row is regular. Element x will be assigned to exactly one of the buckets with respect to which the memory row containing x is special. Assume that the memory row containing x is special with respect to buckets with be the smallest index, 1 - n - l(x), for which one of the equations (11) or (12) holds with j n in place of j. Now, x is assigned to bucket B jn . The next result shows that the buckets we just defined satisfy the conditions (b1)-(b3). the conditions (b1), (b2), and (b3). Proof. Clearly, our assignment scheme guarantees that every element of A gets assigned to some bucket and that no element of A gets assigned to more than one bucket. Thus, condition (b1) is verified. Further, notice that by (9) and (10), combined, for every j, (1 2 ), the candidate set C j with respect to bucket B j contains at most 2m memory rows, and, therefore, at most 2mp elements of A. Moreover, as indicated, the elements actually assigned to bucket B j are a subset of C j , proving that (b2) is Finally, equations (11) and (12) guarantee that if an element x belongs to some bucket b j then it cannot be strictly larger than any element in a bucket B k with k. Thus, condition (b3) holds as well. 2 It is worth noting that the preceding definition of buckets works perfectly well even if all the input elements are identical. In fact, if all elements are distinct, one can define buckets in a simpler way. Moreover, in the case of distinct elements, Steps 1-3 of procedure MULTIWAY MERGE can be further simplified. We now present the implementation details of the assignment of elements to buckets. Write s and denote, for every j, (1 2 ), the ordered pair (s (j \Gamma1)m ; s jm ) as the j-th bounding pair. Notice that equations (11) and (12) amount to testing whether a given element lies between a bounding pair. By (b2), no bucket contains more than 2mp elements from A. This motivates us to set aside 2m memory rows for each bucket B j . Out of these, we allocate the first m memory rows to elements assigned to B j coming from regular memory rows with respect to B j ; we allocate the last m memory rows to elements assigned to bucket B j that reside in special memory rows with respect to B j . In addition, we find it convenient to initialize the contents of the 2m memory rows allocated to B j to all +1's. It is important to note that the regular memory rows with respect to a bucket B j are naturally ordered from 1 to m by the order of the corresponding leaders in S. To clarify this last point, recall that by (9) the leaders belonging to bucket B j are s (j \Gamma1)m+1 ; s (j Accordingly, the memory row whose leader is s (j \Gamma1)m+1 is the first regular row with respect to B j , the memory row whose leader is s (j \Gamma1)m+2 is the second regular row with respect to B j , and so on. Similarly, the fact that each sequence A k is sorted guarantees that it may contain at most one special memory row with respect to bucket B j . Now, in case such a special row exists it will be termed the k-th special memory row with respect to B j , to distinguish it from the others. In order to move the elements to their buckets, the CU scans the memory rows in A one by one. Suppose that the current memory row being scanned is row r in some sequence A k . We assume that the leader of row r is s b and that the leader of row r \Gamma 1 is s a . Using equation (9), the CU establishes that row r is regular with respect to bucket B j , where similarly, that the previous memory row is regular with respect to bucket In case row r is the first row of A k , j 0 is set to 1. Next, the elements in memory row r are read into the element fields of the data registers the CU broadcasts to these registers the bounding pair (s (j \Gamma1)m ; s jm ). Using compare-and-set, each register stores in the short auxiliary field a 1 if the corresponding element is assigned to bucket B j by virtue of (11) or (12) and a 0 otherwise. We say that an element x in some data register is marked if the value in the short auxiliary field is otherwise, x is unmarked. Clearly, every element x that is marked at the end of this first broadcast has been assigned to bucket . In a parallel write operation, the CU copies all the marked elements to the corresponding words of the row allocated to bucket B j . Once this is done, using compare-and-set, all the marked elements in the data registers are set to +1 and the short auxiliary fields are cleared. Further, the CU broadcasts to the data registers, in increasing order, the bounding pairs of the buckets us follow the processing specific to bucket B j 0 . Having received the bounding pair data register determines whether the value x stored in its element field satisfies (11) or (12) with j 0 in place of j and marks x accordingly. In a parallel write operation, the CU copies all the marked elements to the corresponding words of the next available memory row allocated to bucket B j . Next, using compare-and-set all the marked elements in the data registers are set to +1, and the short auxiliary fields are cleared. The same process is then repeated for all the remaining buckets with respect to which row r is special. The reader will not fail to note that when the processing of row r is complete, each of its elements has been moved to the bucket to which it has been assigned. Moreover, by (9) and (10) there are, altogether, at most mp i\Gamma2 regular rows and at most mp i\Gamma2 special rows, and so the total time involved in assigning the elements of A to buckets is bounded by O(mp i\Gamma2 no calls to the sorting device. In summary, Step 3 can be implemented in O(mp i\Gamma2 data movement and no calls to the sorting device. In Step 4, the buckets are sorted independently. If a bucket has no more than p 1 memory rows, it can be sorted in one call to the basic algorithm. Otherwise, the bucket is partitioned in two halves, each sorted in one call to the basic algorithm. Finally, the two sorted halves are merged using procedure GROUPS. By Theorem 4, the task of sorting all the buckets individually can be performed in O(mp i\Gamma2 calls to the sorting device and in O(mp i\Gamma2 data movement not involving sorting. 5.3 Implementing Step 5 To motivate the need for the processing specific to Step 5, we note that after sorting each bucket individually in Step 4, there may be a number of +1's in each bucket. We refer to such elements as empty; memory rows consisting entirely of empty elements will be termed empty rows. A memory row is termed impure if it is partly empty. It is clear that each bucket may have at most one impure row. A memory row that contains no empty elements is referred to as pure. The task of coalescing the non-empty elements in the buckets into mp i\Gamma2 consecutive memory rows will be referred to as compaction. For easy discussion, we assume that all sorted buckets are stored in consecutive rows. That is, the non-empty rows of B 2 follow the non-empty rows of B 1 , the non-empty rows of B 3 follow the non-empty rows of B 2 , and so on, assuming that all empty rows have been removed. The compaction process consists of three phases. Phase 1: Let C be the row sequence obtained by concatenating non-empty rows of B j 's obtained in Step 4 of MULTIWAY MERGE in the increasing order of their indices. We partition sequence C into subsequences x such that each C j contains p 1 consecutive rows of C, except the last subsequence C x , which may contain fewer rows. Clearly, x - 2mp i\Gamma3 2 . We use the basic algorithm to sort these subsequences independently. Let the sorted subsequence corresponding to C i be C 0 i with empty rows eliminated for future consideration. Let D be the row sequence obtained by concatenating rows of C 0 's in the increasing order of their indices. We partition sequence D into subsequences y such that each D j contains p 1consecutive rows of D, except the last subsequence D y , which may contain fewer rows. We then use the basic algorithm to sort these subsequences independently. Let the sorted subsequence corresponding to D i be D 0 i with empty rows eliminated. Let E be the row sequence obtained by concatenating rows of D 0 's in the increasing order of their indices. Lemma 11 The preceding row of every impure row, except the last row, of E is a pure row. Proof. We notice the following fact: except the last row of D, every row of D either contains at least p 1non-empty elements, or if it contains fewer than p 1 non-empty elements then its preceding row must be a pure row. This is because that each row of C contains at least one non-empty element. An impure row of D can be generated under one of two conditions: (a) if C j contains fewer than p non-empty elements, then contains only one row, an impure row, with its non-empty elements coming from p 1 impure rows of C j , and (b) if a C j contains more than p non-empty elements, then C 0 contains only one impure row, and its preceding row is a pure row. The lemma directly follows from this fact. 2 Phase 2: This phase computes a set of parameters, which will be used in the next phase. Let w be the total number of (non-empty) rows in E. Assume that the rows of E are located from row 1 through row w. For every j, (1 stand for the number of non-empty elements in the impure memory row c j . The first subtask of Phase 2 is to determine i\Gamma2. Consider a generic impure row c j . To determine n j the CU reads the entire row c j into the data registers R 1 for every k, (1 - k - p), the c j -th word of memory module M k is read into register R k . The long auxiliary field of data register R k is set to k. By using the compare-and-set feature, the CU instructs each register R k to reset this auxiliary field to \Gamma1 if the element it holds is +1 (i.e. empty). Next, the data registers are loaded into the sorting device and sorted in increasing order of their long auxiliary fields. It is easy to confirm that, after sorting, the largest such value k j is precisely the position of the rightmost non-empty element in memory row c j . Therefore, the CU sets . Consequently, the task of computing all the numbers calls to the sorting device and O(p i\Gamma2 read/write operations and does not involve sorting. Once the numbers are available, the CU computes the prefix sums oe This, of course, involves only additions and can be performed by the CU in O(mp i\Gamma2 call to the sorting device. Let e. mod g. Define Phase 3: Construct row group g, of consecutive rows as follows: if ff k\Gamma2 ? 0 then row is the starting row of E k , else row k(p 1 is the starting row of E k ; the ending row of E k , is row k(p 1 and the ending row of E g is row w. Note that E k and E k+1 may share at most two rows. By Lemma 11, for contains is at least (p+1)p2elements, and the last two rows of contains at least p elements. For each g, perform the following operations: (a) sort using the basic algorithm; (b) replace the fi smallest elements by +1's; (c) sort using the basic algorithms; and (d) if ff k ? 0 and k ! g, eliminate the last row. For E g , perform (a), (b) and (c) only. Let k be the row group obtained from E k , and let F be the row sequence obtained by concatenating rows of 's in the increasing order of their indices. F is the compaction of C. Setting selected elements in a row to +1's can be done in O(1) time by a compare-and-set operation. For example, setting the leftmost s elements of a row to +1's can be carried out as follows: read the row into R i 's, then CU broadcast s to all R i 's, and each R i compare i with s and set its content to +1 if i - s; then the modified row is written back to the memory array. Based on Lemma 10, it is easy to verify that elements in F are in sorted order after Step 5, which can be implemented in O(mp i\Gamma2 calls to the sorting device and O(mp i\Gamma2 data movement not involving sorting. 5.4 Complexity Analysis With the correctness of our multiway merge algorithm being obvious, we now turn to the complexity. Specifically, we are interested in assessing the total amount of data movement, not involving sorting, that is required by procedure MULTIWAY MERGE. Specifically, let J(mp i stand for the time spent on data movement tasks that do not involve the use of the sorting device. If takes O(1) time. In case takes O(m) time (refer to Theorem 3). Finally, if i ? 4, our previous discussion shows that each of Step 1, Step 3, Step 4, and Step 5 require at most O(mp i\Gamma2 recursively, J(mp i\Gamma2 time. Thus, we obtain the following recurrence system: It is easy to confirm that, for p - 4, the solution of the above recurrence satisfies J(mp i A similar analysis, that is not repeated, shows that the total number of calls to a sorting device of I/O size performed by procedure MULTIWAY MERGE for merging m, sequences, each of 2 , is bounded by O(mp i\Gamma2 To summarize our discussion we state the following important result. Theorem 5 Procedure MULTIWAY MERGE performs the task of merging m, quences, each of size p i 2 , in our architecture, using O(mp i\Gamma2 calls to the sorting device of I/O size p, and O(mp i\Gamma2 data movement not involving sorting. 6 The Sorting Algorithm With the basic algorithm and the multiway merge at our disposal, we are in a position to present the details of our sorting algorithm using a sorting device of fixed I/O size p. The input is a set \Sigma of N items stored, as evenly as possible, in p memory modules. Dummy elements of value +1 are added, if necessary, to ensure that all memory modules contain d N e elements: these dummy elements will be removed after sorting. Our goal is to show that using our architecture-algorithm combination the input can be sorted in O N log N log p time and O(N ) data space. We assume that p - 16, which along with (1) implies that log Equation (13) will be important in the analysis of this section, as our discussion will focus on the case where a sorting network of I/O size p and depth O(log 2 p) is used as the sorting device 3 . A natural candidate for such a network is Batcher's classic bitonic sorting network [1] that we shall tacitly assume. Recall that by virtue of (1) we have, for some t; t - 4, In turn, equation (14) guarantees that log N log p 1- At this point we note that (14) and (15), combined, guarantee that log 2: (16) Write and observe that by (14), For reasons that will become clear later, we pad \Sigma with an appropriate number of +1 elements in such a way that, with N 0 standing for the length of the resulting sequence \Sigma 0 , we have It is important to note that (14), (17), and (19), combined, guarantee that suggesting that the number of memory rows used by the sorting algorithm is bounded by O( N will show that this is, indeed, the case. The sorting algorithm consists of iterations. In order to guarantee an overall running time of O( N log N log p ), we ensure that each iteration can be performed in O( N As we will see shortly, the sorting network will be used in the following three contexts: (i) to sort, individually, M memory rows; 3 As it turns out, the same complexity claim holds if the sorting device used is, instead, a p-sorter. (ii) to sort, individually, M groups, each consisting of m consecutive memory rows, where m - (iii) to sort, individually, M groups, each consisting of 2m consecutive memory rows, where m - . For an efficient implementation of (i) we use simple pipelining: the M memory rows to sort are input to the sorting network, one after the other. After an initial overhead of O(log 2 p) time, each subsequent time unit produces a sorted memory row. Clearly, the total sorting time is bounded by O(log Our efficient implementation of (ii) uses interleaved pipelining. Let G GM be the groups we wish to sort. In the interleaved pipelining we begin by running Step 1 of the basic algorithm in pipelined fashion on group G 1 , then on group G 2 , and on so. In other words, Step 1 of the basic algorithm is performed on all groups using simple pipelining. Then, in a perfectly similar fashion, simple pipelining is used to carry out Step 2 of the basic algorithm on all the groups G 1 . The same strategy is used with all the remaining steps of the basic algorithm that require the use of the sorting device. Consequently, the total amount of time needed to sort all the groups using interleaves pipelining is bounded by O(log An efficient implementation of (iii) relies on extended interleaved pipelining. Let G GM be the groups we want to sort. Recall that Theorem 4 states that sorting a group of 2m consecutive memory rows requires five calls to the basic algorithm. The extended interleaved pipelining consists of five interleaved pipelining steps, each corresponding to one of the five calls to the basic algorithm. Thus, the task of sorting all groups can be performed in O(log time. We now discuss each of the iterations of our sorting algorithm in more detail. Iteration 1 The input is Partitioned into N 0 groups, each involving p 1 memory rows. By using interleaved pipelining with , each such group is sorted individually. As discussed above, the running time of Iteration 1 is bounded by O(log Iteration 1. The input to Iteration k is a collection of N 0 ksorted sequences each of size p stored in consecutive memory rows. The output of iteration k is a collection of N 0 +1sorted sequences, each of size p stored in p consecutive memory rows. Having partitioned these sorted sequences into N 0 consecutive sequences each, we proceed to sort each group G(k; j) by the call MULTIWAY MERGE(S 1 (k; We refer to the call MULTIWAY MERGE(S 1 (k; call of the first level. Observe that, since there are N there will be altogether N WAY MERGE calls of the first, one for each group. In Step 1 of a MULTIWAY MERGE call of the first level we extract a sample S 2 (k; j) of S 1 (k; consisting of p 1 sorted sequences, each of size p stored in consecutive memory rows. In turn, for every j, (1 - j - N the sample S 2 (k; j) is sorted by invoking MULTIWAY MERGE(S 2 (k; which is referred to as a MULTIWAY MERGE call of the second level. Step 1 of a MULTIWAY MERGE call of the second level extracts a sample S 3 (k; j) of For every u, 1 - 2 c, a MULTIWAY MERGE call of level u is of the form MULTIWAY MERGE In Step 2 of the call MULTIWAY MERGE(S u (k; a MULTIWAY MERGE call of level u+1, which is of the form MULTIWAY MERGE(S u+1 (k; Let r k;u denote the total number of rows in all samples S u (k; j) of level u. Clearly, we have r p u . By (13), r k;u - qp, and r when t is even and 2. The recursive calls to MULTIWAY MERGE end at level b i k \Gamma1c, the last call being of the form Note that depending on whether or not i k is odd. We proceed to demonstrate that for takes O(r k;1 ) time. We will do this by showing that the total time required by each of the five steps of the MULTIWAY MERGE calls of each level u is bounded by O(r k;u ). Consider a particular level u. Step 1 of all MULTIWAY MERGE calls of level u is performed on the samples S u (k; j), in increasing order of j, so that all the samples S u+1 (k; are extracted one after the other. Clearly, the total time for these operations is O(r k;u ). We perform Step 3 of all the MULTIWAY MERGE calls of level u, in increasing order of j, to partition into buckets each of the samples S u (k; using the corresponding S u+1 (k; j). By Lemma 10, each sample buckets, and no bucket contains more than 2p 3 elements. As discussed in Subsection 5.2, the task of moving all the elements of each S u (k; j) to their buckets can be carried out in O(p using the sorting device. Thus, the total time for partitioning the samples S u (k; in all the MULTIWAY MERGE calls of level u is bounded by O( N 0 Step 4 of a MULTIWAY MERGE call of level u sorts the buckets (involving the elements of S u (k; j)) obtained in Step 3. We perform Step 4 of all MULTIWAY MERGE calls of level u in increasing order of j, and use extended interleaved pipelining with 2 to sort all buckets of each S u (k; j). There are, altogether, N 0 2u+1buckets in all the S u (k; j)'s. Thus, the total time for sorting all buckets is bounded by and (d), the total time for sorting the buckets in all MULTIWAY MERGE calls of level u is O(r k;u ). Step 5 of a MULTIWAY MERGE call of level u has three phases. As discussed in Subsection 5.3, The operations of Phase 1 and Phase 3 that involve the sorting device can be carried out using interleaved pipelining. The operations of Phase 2 that involves the sorting device can be carried out using simple pipelining. Clearly, the time complexity of Step 5 for all MULTIWAY MERGE calls of level u is bounded by O(log We now evaluate the time needed to perform Step 2 of all the MULTIWAY MERGE calls of level u. First, consider the call of level b i k \Gamma1c, MULTIWAY MERGE(S b 1)). The sample extracted in Step 1 of this call has p elements is odd, we use simple pipelining to sort all the samples S b (k; we use interleaved pipelining with 2 to sort all the samples S b (k; In either case, the time required is bounded by O( N 0 which is no more than O( N 0 Thus, the total time for Steps 1 through 5 of all the MULTIWAY MERGE calls of level b i k \Gamma1c) is no more than O(r k;b ). Next, the time to perform Step 2 of all MULTIWAY MERGE calls of level u is inductively derived as O(r k;u ) using our claim that the total time for Steps 1, 3, 4 and 5 of all MULTIWAY MERGE calls of level u is no more than O(r k;u ), and hypothesis that Step 2 of all MULTIWAY MERGE calls of level This, in turn, proves that the total time required for all the MULTIWAY MERGE calls of level u is bounded by O(r k;u ). Having shown that the time required for all the MULTIWAY MERGE calls of level u of Iteration k is O(r k;u ), we conclude that the total time to perform Iteration k is O(r k;1 ), which is O( N Iteration 2 the N input elements are sorted at the end of iterations. Assume that the algorithm does not terminate in iterations. The input to Iteration t \Gamma 1 is a collection of sorted sequences, 2 . Each such sequence is of size p t 2 , stored in p t\Gamma2 consecutive rows. To complete the sorting, we need to merge these q sequences into the desired sorted sequence. This task is performed by the call MULTIWAY MERGE(\Sigma t). The detailed implementation of MULTIWAY MERGE(\Sigma using a sorting network as the sorting device and the analysis involved are almost the same as that of Iteration 2 to Iteration different parameters are used. If the interleaved pipelining with 2 is used in a step of MULTIWAY MERGE for iterations 2 to t \Gamma 2, then the corresponding step of MULTIWAY MERGE for iteration uses the interleaved pipelining with Similarly, if the extended interleaved pipelining with 2 is used in a step of MULTIWAY MERGE for iterations 2 to t \Gamma 2, then the corresponding step of MULTIWAY MERGE for iteration uses the extended interleaved pipelining with q. The MULTIWAY MERGE call of level b t\Gamma1c is MULTIWAY MERGE(S b If t is odd, then 4. The recursion stops at the (b t\Gamma1c)-th level. The sample set S b obtained in Step 1 of the MULTIWAY MERGE call of level b t\Gamma1c has qp 1 is odd, and it has qp elements if t is even. Let r t\Gamma1;u be the total number of memory rows in S . By a simple induction, we conclude that the MULTIWAY MERGE call of level u, 1 - takes no more than O(r t\Gamma1;u ) time. The running time of Iteration is the running time of the MULTIWAY MERGE call of the first level, and it takes O(r t\Gamma1;1 We have shown that each of the iterations of MULTIWAY MERGE can be implemented with time O( N we conclude that the running time of our sorting algorithm is O N log N log p . Since a p-sorter can be abstracted as a sorting network of I/O size p and depth O(1), this time complexity stands if the sorting device used is a p-sorter. The working data memory for each iteration is O(N ) simply because that the sample size of an MULTIWAY MERGE call of level u is p times the sample size of an MULTIWAY MERGE call of level u+1. Since the working data memory of one iteration can be reused by another iteration, the total data memory required by our sorting algorithm remains to be O(N ). Summarizing all our previous discussions, we have proved the main result of this work. Theorem 6 Using our simple architecture, a set of N items stored in N memory rows can be sorted in row-major order, without any memory access conflicts, in O N log N log p time and O(N ) data space, by using either a p-sorter or a sorting network of I/O size p and depth O(log 2 p) as the sorting device. --R Sorting n objects with a k-sorter An optimal sorting algorithm on reconfigurable mesh The Art of Computer Programming Tight bounds on the complexity of parallel sorting Introduction to Parallel Algorithms and Architectures: Arrays Sorting in O(1) time on a reconfigurable mesh of size n Sorting n numbers on n How to sort N items using a sorting Network of fixed I/O size --TR --CTR Classifying Matrices Separating Rows and Columns, IEEE Transactions on Parallel and Distributed Systems, v.15 n.7, p.654-665, July 2004 Giuseppe Campobello , Marco Russo, A scalable VLSI speed/area tunable sorting network, Journal of Systems Architecture: the EUROMICRO Journal, v.52 n.10, p.589-602, October 2006 Brian Grattan , Greg Stitt , Frank Vahid, Codesign-extended applications, Proceedings of the tenth international symposium on Hardware/software codesign, May 06-08, 2002, Estes Park, Colorado

sorting networks;VLSI;special-purpose architectures;columnsort;hardware-algorithms

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Tracing the lineage of view data in a warehousing environment.

We consider the view data lineageproblem in a warehousing environment: For a given data item in a materialized warehouse view, we want to identify the set of source data items that produced the view item. We formally define the lineage problem, develop lineage tracing algorithms for relational views with aggregation, and propose mechanisms for performing consistent lineage tracing in a multisource data warehousing environment. Our result can form the basis of a tool that allows analysts to browse warehouse data, select view tuples of interest, and then drill-through to examine the exact source tuples that produced the view tuples of interest.

Introduction In a data warehousing system, materialized views over source data are defined, computed, and stored in the warehouse to answer queries about the source data (which may be stored in distributed and legacy systems) in an integrated and efficient way [Wid95]. Typically, on-line analytical processing and mining (OLAP and OLAM) systems operate on the data warehouse, allowing users to perform analysis and predictions [CD97, HCC98]. In many cases, not only are the views themselves useful for analysis, but knowing the set of source data that produced specific pieces of view information also can be useful. Given a data item in a materialized view, determining the source data that produced it and the process by which it was produced is termed the data lineage problem. Some applications of view data lineage are: ffl OLAP and OLAM: Effective data analysis and mining needs facilities for data exploration at different levels. The ability to select a portion of relevant view data and "drill-down" to its origins can be very useful. In addition, an analyst may want to check the origins of suspect or anomalous view data to verify the reliability of the sources or even repair the source data. ffl Scientific Databases: Scientists apply algorithms to commonly understood and accepted source data to derive their own views and perform specific studies. As in OLAP, it can be useful for the scientist to focus on specific view data, then explore how it was derived from the original raw data. This work was supported by Rome Laboratories under Air Force Contract F30602-96-1-0312, and by the Advanced Research and Development Committee of the Community Management Staff as a project in the MDDS Program. ffl On-line Network Monitoring and Diagnosis Systems: From anomalous view data computed by the diagnosis system, the network controller can use data lineage to identify the faulty data within huge volumes of data dumped from the network monitors. ffl Cleansed Data Feedback: Information centers download raw data from data sources and "cleanse" the data by performing various transformations on it. Data lineage helps locate the origins of data items, allowing the system to send reports about the cleansed data back to their sources, and even link the cleansed items to the original items. Materialized View Schema Evolution: In a data warehouse, users may be permitted to change view definitions (e.g., add a column to a view) under certain circumstances. View data lineage can help retrofit existing view contents to the new view definition without recomputing the entire view. ffl View Update Problem: Not surprisingly, tracing the origins of a given view data item is related to the well-known view update problem [BS81]. In Section 8.2, we discuss this relationship, and show how lineage tracing can be used to help translate view updates into corresponding base data updates. In general, a view definition provides a mapping from the base data to the view data. Given a state of the base data, we can compute the corresponding view according to the view definition. However, determining the inverse mapping-from a view data item back to the base data that produced it-is not as straightforward. To determine the inverse mapping accurately, we not only need the view definition, but we also need the base data and some additional information. The warehousing environment introduces certain challenges to the lineage tracing problem, such as how to trace lineage when the base data is distributed among multiple sources, and what to do if the sources are inaccessible or not consistent with the warehouse views. At the same time, the warehousing environment can help the lineage tracing process by providing facilities to merge data from multiple sources, and to store auxiliary information in the warehouse in a consistent fashion. In this paper, we focus on the lineage problem for relational Select-Project-Join views with aggregation (ASPJ views) in a data warehousing environment. Our results extend to additional relational operators as we show in [CWW98]. In summary, we: ffl Formulate the view data lineage problem. We give a declarative definition of tuple derivations for relational operators, and inductively define view tuple derivations based on the tuple derivations for the operators (Section 4). ffl Develop derivation tracing algorithms, including proofs of their correctness (Sections 5 and 6). ffl Discuss issues of derivation tracing in a warehousing environment, and show how to trace tuple derivations for views defined on distributed, legacy sources consistently and efficiently (Section 7). We first discuss related work in Section 2. We then motivate the lineage problem using detailed examples in Section 3. After defining the lineage problem and presenting our solutions as summarized above, in Section 8 we revisit some related issues (e.g., the view update problem) in detail. Conclusions and future work are covered in Section 9. All proofs are provided in the Appendix. Related Work There has been a great deal of work in view maintenance and related problems in data warehous- ing, but to the best of our knowledge the lineage problem has not been addressed. Overviews of research directions and results in data warehousing can be found in [CD97, Wid95, WB97]. specifically covers view maintenance problems in data warehousing. Incremental view maintenance algorithms have been presented for relational algebra views [QW91], for aggregation [Qua96], and for recursive views [GMS93]. View "self-maintainability" issues are addressed in [QGMW96]. Warehouse view consistency is studied in [ZGMW96, ZWGM97], to ensure that views in the warehouse are consistent with each other and reflect consistent states of the sources. All of these papers consider computing warehouse views but do not address the reverse problem of view data lineage. OLAP systems, usually sitting on top of a data warehouse, allow users to perform analysis and make predictions based on the warehouse view information [Col96, HCC98]. The data cube is a popular OLAP structure that facilitates multi-dimensional aggregation over source data [GBLP96]. Cube "rolling-up" and "drilling-down" enable the user to browse the view data at any level and any dimension of the aggregation [HRU96, MQM97]. However, data cubes are based on a restricted form of relational views, and usually only allow drilling down within the warehouse, not to the original data sources. Metadata for warehouse views can be maintained to record lineage information about a particular view column [CM89, HQGW93]. However, this approach only provides schema-level lineage tracing, while many applications require lineage at a finer (instance-level) granularity. Some scientific databases use tuple-level annotations to keep track of lineage [HQGW93], which can introduce high storage overhead in warehousing applications. introduces "weak inversion" to compute fine-grained data lineage. However, not all views have an inverse or weak inverse. Also, the system in [WS97] requires users to provide a view's inverse function in order to compute lineage, which we feel is not always practical. Our algorithms trace tuple-level lineage automatically for the user and maintain the necessary auxiliary information to ensure view invertibility. The problem of reconstructing base data from summary data is studied in [FJS97]. Their statistical approach estimates the base data using only the summary data and certain constraints; it does not guarantee accurate lineage tracing. In this paper, we focus on accurate lineage tracing with the base data available either in remote sources or stored locally in the warehouse. The view update problem [DB78, Kel86] is to translate updates against a view into updates against the relevant base tables, so that the updated base tables will derive the updated view. View data lineage can be used to help solve the view update problem, employing a different approach from previous techniques; see Section 8. Finally, Datalog can perform top-down recursive rule-goal unification to provide proofs for a goal proposition [Ull89]. The provided proofs find the supporting facts for the goal proposition, and therefore also can be thought of as providing the proposition's lineage. However, we take an approach that is very different from rule-goal unification; a detailed comparison is presented in Section 8. item id item name category binder stationery stationery shirt clothing pants clothing pot kitchenware Figure 1: item table store id store name city state 004 Macy's New York City NY Figure 2: store table store id item id price num sold Figure 3: sales table Motivating Examples In this section, we provide examples that motivate the definition of data lineage and how lineage tracing can be useful. Consider a data warehouse with retail store data over the following base tables: item(item id, item name, category) store(store id, store name, city, state) sales(store id, item id, price, num sold) The item and store tables are self-explanatory. The sales table contains sales information, including the number and price of each product sold by each store. Example table contents are shown in Figures 1, 2, and 3. Example 3.1 (Lineage of SPJ View) Suppose a sales department wants to study the selling patterns of California stores. A materialized view Calif can be defined in the data warehouse for the analysis. The SQL definition of the view is: SELECT store.store-name, item.item-name, sales.num-sold FROM store, item, sales The view definition also can be expressed using the relational algebra tree in Figure 4. The materialized view for Calif over our sample data is shown in Figure 5. The analyst browses the view table and is interested in the second tuple !Target, pencil, 3000?. He would like to see the relevant detailed information and asks question Q1: "Which base data produced tuple !Target, pencil, 3000? in Calif?" Using the algorithms we present in Section 5, we obtain the answer in Figure 6. The answer tells us that the Target store in Palo Alto sold 3000 pencils at a price of 1 dollar each. sales store store_name, item_name, num_sold item Calif Figure 4: View definition for Calif store name item name num sold Target binder 1000 Target pencil 3000 Target pants 600 Macy's shirt 1500 Macy's pants 600 Figure 5: Calif table store s id s name city state item stationery sales s id i id price num sold Figure Calif lineage for !Target, pencil, 3000? Example 3.2 (Lineage of Aggregation View) Now let's consider another warehouse view Clothing, for analyzing the total clothing sales of the large stores (which have sold more than 5000 clothing items). Clothing AS SELECT sum(num-sold) as total FROM item, store, sales GROUP BY store-name The relational algebra definition of the view is shown in Figure 7. We extend relational algebra with an aggregation operator, denoted ff G;aggr(B) , where G is a list of groupby attributes, and aggr(B) abbreviates a list of aggregate functions over attributes. (Details are given in Section 4.1.) The materialized view contains one tuple, !5400?, as shown in Figure 8. The analyst may wish to learn more about the origins of this tuple, and asks question Q2: "Which base data produced tuple !5400? in Clothing?" Not surprisingly, due to the more complex view definition, this question is more difficult to answer than Q1. We develop the appropriate algorithms in Section 6, and Figure 9 presents the answer. It lists all the branches of Macy's, the clothing items they sell (but not other items), and the sales information. All of this information is used to derive the tuple !5400? in Clothing. Questions such as Q1 and Q2 ask about the base tuples that derive a view tuple. We call these base tuples the derivation (or lineage) of the view tuple. In the next section, we formally define the concept of derivation. Sections 5, 6, and 7 then present algorithms to compute view tuple derivations. a item sales store_name, sum(num_sold) as total s total total > 5000 store Clothing Figure 7: View definition for Clothing total Figure 8: Clothing table store s id s name city state 004 Macy's New York City NY item shirt clothing pants clothing sales s id i id price num sold Figure 9: Clothing lineage for !5400? 4 View Tuple Derivations In this section, we define the notion of a tuple derivation, which is the set of base relation tuples that produce a given view tuple. Section 4.1 first introduces the views on which we focus in this paper. Tuple derivations for operators and views are then defined in Sections 4.2 and 4.3, respectively. We assume that a table (or relation) R with schema R contains a set of tuples with no duplicates. (Thus, we consider set semantics in this paper. We have adapted our work to bag semantics as well; please see [CWW98].) A database D contains a list of base tables view V is a virtual or materialized result of a query over the base tables in D. The query (or the mapping from the base tables to the view table) is called the view definition, denoted as v. We say that 4.1 Views We consider a class of views defined over base relations using the relational algebra operators selection (oe), projection (-), join (./), and aggregation (ff). Our framework applies as well to set union ([), set intersection ("), and set difference (\Gamma), however these operators are omitted in this paper due to space constraints. Please see [CWW98] for details. We use the standard relational semantics, included here for completeness: Base case: ffl Projection: -A ig. We consider the multi-way natural join Thus, the grammar of our view definition language is as follows: where R is a base table, are views, C is a selection condition (any boolean expression) on attributes of V 1 , A is a projection attribute list from V 1 , G is a groupby attribute list from V 1 , and aggr(B) abbreviates a list of aggregation functions to apply to attributes of V 1 . For convenience in formulation, when a view references the same relation more than once, we consider each relation instance as a separate relation. For example, we treat the self-join R ./ R as (R as R 1 (R as R 2 ), and we consider R 1 and R 2 to be two tables in D. This approach allows view definitions to be expressed using an algebra tree instead of a graph, while not limiting the views we can handle. Any view definition in our language can be expressed using a query tree, with base tables as the leaf nodes and operators as inner nodes. Figures 4 and 7 are examples of query trees. 4.2 Tuple Derivations for Operators To define the concept of derivation, we assume logically that the view contents are computed by evaluating the view definition query tree bottom-up. Each operator in the tree generates its result table based on the results of its children nodes, and passes it upwards. We begin by focusing on each operator, defining derivations of the operator's result tuples based on its input tuples. According to relational semantics, each operator generates its result tuple-by-tuple based on its operand tables. Intuitively, given a tuple t in the result of operator Op, only a subset of the input tuples produce t. We say that the tuples in this subset contribute to t, and we call the entire subset the derivation of t. Input tuples not in t's derivation either contribute to nothing, or only contribute to result tuples other than t. Figure illustrates the derivation of a view tuple. In the figure, operator Op is applied to tables T 1 and T 2 , which may be base tables or temporary results from other operators. (In general, we use R's to denote base tables and T 's to denote tables that may be base or derived.) Table T is the operation result. Given tuple t in T , only subsets T 2 of T 1 and T 2 contribute to t. called t's derivation. The formal definition of tuple derivation for an operator is given next, followed by additional explanation. Definition 4.1 (Tuple Derivation for an Operator) Let Op be any of our relational operator (oe, -, ./, ff) over tables T be the table that results from applying Op to T Given a tuple t 2 T , we define t's derivation in T according to Op to be Op \Gamma1 are maximal subsets of Op T1* T2* Figure 10: Derivation of tuple t (a) Op(T (b) 8T Also, we say that Op \Gamma1 i is t's derivation in T i , and each tuple t in T i contributes to t, can be extended to represent the derivations of a set of tuples: where S represents the multi-way union of relation lists. 1 In Definition 4.1, requirement (a) says that the derivation tuple sets (the T i 's) derive exactly t. From relational semantics, we know that for any result tuple t, there must exist such tuple sets. Requirement (b) says that each tuple in the derivation does in fact contribute something to t. For example, with requirement (b) and given base tuples that do not satisfy the selection condition C and therefore make no contribution to any view tuple will not appear in any view tuple's derivation. By defining the T i \Lambda 's to be the maximal subsets that satisfy requirements (a) and (b), we make sure that the derivation contains exactly all the tuples that contribute to t. Thus, the derivation fully explains why a tuple exists in the view. Theorem 4.2 shows that there is a unique derivation for any given view tuple. Recall that all proofs are provided in the Appendix . Theorem 4.2 (Derivation Uniqueness) Given t 2 Op(T is a tuple in the result of applying operator Op to tables T there exists a unique derivation of t in according to Op. Example 4.3 (Tuple Derivation for Aggregation) Given table R in Figure 11(a), and tuple Figure 11(b), the derivation of t is R sum(Y)1(b) a (a) R X, sum(Y) (R) R (c) a -1 (<2, 8> R Figure Tuple derivations for aggregation shown in Figure 11(c). Notice that R's subset fh2; 3i, h2; 5ig also satisfies requirements (a) and (b) in Definition 4.1, but it is not maximal. Intuitively, h2; 0i also contributes to the result tuple, since computed by adding the Y attributes of h2; 3i, h2; 5i, and h2; 0i in R. From Definition 4.1 and the semantics of the operators in Section 4.1, we now specify the actual tuple derivations for each of our operators. Theorem 4.4 (Tuple Derivations for Operators) Let be tables and t be a result tuple. -A 4.3 Tuple Derivations for Views Now that we have defined tuple derivations for the operators, we proceed to define tuple derivations for views. As mentioned earlier, a view definition can be expressed as a query tree evaluated bottom-up. Intuitively, if a base tuple t contributes to a tuple t 0 in the intermediate result of a view evaluation, and t 0 further contributes to a view tuple t, then t contributes to t. We define a view tuple's derivation to be the set of all base tuples that contribute to the view tuple. The specific process through which the view tuple is derived can be illustrated by applying the view query tree to the derivation tuple sets, and presenting the intermediate results for each operator in the evaluation. Definition 4.5 (Tuple Derivation for a View) Let D be a database with base tables be a view over D. Consider a tuple t 2 V . 1. contributes to itself in V . 2. is a view definition over D, contributes to t according to the operator Op (by Definition 4.1), and t 2 R i contributes to according to the view v j (by this definition recursively). Then t contributes to t according to v. R Y (b)2 Y a (c) s (R) R (a) 0 Figure 12: Tuple derivation for a view We define t's derivation in D according to v to be v \Gamma1 are subsets of R Rm such that t 2 R contributes to t according to v, for Also, we call R derivation in R i according to v, denoted as v Finally, the derivation of a view tuple set T contains all base tuples that contribute to any view tuple in the set T : (v Theorem 4.2 can be applied inductively in the obvious way to show that a view tuple's derivation is unique. Example 4.6 (Tuple Derivation for a View) Given base table R in Figure 12(a), view Figure 12(c), and tuple is easy to see that tuples h2; 3i and h2; 5i in R contribute to h2; 3i and h2; 5i in oe Y 6=0 (R) in Figure 12(b), and further contribute to h2; 8i in V . The derivation of t is v \Gamma1 R as shown in Figure 12(d). We now state some properties of tuple derivations to provide the groundwork for our derivation tracing algorithms. Theorem 4.7 (Derivation Transitivity) Let D be a database with base tables R and let be a view over D. Suppose that v can also be represented as (D) is an intermediate view over D, for derivation in V j according to v 0 . Then t's derivation in D according to v is the concatenation of derivations in D according to v j , (v where represents the multi-way concatenation of relation lists. 2 Theorem 4.7 is a result of Definition 4.5. It shows that given a view V with a complex definition tree, we can break down its definition query tree into intermediate views, and compute a tuple's derivation by recursively tracing the hierarchy of intermediate views. 2 The concatenation of two relation lists relations are renamed so that the same relation never appears twice. Since we define tuple derivations inductively based on the view query tree, an interesting question arises: Are the derivations of two equivalent views also equivalent? Two view definitions (or query trees) v 1 and v 2 are equivalent iff 8D: v 1 We prove in Theorem 4.8 that given any two equivalent Select-Project-Join (SPJ) views, their tuple derivations are also equivalent. Theorem 4.8 (Derivation Equivalence after SPJ Transformation) Tuple derivations of equivalent SPJ views are equivalent. In other words, given equivalent SPJ views v 1 and v 2 , According to Theorem 4.8, we can transform an SPJ view to a simple canonical form before tracing tuple derivations. Unfortunately, views with aggregation do not have this nice property, as shown in the following example. Example 4.9 (Tuple Derivations for Equivalent Views with Aggregation) Let are equivalent, since Given base table R in Figures 11(a) and 12(a), Figures 11(b) and 12(c) show that the contents of the two views are the same. However, the derivation of tuple according to v 1 (shown in Figure 11(c)) is different from that according to v 2 (shown in Figure 12(d)). Given Definition 4.5, a straightforward way to compute a view tuple derivation is to compute the intermediate results for all operators in the view definition and store them as temporary tables, then trace the tuple's derivation in the temporary tables recursively, until reaching the base tables. Obviously, this approach is impractical due to the computation and storage required for all the intermediate results. In the next two sections, we separately consider SPJ views and views with aggregation (ASPJ views). We show in Section 5 that one relational query over the base tables suffices to compute tuple derivations for SPJ views. A recursive algorithm for ASPJ view derivation tracing that requires a modest amount of auxiliary information is described in Section 6. 5 SPJ View Derivation Tracing Derivations for tuples in SPJ views can be computed using a single relational query over the base data. In this section, we first define the general concept of a derivation tracing query, which can be applied directly to the base tables to compute a view tuple's derivation. We then specify tracing queries for SPJ views, and discuss optimization issues for tracing queries. 5.1 Derivation Tracing Queries Sometimes, we can write a query for a specific view definition v and view tuple t, such that if we apply the query to the database D it returns t's derivation in D (based on Definition 4.5). We call such a query a derivation tracing query (or tracing query) for t and v. More formally, we Definition 5.1 (Derivation Tracing Query) Let D be a database with base tables Given view definition v over R a derivation tracing query for t and v iff: D (t) is t's derivation over D according to v, and TQ t;v is independent of database instance D. We can similarly define the tracing query for a view tuple set T , and denote it as TQ T;v (D). 5.2 Tracing Queries for SPJ Views All SPJ views can be transformed into the form -A (oe C (R 1 ./ using a sequence of SPJ algebraic transformations [Ull89]. We call this form the SPJ canonical form. From Theorem 4.8, we know that SPJ transformations do not affect view tuple derivations. Thus, given an SPJ view, we first transform it into SPJ canonical form, so that its tuple derivations can be computed systematically using a single query. We first introduce an additional operator used in tracing queries for SPJ views. Definition 5.2 (Split Operator) Let T be a table with schema T. The operator Split breaks T into a list of tables; each table in the list is a projection of T onto a set of attributes A i ' T, h- A 1 Theorem 5.3 (Derivation Tracing Query for an SPJ View) Let D be a database with base tables R be an SPJ view over D. Given derivation in D according to v can be computed by applying the following query to the base tables: Given a tuple set T ' v(D), T 's derivation tracing query is: where n is the semi-join operator. Example 5.4 (Tracing Query for Calif) Recall Q1 over view Calif in Example 3.1, where we asked about the derivation of tuple !Target, pencil, 3000?. Figure 13(a) shows the tracing query for !Target, pencil, 3000? in Calif according to Theorem 5.3. The reader may verify that by applying the tracing query to the source tables in Figures 1, 2, and 3, we obtain the derivation result in Figure 6. 5.3 Tracing Query Optimization The derivation tracing queries in Section 5.2 clearly can be optimized for better performance. For example, the simple technique of pushing selection conditions below the join operator is especially applicable in tracing queries, and can significantly reduce query cost. Figure 13(b) shows the optimized tracing query for the Calif tuple. If sufficient key information is present, the tracing query is even simpler: store sales store sales item item store, item, sales s TQ <Target, pencil, 3000>, Calif s store, item, sales <Target, pencil, 3000>, Calif s (b) Optimized (a) Unoptimized Figure 13: Derivation tracing query for !Target, pencil, 3000? Theorem 5.5 (Derivation Tracing using Key Information) Let R i be a base table with attributes K i include the base keys (i.e., derivation is hoe K 1 (R According to Theorem 5.5, we can use key information to fetch the derivation of a tuple directly from the base tables, without performing a join. The query complexity is reduced from O(n m ) to O(mn), where n is the maximum size of the base tables, and m is the number of base tables on which v is defined. We have shown that tuple derivations for SPJ views can be traced efficiently. For more complex views with aggregations we cannot compute tuple derivations by a single query over the base tables. In the next section, we present a recursive tracing algorithm for these views. 6 Derivation Tracing Algorithm for ASPJ Views In this section, we consider SPJ views with aggregation (ASPJ views). Although we have shown that no intermediate results are required for SPJ view derivation tracing, some ASPJ views are not traceable without storing certain intermediate results. For example, in Q2 in Example 3.2 the user asks for the derivation of tuple in the view Clothing. It is not possible to compute t's derivation directly from store, item, and sales, because total is the only column of view Clothing, and it is not contained in the base tables at all. Therefore, we cannot find t's derivation by knowing only that t:total = 5400. In order to trace the derivation correctly, we need tuple hMacy's, 5400i in the intermediate aggregation result to serve as a ``bridge'' that connects the base tables and the view table. We introduce a canonical form for ASPJ views in Section 6.1. In Section 6.2, we specify the derivation tracing query for a simple one-level ASPJ view. We then develop a recursive tracing algorithm for complex ASPJ views and justify its correctness in Section 6.3. As mentioned above, intermediate (aggregation) results in the view evaluation are needed for derivation tracing. These intermediate results can either be recomputed from the base tables when needed, or they can be stored as materialized auxiliary views in a warehouse; this issue is further discussed in Section 7.1. In the remainder of this section we simply assume that all intermediate aggregation results are available. 6.1 ASPJ Canonical Form Unlike SPJ views, ASPJ views do not have a simple canonical form, because in an ASPJ view definition some selection, projection, and join operators cannot be pushed above or below the aggregation operators. View Clothing in Figure 7 is such an example, where the selection total ? 5000 cannot be pushed below the aggregation, and the selection category = "clothing" cannot be pulled above the aggregation. However, by commuting and combining some SPJ operators [Ull89], it is possible to transform a general ASPJ view query tree into a form composed of ff-oe ./ operator sequences, which we call ASPJ segments. Each segment in the query tree except the topmost must include a non-trivial aggregation operator. We call this form the ASPJ canonical form. Definition 6.1 (ASPJ Canonical Form) Let v be an ASPJ view definition over database D. 1. R is a base table in D, is in ASPJ canonical form. 2. is in ASPJ canonical form if v j is an ASPJ view in ASPJ canonical form, 6.2 Derivation Tracing Queries for One-level ASPJ Views A view defined by one ASPJ segment is called a one-level ASPJ view. Similar to SPJ views, we can use one query to compute a tuple derivation for a one-level ASPJ view. Theorem 6.2 (Derivation Tracing Query for a One-Level ASPJ View) Given a one-level derivation in T according to v can be computed by applying the following query to the base tables: Given tuple set derivation tracing query is: Here too, evaluation of the actual tracing query can be optimized in various ways as discussed in Section 5.3. 6.3 Recursive Derivation Tracing Algorithm for Multi-level ASPJ Views Given a general ASPJ view definition, we first transform the view into ASPJ canonical form, divide it into a set of ASPJ segments, and define an intermediate view for each segment. Example 6.3 (ASPJ Segments and Intermediate Views for Clothing) Recall the view Clothing in Example 3.2. We can rewrite its definition in ASPJ canonical form with two seg- ments, and introduce an intermediate view AllClothing as shown in Figure 14. We then trace a tuple's derivation by recursively tracing through the hierarchy of intermediate views top-down. At each level, we use the tracing query for a one-level ASPJ view to compute derivations for the current tracing tuples with respect to the view or base tables at the next level below. Details follow. Clothing a store_name, sum(num_sold) as total AllClothing s total > 5000 total segment 1 segment 2 item store sales Figure 14: ASPJ segments and intermediate views for Clothing 6.3.1 Algorithm Figure presents our recursive derivation tracing algorithm for a general ASPJ view. Given a view definition v in ASPJ canonical form, and tuple t 2 v(D), procedure TupleDerivation(t; v; D) computes the derivation of tuple t according to v over D. The main al- gorithm, Procedure TableDerivation(T ; v; D), computes the derivation of a tuple set T ' v(D) according to v over D. As discussed earlier, we assume that a one-level ASPJ view, and available as a base table or an intermediate view, 1::k. The procedure first computes T 's derivation hV using the one-level ASPJ view tracing query TQ(T i) from Theorem 6.2. It then calls procedure which computes (recursively) the derivation of each tuple set V j according to v j , concatenates the results to form the derivation of the entire list of view tuple sets. Example 6.4 (Recursive Derivation Tracing) We divided the view Clothing into two segments in Example 6.3. We assume that the contents of the intermediate view AllClothing are available (shown in Figure 16). According to our algorithm, we first compute the derivation T of !5400? in AllClothing to obtain T trace T 's derivation to the base tables to obtain the derivation result in Figure 9. Note that we do not necessarily materialize complete intermediate aggregation views such as AllClothing. In fact, there are many choices of what (if anything) to store. The issue of storing versus recomputing the intermediate information needed for derivation tracing is discussed in Section 7.1. 6.3.2 Correctness To justify the correctness of our algorithm, we claim the following: 1. Transforming a view into ASPJ canonical form does not affect its derivation. We can "canonicalize" an ASPJ view by transforming each segment between adjacent aggregation operators into its SPJ canonical form. The process consists only of SPJ transformations [Ull89]. Theorem 4.8 shows that derivations are unchanged by SPJ transformations. procedure TupleDerivation(t; v; D) begin return (TableDerivation(ftg;v; D)); procedure TableDerivation(T; v; D) begin is a one-level ASPJ view, (D) is an intermediate view or a base table, return (TableListDerivation(hV procedure begin do return (D Figure 15: Algorithm for ASPJ view tuple derivation tracing store name total Target 1400 Macy's 5400 Figure AllClothing table 2. It is correct to trace derivations recursively down the view definition tree. From Theorem 4.7, we know that derivations are transitive through levels of the view definition tree. Thus, when tracing tuple derivations for a canonicalized ASPJ view, we can first divide its definition into one-level ASPJ views, and then compute derivations for the intermediate views in a top-down manner. Our recursive derivation tracing algorithm can be used to trace the derivation of any tuple in any ASPJ view in a conventional database. However, certain additional issues arise when performing derivation tracing in a multi-source warehousing environment, which we proceed to discuss in the next section. 7 Derivation Tracing in a Warehousing Environment In Sections 5 and 6 we presented algorithms to trace view tuple derivations. Our algorithms assume that all of the base tables as well as the intermediate views are accessible, and that they are consistent with the view being traced. These assumptions may not hold when we are tracing derivations for a warehouse view defined on remote distributed sources. The following problems may arise: 1. Efficiency problem: Querying remote sources and performing selections and joins (in the tracing queries) over them for each tuple derivation trace can be very inefficient. Also, recomputing intermediate views for tracing multi-level ASPJ views can be expensive. 2. Consistency problem: The warehouse may not refresh its views in real time, which means that warehouse views can become out of date. Thus, we cannot always compute the derivation of tuples in the "old" view from the "new" source base tables. For example, if a base tuple in the derivation of a view tuple has just been deleted from the source, but the change has not yet been propagated to the view, then the user sees the view tuple but cannot correctly trace its derivation since a relevant base tuple is gone. 3. Legacy source problem: Views defined on inaccessible legacy sources are not traceable because the base tables are not available. Storing auxiliary views in the warehouse to reduce computation cost and to avoid querying the sources [QGMW96] is a solution that solves all three of the above problems. The price is extra storage and view maintenance costs. In Section 7.1, we consider the trade-offs between materializing and recomputing intermediate results, and propose to store intermediate aggregation results to improve overall performance. In Section 7.2, we introduce derivation views, which store information about sources so that view derivations always can be computed without querying the sources. 7.1 Materializing vs. Recomputing Intermediate Aggregate Results In Section 6, we saw that intermediate aggregation results are needed for derivation tracing in the general case. There are two ways to obtain such information. One is to recompute the intermediate result during the tracing process. This approach requires no permanent extra storage, but the tracing process takes longer, especially when the recomputation may require querying the sources. The other way is to maintain materialized auxiliary views containing the intermediate results. In this case, less computation is required at tracing time, but the auxiliary views must be stored and kept up-to-date. Due to the characteristics of warehousing environments, we suggest warehouses maintain the intermediate aggregation results as materialized auxiliary views rather than recomputing them [QGMW96]. Example 7.1 (Materialized View for AllClothing) To improve the efficiency of the tracing process in Example 6.4, we materialize auxiliary view AllClothing (in Figure 14) with the contents shown in Figure 16. Note that when materializing AllClothing, tuple hTarget, 1400i is not used when tracing tuple derivations for Clothing. In fact it is filtered out by the selection condition oe total?5000 in Clothing's definition. In this case, materializing the result of V instead of storing AllClothing seems to be a better choice. However, notice that V 0 is not incrementally maintainable without storing AllClothing. Thus we would either need to recompute for each relevant update, which would incur a high maintenance cost, or we need to store AllClothing in any case in order to maintain V 0 . Therefore, materializing V 0 is not actually likely to be an improvement. In general, given a view definition tree where selections are pushed down as far as possible, all selection conditions above an aggregate must be on the summary at- tributes, and therefore are not incrementally maintainable without storing the entire aggregation results. 7.2 Storing Derivation Views in a Multi-Source Warehouse The problems described earlier (e.g., inefficiency, inconsistency) in warehouse view derivation tracing arise when we apply our tracing queries to remote sources. We therefore may prefer to store auxiliary information about the sources in the warehouse, in order to avoid source queries during each derivation trace. Note that these auxiliary views may be in addition to the intermediate views for aggregate view tracing discussed in Section 7.1. There are various strategies for storing such auxiliary views. A simple extreme solution is to store a copy of each source table in the warehouse. Our algorithm will then query the base views as if they are the sources. However, this solution can be costly and wasteful if a source is large, especially if much of its data does not contribute to the view. Also, computing selections and joins over large base views each time a tuple's derivation is traced can be expensive. Other solutions can in some cases store much less information and still enable derivation tracing without accessing the sources, but the maintenance cost is much higher. We propose an intermediate scheme that achieves low tracing query cost with modest extra storage and maintenance cost. After adding auxiliary views as described in Section 7.1, the view definition is broken down into multiple ASPJ segments. Only views defined by the lowest-level segments are directly over the source tables, and it is these views that are of concern. Let be such a view. Based on V , we introduce an auxiliary view, called the derivation view for V . It contains information about the derivation of each tuple in V over R as specified in Definition 7.2 given next. Theorem 7.3 then shows that any V tuple's derivation in R can be computed with a simple selection and split operation over the derivation view. Definition 7.2 (Derivation View) Let Rm ))) be a one-level ASPJ view over the source tables. The derivation view for V , denoted as DV (v), is Theorem 7.3 (Derivation Tracing using the Derivation View) Let V be a one-level ASPJ view over base tables: DV (v) be v's derivation view as defined in Definition 7.2. Given a tuple t derivation can be computed using DV (v) as follows: Given tuple set derivation is: Example 7.4 (Derivation View for AllClothing) The view AllClothing in Example 7.1 is defined on base tables store, item, and sales. Suppose these base tables are located in remote sources that we cannot or do not wish to access. In order to trace tuple derivations for AllClothing, we maintain a derivation view DV AllClothing. Figure 17 shows the derivation view definition, and Figure shows its contents. The derivation tracer need only to query DV AllCothing to compute view AllClothing's tuple derivations, as shown in Figure 17. Using known techniques, the auxiliary intermediate views and derivation views can be maintained consistently with other views in the warehouse [ZGMW96, ZWGM97]. Note that in cases of warehousing environments where the sources are inaccessible, the auxiliary views themselves need to be made self-maintainable. Known techniques can be used here as well [GJM96, QGMW96]. s a store_name, s sum(num_sold) as total DV_AllClothing AllClothing Figure 17: View definition for DV AllClothing store id store name city state item id item name category price num sold 001 Target Palo Alto CA 0004 pants clothing Albany NY 0004 pants clothing 35 800 003 Macy's San Francisco CA 0003 shirt clothing 45 1500 003 Macy's San Francisco CA 0004 pants clothing 004 Macy's New York City NY 0003 shirt clothing 50 2100 004 Macy's New York City NY 0004 pants clothing 70 1200 Figure AllClothing table 7.3 A Warehousing System Supporting Derivation Tracing Figure 19 illustrates an overall warehouse structure that supports tuple derivation tracing with materialized intermediate aggregation results and derivation views. Recall question Q2 from Example 3.2. The query tree on the left side of Figure 19 is the original definition of view Clothing. In order to trace Clothing's tuple derivations, an auxiliary view AllClothing is maintained to record the intermediate aggregation results (as discussed in Section 7.1). Furthermore, to trace tuples in AllClothing, derivation view DV AllClothing is maintained (as discussed in Section 7.2). The final set of materialized views is: AllClothing = ff store name, sum(num sold) as total (DV AllClothing) Each view can be computed and maintained based on the views (or base tables) directly beneath it using warehouse view maintenance techniques [QGMW96, ZWGM97]. Solid arrows on the right side of Figure 19 show the query and answer data flow. Ordinary view queries are sent to the view Clothing, while derivation queries are sent to the Derivation Tracer module. The tracer takes a request for the derivation of a tuple t in Clothing and queries auxiliary view AllClothing for t's derivation T 1 in AllClothing as specified in Theorem 5.3. The tracer then queries DV AllClothing for the derivation T 2 of T 1 over D as specified in Theorem 7.3. T 2 is t's derivation over D. There are alternative derivation views to the one we described that trade tracing query cost for store item sales Sources Warehouse Clothing User s store item sales AllClothing derivation over D derivation over AllClothing Clothing DV_AllClothing Derivation Queries: View Queries Derivation Tracer s total > 5000 store_name, sum(num_sold) as total a total total total > 5000 store_name, sum(num_sold) as total a Figure 19: Derivation tracing in the warehouse system storage or maintenance cost. One simple option is to split the derivation view into separate tables that contain the base tuples of each source relation that contribute to the view. This scheme may reduce the storage requirement, but tracing queries must then recompute the join. Of course if accessing the sources is cheap and reliable, then it may be preferable to query the sources directly. However, a compensation log [HZ96] may be needed to keep the tracing result consistent with the warehouse views. Determining whether it is better to materialize the necessary information for derivation tracing or query the sources and recompute information during tracing time in a given setting (based on query cost, update cost, storage constraints, source availability, etc.) is an interesting question left open for future work, and is closely related to results in [Gup97, LQA97]. 8 Related Work Revisited In this section, we revisit some related topics, including top-down Datalog query processing and the view update problem, and examine the differences between those problems and ours. We also show how lineage tracing can be applied to the view update problem. 8.1 Top-down Datalog Query Processing In Datalog, relations are represented as predicates, tuples are atoms (or facts), and queries or views are represented by logical rules. Each rule contains a head (or goal) and a body with some subgoals that can (possibly recursively) derive the head [Ull89]. There are two modes of reasoning in Datalog: the bottom-up (or forward-chaining) mode and the top-down (or backward-chaining) mode. The top-down mode proves a goal by constructing a rule-goal graph with the goal as the top node, scanning the graph top-down, and recursively applying rule-goal unification and atom matching until finding an instantiation of all of the subgoals in the base data. Backtracking is used if a dead-end is met in the searching (proving) process. Top-down evaluation of a Datalog goal thus provides information about the facts in the base data that yield the goal; in other words, it provides the lineage of the goal tuple. Our approach to tracing tuple lineage is obviously different from Datalog top-down processing. Instead of performing rule-goal unification and atom matching one tuple at a time, we generate a single query to retrieve all lineage tuples of a given tuple (or tuple set) in an SPJ or one-level ASPJ view. Our approach is better suited to tracing query optimization (as described in Section 5.3), and we support lineage tracing for aggregation views, which are not handled in Datalog. We do not handle recursion in this paper, although we believe our approach can be extended to recursive views while maintaining efficiency. 8.2 View Update Problem The well-known view update problem is to transform updates on views into updates against the base tables, so that the new base tables will continue to derive the updated view. The problem was first formulated in [DB78]. A view update can be an insertion, deletion, or modification of a view tuple. [Mas84] deduced a set of view update translation rules for different view update commands against select, project, and join views. Given a view update command, in some cases more than one set of base table updates can achieve the same view update effect. Much effort has been focused on finding appropriate translations for specific cases [BS81, CM89, Kel86, LS91]. In general, extra semantic information is needed to choose a translation. View update algorithms cannot be used directly to compute view tuple derivations. First, none of the algorithms we know of consider aggregation. Second, although the algorithms do identify a set of base tuples that can affect a view tuple, the base tuples identified may not even derive the view tuple, and therefore do not satisfy our Definition 4.5 of a view tuple derivation. In general, the view update approach and the derivation tracing approach answer two different questions: "Which base tuples can affect a view tuple?" and "Which base tuples exactly derive a view tuple?", respectively. The two questions are not equivalent, but they are related to each other. Our derivation tracing algorithms can be used to guide the view update process to find an appropriate view update translation in some cases. For deletions and modifications, our derivation tracing algorithms can directly identify an appropriate set of base relation tuples to modify, as shown in the following example. Example 8.1 (View Update: Deletion) In Example 4.6 (Figure 12), we illustrated the derivation for h2; 8i in view (R)). When the view update command "delete h2; 8i from V " is issued, we can use the tuple derivation to determine that h2; 3i and h2; 5i should be deleted, and these should be the only changes. The updated base table will be which derives the updated view fh1; 6ig. Note that without using tuple derivation tracing, a more naive algorithm might choose also to delete h2; 0i, which maintains "correctness" but deletes more than necessary. For insertions, the problem is harder, since the view tuple being inserted as well as its derivation do not currently exist. Our derivation tracing algorithms can be adapted to identify some components of the possible derivations of a view tuple being inserted, thereby guiding the base tuple insertions. Any attribute that is not projected into the view must be guessed using extra semantics, such as user instructions or base table constraints, or left null. Even here, derivation tracing can help in the "guessing" process in certain cases, as shown in the following example. Example 8.2 (View Update: Insertion) Suppose view update "insert h3; 2i into V " is issued to the view in Example 4.6 (Figure 12). Since h3; 2i is not in view V , we cannot ask for its current derivation. We only know that after we update R, R should produce a new view with h3; 2i in it. According to the tracing query for V (as specified in Theorem 6.2), we can guess that after the update, the derivation R of h3; 2i must satisfy the condition: 8t 2 R Assuming a constraint that R:Y attributes are positive integers, and considering the requirement sum(R :Y 2, we can also assert that 8t 2 R 2. By further assuming a constraint that R has no duplicates, we can assert that Putting all these assertions together, the only potential derivation of h3; 2i is h3; 2i, so an appropriate base table update is to insert h3; 2i into R. Notice that the inserted tuple in Example 8.2 was carefully chosen. If h3; 8i were inserted instead, we would have to randomly pick a translation from the reasonable ones or ask the user to choose. Even in this case, lineage tracing techniques incorporated with base table constraints can be very useful in reducing the number of possible translations. 9 Conclusions and Future Work We formulated the view data lineage problem and presented lineage tracing algorithms for relational views with aggregation. Our algorithms identify the exact subset of base data that produced a given view data item. We also presented techniques for efficient and consistent lineage tracing in a multi-source warehousing system. Our results can form the basis of a tool by which an analyst can browse warehouse data, then "drill-down" to the source data that produced warehouse data of interest. Follow-on and future work includes the following. ffl We have extended the results in this paper to view definitions that use bag (instead of set) semantics, and to additional relational operators including [ and \Gamma. Please see [CWW98]. We also plan to extend our work to handle recursive views. ffl Tuple derivations as defined in this paper explain how certain base relation tuples cause certain view tuples to exist. As such, derivation tracing is a useful technique for investigating the origins of potentially erroneous view data. However, in some cases a view tuple may be erroneous not (only) because the base tuples that derive it are erroneous, but because base relation tuples that should appear in the derivation are missing. For example, a base tuple may contribute to the wrong group in an aggregate view because its grouping value is incorrect. We plan to explore how this "missing derivation data" problem can be addressed in our lineage framework. ffl In Sections 7.1 and 7.2 we discussed trade-offs associated with materializing versus recomputing intermediate and derivation views, and we mentioned briefly self-maintainability of auxiliary views. We are in the process of conducting a comparative performance study of the various options. ffl We will apply our derivation tracing techniques to the view schema evolution problem, as motivated in Section 1. ffl We will further study how lineage tracing can be incorporated with existing techniques to help solve the view update problem. As seen in Example 8.2, one interesting problem is to extend our derivation tracing algorithms to handle tuples not yet in the view. Most importantly, we are implementing a lineage tracing package within the WHIPS data warehousing prototype at Stanford[WGL + 96]. Once the basic algorithms are completed, we plan to experiment with appropriate user interface tools through which an analyst can obtain and browse derivation information. For example, the analyst may wish to see not only the base derivation data itself, but also a representation of the process by which the view data item was derived. In summary, data lineage is a rich problem with many interesting applications. In this paper we provide an initial practical solution for lineage tracing in data warehouses, and we plan to extend our work in the many directions outlined above. Acknowledgements We are grateful to Sudarshan Chawathe, Himanshu Gupta, Jeff Ullman, Vasilis Vassalos, Yue Zhuge, and all of our WHIPS group colleagues for helpful and enlightening discussions. --R Update semantics of relational views. An overview of data warehousing and OLAP technology. Derived data update in semantic databases. A complete solution for tracing the lineage of relational view data. On the updatability of relational views. Recovering information from summary data. Data cube: A relational aggregation operator generalizing group-by Data integration using self-maintainable views Maintenance of materialized views: Problems Maintaining views incrementally. Selection of views to materialize in a data warehouse. Issues for on-line analytical mining of data warehouses Managing derived data in the Gaea scientific DBMS. Implementing data cubes effi- ciently A framework for supporting data integration using the materialized and virtual approaches. Choosing a view update translator by Dialog at view definition time. Physical database design for data ware- housing Updating relational views using knowledge at view definition and view update time. A relational database view update translation mechanism. Maintenance of data cubes and summary tables in a warehouse. Making views self-maintainable for data warehousing Maintenance expressions for views with aggregation. Incremental recomputation of active relational expres- sions Database and Knowledge-base Systems (Vol <Volume>2</Volume>) Research issues in data warehousing. A system prototype for warehouse view maintenance. Research problems in data warehousing. Supporting fine-grained data lineage in a database visualization environment The Strobe algorithms for multi-source warehouse consistency Multiple view consistency for data warehousing. --TR Principles of database and knowledge-base systems, Vol. I Research problems in data warehousing A framework for supporting data integration using the materialized and virtual approaches An overview of data warehousing and OLAP technology Update semantics of relational views The Strobe algorithms for multi-source warehouse consistency Making views self-maintainable for data warehousing Implementation of integrity constraints and views by query modification Data Integration using Self-Maintainable Views Supporting Fine-grained Data Lineage in a Database Visualization Environment Multiple View Consistency for Data Warehousing Physical Database Design for Data Warehouses Data Cube Concurrency Control Theory for Deferred Materialized Views Selection of Views to Materialize in a Data Warehouse Managing Derived Data in the Gaea Scientific DBMS Aggregate-Query Processing in Data Warehousing Environments Recovering Information from Summary Data --CTR Ling Wang , Elke A. 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Green , Grigoris Karvounarakis , Val Tannen, Provenance semirings, Proceedings of the twenty-sixth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, June 11-13, 2007, Beijing, China James Annis , Yong Zhao , Jens Voeckler , Michael Wilde , Steve Kent , Ian Foster, Applying Chimera virtual data concepts to cluster finding in the Sloan Sky Survey, Proceedings of the 2002 ACM/IEEE conference on Supercomputing, p.1-14, November 16, 2002, Baltimore, Maryland Peter Buneman , Sanjeev Khanna , Wang-Chiew Tan, On propagation of deletions and annotations through views, Proceedings of the twenty-first ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, June 03-05, 2002, Madison, Wisconsin Alon Halevy , Michael Franklin , David Maier, Principles of dataspace systems, Proceedings of the twenty-fifth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, p.1-9, June 26-28, 2006, Chicago, IL, USA Michael Bhlen , Johann Gamper , Christian S. 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lineage;materialized views;data warehouse;derviation

358017

A method for obtaining digital signatures and public-key cryptosystems.

An encryption method is presented with the novel property that publicly revealing an encryption key does not thereby reveal the corresponding decryption key. This has two important consequences: Couriers or other secure means are not needed to transmit keys, since a message can be enciphered using an encryption key publicly revealed by the intended recipient. Only he can decipher the message, since only he knows the corresponding decryption key. A message can be signed using a privately held decryption key. Anyone can verify this signature using the corresponding publicly revealed encryption key. Signatures cannot be forged, and a signer cannot later deny the validity of his signature. This has obvious applications in electronic mail and electronic funds transfer systems. A message is encrypted by representing it as a number M, raising M to a publicly specified power e, and then taking the remainder when the result is divided by the publicly specified product, n, of two large secret prime numbers p and q. Decryption is similar; only a different, secret, power d is used, where e * 1)). The security of the system rests in part on the difficulty of factoring the published divisor, n.

Introduction The era of "electronic mail" [10] may soon be upon us; we must ensure that two important properties of the current "paper mail" system are preserved: (a) messages are private, and (b) messages can be signed . We demonstrate in this paper how to build these capabilities into an electronic mail system. At the heart of our proposal is a new encryption method. This method provides an implementation of a "public-key cryptosystem," an elegant concept invented by Diffie and Hellman [1]. Their article motivated our research, since they presented the concept but not any practical implementation of such a system. Readers familiar with [1] may wish to skip directly to Section V for a description of our method. II Public-Key Cryptosystems In a "public key cryptosystem" each user places in a public file an encryption procedure That is, the public file is a directory giving the encryption procedure of each user. The user keeps secret the details of his corresponding decryption procedure D. These procedures have the following four properties: (a) Deciphering the enciphered form of a message M yields M . Formally, (b) Both E and D are easy to compute. (c) By publicly revealing E the user does not reveal an easy way to compute D. This means that in practice only he can decrypt messages encrypted with E, or compute D efficiently. (d) If a message M is first deciphered and then enciphered, M is the result. Formally An encryption (or decryption) procedure typically consists of a general method and an encryption key. The general method, under control of the key, enciphers a message M to obtain the enciphered form of the message, called the ciphertext C. Everyone can use the same general method; the security of a given procedure will rest on the security of the key. Revealing an encryption algorithm then means revealing the key. When the user reveals E he reveals a very inefficient method of computing D(C): testing all possible messages M until one such that E(M) = C is found. If property (c) is satisfied the number of such messages to test will be so large that this approach is impractical. A function E satisfying (a)-(c) is a "trap-door one-way function;" if it also satisfies (d) it is a "trap-door one-way permutation." Diffie and Hellman [1] introduced the concept of trap-door one-way functions but did not present any examples. These functions are called "one-way" because they are easy to compute in one direction but (apparently) very difficult to compute in the other direction. They are called "trap- door" functions since the inverse functions are in fact easy to compute once certain private "trap-door" information is known. A trap-door one-way function which also satisfies (d) must be a permutation: every message is the cipertext for some other message and every ciphertext is itself a permissible message. (The mapping is "one- to-one" and "onto"). Property (d) is needed only to implement "signatures." The reader is encouraged to read Diffie and Hellman's excellent article [1] for further background, for elaboration of the concept of a public-key cryptosystem, and for a discussion of other problems in the area of cryptography. The ways in which a public-key cryptosystem can ensure privacy and enable "signatures" (described in Sections III and IV below) are also due to Diffie and Hellman. For our scenarios we suppose that A and B (also known as Alice and Bob) are two users of a public-key cryptosystem. We will distinguish their encryption and decryption procedures with subscripts: EA III Privacy Encryption is the standard means of rendering a communication private. The sender enciphers each message before transmitting it to the receiver. The receiver (but no unauthorized person) knows the appropriate deciphering function to apply to the received message to obtain the original message. An eavesdropper who hears the transmitted message hears only "garbage" (the ciphertext) which makes no sense to him since he does not know how to decrypt it. The large volume of personal and sensitive information currently held in computerized data banks and transmitted over telephone lines makes encryption increasingly important. In recognition of the fact that efficient, high-quality encryption techniques are very much needed but are in short supply, the National Bureau of Standards has recently adopted a "Data Encryption Standard" [13, 14], developed at IBM. The new standard does not have property (c), needed to implement a public-key cryptosystem. All classical encryption methods (including the NBS standard) suffer from the "key distribution problem." The problem is that before a private communication can begin, another private transaction is necessary to distribute corresponding encryption and decryption keys to the sender and receiver, respectively. Typically a private courier is used to carry a key from the sender to the receiver. Such a practice is not feasible if an electronic mail system is to be rapid and inexpensive. A public-key cryptosystem needs no private couriers; the keys can be distributed over the insecure communications channel. How can Bob send a private message M to Alice in a public-key cryptosystem? First, he retrieves EA from the public file. Then he sends her the enciphered message EA (M ). Alice deciphers the message by computing DA (EA . By property (c) of the public-key cryptosystem only she can decipher EA (M ). She can encipher a private response with EB , also available in the public file. Observe that no private transactions between Alice and Bob are needed to establish private communication. The only "setup" required is that each user who wishes to receive private communications must place his enciphering algorithm in the public file. Two users can also establish private communication over an insecure communications channel without consulting a public file. Each user sends his encryption key to the other. Afterwards all messages are enciphered with the encryption key of the recipient, as in the public-key system. An intruder listening in on the channel cannot decipher any messages, since it is not possible to derive the decryption keys from the encryption keys. (We assume that the intruder cannot modify or insert messages into the channel.) Ralph Merkle has developed another solution [5] to this problem. A public-key cryptosystem can be used to "bootstrap" into a standard encryption scheme such as the NBS method. Once secure communications have been established, the first message transmitted can be a key to use in the NBS scheme to encode all following messages. This may be desirable if encryption with our method is slower than with the standard scheme. (The NBS scheme is probably somewhat faster if special-purpose hardware encryption devices are used; our scheme may be faster on a general-purpose computer since multiprecision arithmetic operations are simpler to implement than complicated bit manipulations.) IV Signatures If electronic mail systems are to replace the existing paper mail system for business transactions, "signing" an electronic message must be possible. The recipient of a signed message has proof that the message originated from the sender. This quality is stronger than mere authentication (where the recipient can verify that the message came from the sender); the recipient can convince a "judge" that the signer sent the message. To do so, he must convince the judge that he did not forge the signed message himself! In an authentication problem the recipient does not worry about this possibility, since he only wants to satisfy himself that the message came from the sender. An electronic signature must be message-dependent, as well as signer-dependent. Otherwise the recipient could modify the message before showing the message-signature pair to a judge. Or he could attach the signature to any message whatsoever, since it is impossible to detect electronic "cutting and pasting." To implement signatures the public-key cryptosystem must be implemented with trap-door one-way permutations (i.e. have property (d)), since the decryption algorithm will be applied to unenciphered messages. How can user Bob send Alice a "signed" message M in a public-key cryptosystem? He first computes his "signature" S for the message M using DB : (Deciphering an unenciphered message "makes sense" by property (d) of a public-key cryptosystem: each message is the ciphertext for some other message.) He then encrypts S using EA (for privacy), and sends the result EA (S) to Alice. He need not send M as well; it can be computed from S. Alice first decrypts the ciphertext with DA to obtain S. She knows who is the presumed sender of the signature (in this case, Bob); this can be given if necessary in plain text attached to S. She then extracts the message with the encryption procedure of the sender, in this case EB (available on the public file): She now possesses a message-signature pair (M;S) with properties similar to those of a signed paper document. Bob cannot later deny having sent Alice this message, since no one else could have created Alice can convince a "judge" that EB so she has proof that Bob signed the document. Alice cannot modify M to a different version M 0 , since then she would have to create the corresponding signature S Therefore Alice has received a message "signed" by Bob, which she can "prove" that he sent, but which she cannot modify. (Nor can she forge his signature for any other message.) An electronic checking system could be based on a signature system such as the above. It is easy to imagine an encryption device in your home terminal allowing you to sign checks that get sent by electronic mail to the payee. It would only be necessary to include a unique check number in each check so that even if the payee copies the check the bank will only honor the first version it sees. Another possibility arises if encryption devices can be made fast enough: it will be possible to have a telephone conversation in which every word spoken is signed by the encryption device before transmission. When encryption is used for signatures as above, it is important that the encryption device not be "wired in" between the terminal (or computer) and the communications channel, since a message may have to be successively enciphered with several keys. It is perhaps more natural to view the encryption device as a "hardware subroutine" that can be executed as needed. We have assumed above that each user can always access the public file reliably. In a "computer network" this might be difficult; an "intruder" might forge messages purporting to be from the public file. The user would like to be sure that he actually obtains the encryption procedure of his desired correspondent and not, say, the encryption procedure of the intruder. This danger disappears if the public file "signs" each message it sends to a user. The user can check the signature with the public file's encryption algorithm E PF . The problem of "looking up" E PF itself in the public file is avoided by giving each user a description of E PF when he first shows up (in person) to join the public-key cryptosystem and to deposit his public encryption procedure. He then stores this description rather than ever looking it up again. The need for a courier between every pair of users has thus been replaced by the requirement for a single secure meeting between each user and the public file manager when the user joins the system. Another solution is to give each user, when he signs up, a book (like a telephone directory) containing all the encryption keys of users in the system. Our Encryption and Decryption Methods To encrypt a message M with our method, using a public encryption key (e; n), proceed as follows. (Here e and n are a pair of positive integers.) First, represent the message as an integer between 0 and n \Gamma 1. (Break a long message into a series of blocks, and represent each block as such an integer.) Use any standard representation. The purpose here is not to encrypt the message but only to get it into the numeric form necessary for encryption. Then, encrypt the message by raising it to the eth power modulo n. That is, the result (the ciphertext C) is the remainder when M e is divided by n. To decrypt the ciphertext, raise it to another power d, again modulo n. The encryption and decryption algorithms E and D are thus: Note that encryption does not increase the size of a message; both the message and the ciphertext are integers in the range 0 to n \Gamma 1. The encryption key is thus the pair of positive integers (e; n). Similarly, the decryption key is the pair of positive integers (d; n). Each user makes his encryption public, and keeps the corresponding decryption key private. (These integers should properly be subscripted as in nA ; e A , and dA , since each user has his own set. However, we will only consider a typical set, and will omit the subscripts.) How should you choose your encryption and decryption keys, if you want to use our method? You first compute n as the product of two primes p and q: These primes are very large, "random" primes. Although you will make n public, the factors p and q will be effectively hidden from everyone else due to the enormous difficulty of factoring n. This also hides the way d can be derived from e. You then pick the integer d to be a large, random integer which is relatively prime to 1). That is, check that d satisfies: ("gcd" means "greatest common divisor"). The integer e is finally computed from p; q, and d to be the "multiplicative inverse" of d, modulo (p 1). Thus we have e We prove in the next section that this guarantees that (1) and (2) hold, i.e. that E and D are inverse permutations. Section VII shows how each of the above operations can be done efficiently. The aforementioned method should not be confused with the "exponentiation" technique presented by Diffie and Hellman [1] to solve the key distribution problem. Their technique permits two users to determine a key in common to be used in a normal cryptographic system. It is not based on a trap-door one-way permutation. Pohlig and Hellman [8] study a scheme related to ours, where exponentiation is done modulo a prime number. VI The Underlying Mathematics We demonstrate the correctness of the deciphering algorithm using an identity due to Euler and Fermat [7]: for any integer (message) M which is relatively prime to n, Here OE(n) is the Euler totient function giving number of positive integers less than n which are relatively prime to n. For prime numbers p, In our case, we have by elementary properties of the totient function [7]: Since d is relatively prime to OE(n), it has a multiplicative inverse e in the ring of integers modulo OE(n): e We now prove that equations (1) and (2) hold (that is, that deciphering works correctly if e and d are chosen as above). Now and From (3) we see that for all M such that p does not divide M and since (p \Gamma 1) divides OE(n) This is trivially true when so that this equality actually holds for all M . Arguing similarly for q yields Together these last two equations imply that for all M , This implies (1) and (2) for all M; are inverse permutations. (We thank Rich Schroeppel for suggesting the above improved version of the authors' previous proof.) VII Algorithms To show that our method is practical, we describe an efficient algorithm for each required operation. A How to Encrypt and Decrypt Efficiently Computing M e (mod n) requires at most 2 \Delta log 2 (e) multiplications and 2 \Delta log 2 (e) divisions using the following procedure (decryption can be performed similarly using d instead of e): Step 1. Let e k e be the binary representation of e. Step 2. Set the variable C to 1. Step 3. Repeat steps 3a and 3b for Step 3a. Set C to the remainder of C 2 when divided by n. Step 3b. If e set C to the remainder of C \Delta M when divided by n. Step 4. Halt. Now C is the encrypted form of M . This procedure is called "exponentiation by repeated squaring and multiplication." This procedure is half as good as the best; more efficient procedures are known. Knuth [3] studies this problem in detail. The fact that the enciphering and deciphering are identical leads to a simple implementation. (The whole operation can be implemented on a few special-purpose integrated circuit chips.) A high-speed computer can encrypt a 200-digit message M in a few seconds; special-purpose hardware would be much faster. The encryption time per block increases no faster than the cube of the number of digits in n. B How to Find Large Prime Numbers Each user must (privately) choose two large random numbers p and q to create his own encryption and decryption keys. These numbers must be large so that it is not computationally feasible for anyone to factor (Remember that n, but not or q, will be in the public file.) We recommend using 100-digit (decimal) prime numbers p and q, so that n has 200 digits. To find a 100-digit "random" prime number, generate (odd) 100-digit random numbers until a prime number is found. By the prime number theorem [7], about will be tested before a prime is found. To test a large number b for primality we recommend the elegant "probabilistic" algorithm due to Solovay and Strassen [12]. It picks a random number a from a uniform distribution on tests whether a (b\Gamma1)=2 (mod b); (6) where J(a; b) is the Jacobi symbol [7]. If b is prime (6) is always true. If b is composite will be false with probability at least 1=2. If (6) holds for 100 randomly chosen values of a then b is almost certainly prime; there is a (negligible) chance of one in 2 100 that b is composite. Even if a composite were accidentally used in our system, the receiver would probably detect this by noticing that decryption didn't work correctly. When b is odd, a - b, and gcd(a; 1, the Jacobi symbol J(a; b) has a value in f\Gamma1; 1g and can be efficiently computed by the program: if a is even then J(a=2; b) \Delta (\Gamma1) (b 2 \Gamma1)=8 else J(b (mod a); a) \Delta (\Gamma1) (a\Gamma1)\Delta(b\Gamma1)=4 (The computations of J(a; b) and gcd(a; b) can be nicely combined, too.) Note that this algorithm does not test a number for primality by trying to factor it. Other efficient procedures for testing a large number for primality are given in [6,9,11]. To gain additional protection against sophisticated factoring algorithms, p and q should differ in length by a few digits, both (p \Gamma 1) and (q \Gamma 1) should contain large prime factors, and gcd(p \Gamma should be small. The latter condition is easily checked. To find a prime number p such that (p \Gamma 1) has a large prime factor, generate a large random prime number u, then let p be the first prime in the sequence i \Delta Additional security is provided by ensuring that has a large prime factor. A high-speed computer can determine in several seconds whether a 100-digit number is prime, and can find the first prime after a given point in a minute or two. Another approach to finding large prime numbers is to take a number of known factorization, add one to it, and test the result for primality. If a prime p is found it is possible to prove that it really is prime by using the factorization of p \Gamma 1. We omit a discussion of this since the probabilistic method is adequate. C How to Choose d It is very easy to choose a number d which is relatively prime to OE(n). For example, any prime number greater than max(p; q) will do. It is important that d should be chosen from a large enough set so that a cryptanalyst cannot find it by direct search. D How to Compute e from d and OE(n) To compute e, use the following variation of Euclid's algorithm for computing the greatest common divisor of OE(n) and d. (See exercise 4.5.2.15 in [3].) Calculate d) by computing a series x x until an x k equal to 0 is found. Then gcd(x for each x i numbers a i and b i such that x is the multiplicative inverse of x 1 (mod x 0 ). Since k will be less than 2 log 2 (n), this computation is very rapid. If e turns out to be less than log 2 (n), start over by choosing another value of d. This guarantees that every encrypted message (except some "wrap-around" (reduction modulo n) . VIII A Small Example Consider the case can be computed as follows: Therefore the multiplicative inverse (mod 2668) of 2773 we can encode two letters per block, substituting a two-digit number for each letter: 26. Thus the message (Julius Caesar, I, ii, 288, paraphrased) is encoded: in binary, the first block The whole message is enciphered as: 0948 2342 1084 1444 2663 2390 0778 0774 0219 1655 . The reader can check that deciphering works: 948 157 j 920 (mod 2773), etc. IX Security of the Method: Cryptanalytic Ap- proaches Since no techniques exist to prove that an encryption scheme is secure, the only test available is to see whether anyone can think of a way to break it. The NBS standard was "certified" this way; seventeen man-years at IBM were spent fruitlessly trying to break that scheme. Once a method has successfully resisted such a concerted attack it may for practical purposes be considered secure. (Actually there is some controversy concerning the security of the NBS method [2].) We show in the next sections that all the obvious approaches for breaking our system are at least as difficult as factoring n. While factoring large numbers is not provably difficult, it is a well-known problem that has been worked on for the last three hundred years by many famous mathematicians. Fermat (1601?-1665) and Legendre developed factoring algorithms; some of today's more efficient algorithms are based on the work of Legendre. As we shall see in the next section, however, no one has yet found an algorithm which can factor a 200-digit number in a reasonable amount of time. We conclude that our system has already been partially "certified" by these previous efforts to find efficient factoring algorithms. In the following sections we consider ways a cryptanalyst might try to determine the secret decryption key from the publicly revealed encryption key. We do not consider ways of protecting the decryption key from theft; the usual physical security methods should suffice. (For example, the encryption device could be a separate device which could also be used to generate the encryption and decryption keys, such that the decryption key is never printed out (even for its owner) but only used to decrypt messages. The device could erase the decryption key if it was tampered with.) A Factoring n Factoring n would enable an enemy cryptanalyst to "break" our method. The factors of n enable him to compute OE(n) and thus d. Fortunately, factoring a number seems to be much more difficult than determining whether it is prime or composite. A large number of factoring algorithms exist. Knuth [3, Section 4.5.4] gives an excellent presentation of many of them. Pollard [9] presents an algorithm which factors a number n in time O(n 1=4 ). The fastest factoring algorithm known to the authors is due to Richard Schroeppel (unpublished); it can factor n in approximately exp steps (here ln denotes the natural logarithm function). Table 1 gives the number of operations needed to factor n with Schroeppel's method, and the time required if each operation uses one microsecond, for various lengths of the number n (in decimal digits). Table Digits Number of operations Time hours We recommend that n be about 200 digits long. Longer or shorter lengths can be used depending on the relative importance of encryption speed and security in the application at hand. An 80-digit n provides moderate security against an attack using current technology; using 200 digits provides a margin of safety against future developments. This flexibility to choose a key-length (and thus a level of security) to suit a particular application is a feature not found in many of the previous encryption schemes (such as the NBS scheme). Computing OE(n) Without Factoring n If a cryptanalyst could compute OE(n) then he could break the system by computing d as the multiplicative inverse of e modulo OE(n) (using the procedure of Section VII D). We argue that this approach is no easier than factoring n since it enables the cryptanalyst to easily factor n using OE(n). This approach to factoring n has not turned out to be practical. How can n be factored using OE(n)? First, (p + q) is obtained from n and is the square root of (p Finally, q is half the difference of (p Therefore breaking our system by computing OE(n) is no easier than breaking our system by factoring n. (This is why n must be composite; OE(n) is trivial to compute if n is prime.) C Determining d Without Factoring n or Computing OE(n). Of course, d should be chosen from a large enough set so that a direct search for it is unfeasible. We argue that computing d is no easier for a cryptanalyst than factoring n, since once d is known n could be factored easily. This approach to factoring has also not turned out to be fruitful. A knowledge of d enables n to be factored as follows. Once a cryptanalyst knows d he can calculate e which is a multiple of OE(n). Miller [6] has shown that n can be factored using any multiple of OE(n). Therefore if n is large a cryptanalyst should not be able to determine d any easier than he can factor n. A cryptanalyst may hope to find a d 0 which is equivalent to the d secretly held by a user of the public-key cryptosystem. If such values d 0 were common then a brute-force search could break the system. However, all such d 0 differ by the least common multiple of (p \Gamma 1) and (q \Gamma 1), and finding one enables n to be factored. (In (3) and (5), OE(n) can be replaced by lcm(p \Gamma Finding any such d 0 is therefore as difficult as factoring n. Computing D in Some Other Way Although this problem of "computing e-th roots modulo n without factoring n" is not a well-known difficult problem like factoring, we feel reasonably confident that it is computationally intractable. It may be possible to prove that any general method of breaking our scheme yields an efficient factoring algorithm. This would establish that any way of breaking our scheme must be as difficult as factoring. We have not been able to prove this conjecture, however. Our method should be certified by having the above conjecture of intractability withstand a concerted attempt to disprove it. The reader is challenged to find a way to "break" our method. Avoiding "Reblocking" When Encrypting A Signed Message A signed message may have to be "reblocked" for encryption since the signature n may be larger than the encryption n (every user has his own n). This can be avoided as follows. A threshold value h is chosen (say for the public-key cryptosystem. Every user maintains two public (e; n) pairs, one for enciphering and one for signature- verification, where every signature n is less than h, and every enciphering n is greater than h. Reblocking to encipher a signed message is then unnecessary; the message is blocked according to the transmitter's signature n. Another solution uses a technique given in [4]. Each user has a single (e; n) pair where n is between h and 2h, where h is a threshold as above. A message is encoded as a number less than h and enciphered as before, except that if the ciphertext is greater than h, it is repeatedly re-enciphered until it is less than h. Similarly for decryption the ciphertext is repeatedly deciphered to obtain a value less than h. If n is near h re-enciphering will be infrequent. (Infinite looping is not possible, since at worst a message is enciphered as itself.) XI Conclusions We have proposed a method for implementing a public-key cryptosystem whose security rests in part on the difficulty of factoring large numbers. If the security of our method proves to be adequate, it permits secure communications to be established without the use of couriers to carry keys, and it also permits one to "sign" digitized documents. The security of this system needs to be examined in more detail. In particular, the difficulty of factoring large numbers should be examined very closely. The reader is urged to find a way to "break" the system. Once the method has withstood all attacks for a sufficient length of time it may be used with a reasonable amount of confidence. Our encryption function is the only candidate for a "trap-door one-way permuta- tion" known to the authors. It might be desirable to find other examples, to provide alternative implementations should the security of our system turn out someday to be inadequate. There are surely also many new applications to be discovered for these functions. Acknowledgments . We thank Martin Hellman, Richard Schroeppel, Abraham Lempel, and Roger Needham for helpful discussions, and Wendy Glasser for her assistance in preparing the initial manuscript. Xerox PARC provided support and some marvelous text-editing facilities for preparing the final manuscript. Received April 4, 1977; revised September 1, 1977. --R New directions in cryptography. Exhaustive cryptanalysis of the NBS data encryption standard. The Art of Computer Programming Some cryptographic applications of permutation polynomials. Secure communications over an insecure channel. Riemann's hypothesis and tests for primality. An Introduction to the Theory of Numbers. An improved algorithm for computing logarithms over GF (p) and its cryptographic significance. Theorems on factorization and primality testing. Electronic mail. Probabilistic algorithms. A Fast Monte-Carlo test for primality Federal Register Federal Register --TR The art of computer programming, volume 2 (3rd ed.) Secure communications over insecure channels Riemann''s Hypothesis and tests for primality --CTR W. Kchlin, Public key encryption, ACM SIGSAM Bulletin, v.21 n.3, p.69-73, Aug. 1987 Reena Bhaindarkar , Sandhya Suman , Reema Raghava , T. J. Mathew, Design of RSRM protocol and analysis of the computational complexity of RLR algorithm for communication over the internet, Proceedings of the 15th international conference on Computer communication, p.1050-1060, August 12-14, 2002, Mumbai, Maharashtra, India G. C. Meletiou , D. K. Tasoulis , M. N. Vrahatis, Transformations of two cryptographic problems in terms of matrices, ACM SIGSAM Bulletin, v.39 n.4, December 2005 Mihir Bellare , Silvio Micali, How to sign given any trapdoor permutation, Journal of the ACM (JACM), v.39 n.1, p.214-233, Jan. 1992 Ray Bird , Inder Gopal , Amir Herzberg , Phil Janson , Shay Kutten , Refik Molva , Moti Yung, The KryptoKnight family of light-weight protocols for authentication and key distribution, IEEE/ACM Transactions on Networking (TON), v.3 n.1, p.31-41, Feb. 1995 Amitanand S. Aiyer , Lorenzo Alvisi , Allen Clement , Mike Dahlin , Jean-Philippe Martin , Carl Porth, BAR fault tolerance for cooperative services, ACM SIGOPS Operating Systems Review, v.39 n.5, December 2005

electronic funds transfer;privacy;digital signatures;public-key cryptosystems;cryptography;electronic mail;factorization;authentication;prime number;security;message-passing

358684

Efficient Address Generation for Affine Subscripts in Data-Parallel Programs.

Address generation for compiling programs, written in HPF, to executable SPMD code is an important and necessary phase in a parallelizing compiler. This paper presents an efficient compilation technique to generate the local memory access sequences for block-cyclically distributed array references with affine subscripts in data-parallel programs. For the memory accesses of an array reference with affine subscript within a two-nested loop, there exist repetitive patterns both at the outer and inner loops. We use tables to record the memory accesses of repetitive patterns. According to these tables, a new start-computation algorithm is proposed to compute the starting elements on a processor for each outer loop iteration. The complexities of the table constructions are O(k+s2), where k is the distribution block size and s2 is the access stride for the inner loop. After tables are constructed, generating each starting element for each outer loop iteration can run in O(1) time. Moreover, we also show that the repetitive iterations for outer loop are Pk/gcd(Pk,s1), where P is the number of processors and s1 is the access stride for the outer loop. Therefore, the total complexity to generate the local memory access sequences for a block-cyclically distributed array with affine subscript in a two-nested loop is O(Pk/gcd(Pk,s1)+k+s2).

Introduction Generally speaking, data-parallel languages support three regular data distributions: block, cyclic, and block-cyclic data distributions. The block distribution is to distribute contiguous array elements evenly onto processors. The cyclic distribution is to distribute each array element onto processors one at a time and in a round-robin fashion. The distribution that blocks of size k are distributed onto processors in a round-robin fashion is the block-cyclic distribution and is denoted as cyclic(k). The block-cyclic distribution is known to be the most general data distribution. The block and cyclic distributions can be represented by the block-cyclic distribution as cyclic(d NA e) and cyclic(1), respectively, where NA is the number of array elements and P is the number of processors. The address generation problems for compiling array references with block or cyclic distributions have been studied thoroughly [5, 12, 13, 21]. The more general problems for compiling array references with block-cyclic distribution also have been studied extensively [3, 7, 9, 11, 11, 14, 16, 19, 20, 22]. A finite state machine (FSM) approach is proposed to traverse the local memory access sequence of each processor [3]. The method is a table-based approach. The table construction needs to solve k linear Diophantine equations and incurs a sorting operation. The work improving the FSM approach [3] is proposed in [10, 11, 20]. Efficient FSM table generation is proposed. The improved work enumerates the local memory access sequences by viewing the accessed elements an integer lattice. The sorting step in [3] is avoided in the improved work. In [7], the authors use the virtual processors to generate communication sets for each processor. From different viewpoint of a block-cyclic distribution, the virtual processor approach actually contains two approaches, one is termed the virtual block approach and the other is termed the virtual cyclic approach. The virtual block approach views a block-cyclic distribution as a block distribution on a set of virtual processors, which are then cyclically mapped to processors. On the con- trary, the virtual cyclic approach views a block-cyclic distribution as a cyclic distribution on a set of virtual processors, which are then block-wise mapped to processors. The other approaches are similar to either FSM approach or virtual processor approach except some modifications. However, most of them consider the simple array subscript. That is, the array subscripts contain only one induction variable. Recently, several efforts on compiling array references with affine array subscripts are proposed [1, 10, 11, 15, 17, 22]. Affine array subscript means the array subscript is a linear combination of multiple induction variables (MIVs). In [1], the authors use a linear algebra framework to generate communication sets for affine array subscripts. Complex loop bounds and local array subscripts of the generated code will incur significant overhead. A table-based approach is proposed in [22]. The authors classify all blocks into classes and use a class table to record the memory accesses of the first repetitive pattern. By using the class table, they derived the communication sets for each processor. Both [1] and [22] are addressing the compilation of array references with affine subscripts within a multi-nested loop. How- ever, the proposed methods are not efficient enough for dealing with some special case. For compiling array references with affine sub- scripts, some researchers pay their attention on the array reference enclosed within a two-nested loop to find a better result [10, 11, 17]. Based on FSM approach [3], Kennedy et al. proposed another approach to solving the compilation of array references with affine subscripts within a two-nested loop [10, 11]. For the memory accesses of an array reference with affine subscript within a two-nested loop, there exist repetitive patterns both at the outer and inner loops. Moreover, to fix each iteration of the outer loop, the affine subscript is reduced to a simple subscript. Therefore, for each iteration in the outer repetitive pattern, they use FSM approach to generate the local memory access sequences for the inner loop. To use FSM approach at the inner loop, start- computations to decide the initial state of FSM for each iteration of the outer loop are necessary. They proposed an O(Pk) algorithm for a start-computation, where P is the number of processors and k is the distribution block size. For the outer loop, they found that the repetitive iterations are Pk iterations. Hence, the total complexity to generate the local memory access sequence for an array reference with affine subscript within a two-nested loop is O(P Ramanujam et al. proposed an improved work to find the local starting elements on each processor [17]. They first find a factor of basis vectors to jump from a global start to a processor's space and then traverse the lattice until hitting the starting element. ENDDO ENDDO Fig. 1: HPF-like program model considered in the paper. Since a traverse step is incurred, the complexity of their start-computation algorithm is O(k). Thus the total complexity of Ramanujam's algorithm is turned out to be O(Pk 2 ). In this paper, we also propose another new start- computation algorithm. A preprocessing step is required before we compute the starting elements. The complexity of the preprocessing step is O(k where s 2 is the access stride for the inner loop. After preprocessing is done, the time complexity to generate each starting element on a processor is O(1). In addition, we also discover that the outer loop repetitive iterations are Pk= gcd(Pk; s 1 ) instead of Pk, where s 1 is the access stride for the outer loop. Therefore, the total complexity of our proposed approach is O(Pk= gcd(Pk; s 1 which is asymptotical to O(Pk). The proposed approach is not only correct but also efficient. The paper is organized as follows. Section 2 formulates the problem and describes the traditionally techniques to generate local memory access sequences for compiling the array references with affine subscripts within a two-nested loop. An efficient approach to finding the starting elements from a global start is proposed in Section 3. The performance analyses and comparisons with the related work are demonstrated in Section 4. Section 5 concludes the paper. Address Generation for Affine Subscripts 2.1 Problem Formulation Specifically, Fig. 1 illustrates the program model considered in the paper. Array A is distributed onto processors with cyclic(k) distribution. The array reference contains two induction variables i 1 and i 2 . The access strides of the array reference with respect to i 1 and i 2 are s 1 and s 2 , respectively. The access offset of the array reference is o. Fig. 2 is an example of the program model shown in Fig. 1, where Fig. 2(a) is the layout of array elements on processors. The colored elements are the array elements accessed by the array reference in the two-nested loop. Fig. 2(b) shows the global addresses of array elements accessed by every processor. However, data distribution transfers the global addressing space to processors' local spaces. Therefore, what we care is to generate the local addresses on a processor for the accessed ele- ments. Thus the MIV address generation problem is to generate the local addresses of array elements accessed by each processor, just like Fig. 2(c) shows. 2.2 Table -Based Address Generation for Affine Subscripts For the case of the array reference containing single induction variable (SIV), as well-known, the memory accesses on processors have a repetitive pattern. In [3], a finite state machine (FSM) is built to orderly iterate the local memory access sequence on a pro- cessor. Similarly, for the array reference containing multiple induction variables (MIVs), we can extend the technique used for SIV to orderly enumerate the local memory access sequences on a processor. Consider the program model shown in Fig. 1. For each outer iteration, the MIV address generation problem can be reduced to an SIV problem. Thus we can utilize the FSM approach to generate the local memory access sequence for an SIV problem. Generating the local memory access sequence for an MIV problem can, therefore, be easily solved by enumerating the sequence for each outer loop iteration until reaching the outer loop bound. For example, consider the example illustrated in Fig. 2. Suppose the outer loop iteration i 1 0. The two-nested loop is reduced to a single nested loop and the array reference turns out to be A(2i 2 ). Thus a finite state machine (FSM) can be built to enumerate the local memory access sequences for the SIV problem. Fig. 3 illustrates the FSM to generate the local memory access sequences when the array reference contains a single induction variable and the access stride is 2. Fig. 3(a) shows the FSM table, where Next records the next transition state and \DeltaM records the local memory gaps of successive array elements from current state transmitting to the next state. Fig. 3(b) is the transition diagram of the FSM. The initial state of the FSM depends on the position of the starting array element in a block on the processor. For instance, when the starting element on processor p 0 is 0 and its position in a (a) (b) (c) Fig. 2: An MIV address generation example, where Layout of array elements on processors. (b). Global addresses of array elements accessed by every proces- sor. (c). Local addresses of array elements accessed by every processor. State Next \DeltaM (a) (b) Fig. 3: A finite state machine (FSM) to generate the local memory access sequences for an SIV problem with access stride 2. block is 0, thus the initial state of the FSM for the case when i 1 0 is at state 0. In addition to the initial state of the FSM, we also need to know the local address of the starting element since FSM only records the local memory gaps between successive array elements allocated on the processor. FSM has no enough information to show where to start in terms of local address. In other words, although we have the FSM and its initial state, we still do not know the starting local address in this case. For exam- ple, besides the initial state of the FSM being state 0 that we should know, we have to figure out the local address of the starting element. Obviously, when the local address of the starting element 0 on processor p 0 is 0. Therefore, when the local memory access sequence for processor p 0 is 0; 2; 4; and so on. Likewise, if the two-nested loop is reduced to a single nested loop and the array reference is simplified to 37). The finite state machine built for the case of i 1 can still be reused since the memory access stride is still the same (2, for this example). When i 1 1, the starting element on processor is 49 and its position in a block is 1. Thus the initial state of the FSM for the case of i 1 1 is at state 1. However, where to start in terms of local address is the key point now. When i 1 1, the starting element on processor p 0 is its local address on processor p 0 is 13. Therefore, the local memory access sequence for processor p 0 is 13; 15; and so on. Similarly, it is done likewise when the outer loop iteration Accordingly, we can obtain the local memory access sequences on processors as Fig. 2(c) shows. Actually, there is no need to iterate all of the outer loop iterations from 0 to n 1 . We have found out that iterating the outer loop iterations Pk= gcd(Pk; s 1 times is enough because there is a repetitive pattern in the outer loop. Having this discovery can save a lot of time due to the avoidance of recomputation for repetitive patterns. The following theorem demonstrates that the repetitive period of the outer loop is Pk= gcd(Pk; s 1 iterations. Since the space is lim- ited, the proof of the theorem is omitted in the paper. One can refer to [18] for more details. Theorem 1 For the program model shown in Fig. 1, the memory accesses of the array reference have a repetitive pattern and the repetitive period in respect of the outer loop iteration is Pk= gcd(Pk; s 1 tions. 2 According to the above description, evidently, determining the local address of the starting element for each outer loop iteration is the primary step to solve the MIV address generation problem. The problem to find the local address of starting element for each outer loop iteration will be described in the next section. A new approach to generating the local addresses of the starting elements will be presented in the next section as well. Generating Starting Elements The findings of the starting elements for outer loop iterations are important for solving the MIV address generation problem. It is obvious that for a given outer loop iteration the memory accesses just depend on the inner loop access stride s 2 . Therefore, in this section, we use s to indicate the inner loop access stride s 2 except otherwise notified. The method to find the starting elements in case of s - k can be found in [10, 17]. Both of them are O(1) in com- plexity. However, their methods to find the starting elements in case of s ? k are O(Pk) and O(k), re- spectively. We propose a new approach to find the starting elements in case of s ? k and the time complexity of the algorithm is O(1). The problem and its solution are described as follows. 3.1 Problem Description We have given an overall description of the MIV address generation problem in Section 2.2. Finding the starting element on a processor from a given outer loop iteration plays an important role in dealing with the MIV address generation problem. The following we formally describe the induced problem. Let the accessed element for some fixed outer loop iteration (a) (b) Fig. 4: Starting elements on every processor for the example shown in Fig. 2. (a). Global addresses of starting elements on every processor. (b). Local addresses of starting elements on every processor. be a global start and G be the local address of the global start. Specifically, given a global start G, the processor p on which G is allocated and the processor q which we would like to find its starting element, the problem is to figure out S q , the local address of the starting element, for processor q. For example, consider the example shown in Fig. 2(a). The gray colored elements are the elements accessed by the array reference, in which the deep-colored shaded elements are the global starts corresponding to every outer loop iterations and the light-colored shaded elements on each processor are the starting elements corresponding to every global starts. Suppose a given global start is 37, which local address is 9 on processor . The starting elements on processors p 0 , and p 3 are 49, 41, and 45, respectively, in terms of global addresses. The problem is to figure out the local addresses of these starting elements. That is, 13, 9, and 9, respectively. The starting elements for every outer loop iteration are shown in Fig. 4. The global and local addresses of the starting elements on every processor are listed in Figs. 4(a) and 4(b), respectively. The goal of the induced problem is to obtain a table containing the local addresses of the starting elements on some required processor for every global start, as Fig. 4(b) shows for that required processor. 3.2 Preprocessing Given a global start G, we propose a new approach to find the local address of the starting element S q for processor q in case of s ? k. The proposed approach is a table-based approach. In our approach, it is necessary to pre-compute a few tables in order to evaluate the starting elements for a given global start. In this section, we only describe the character- Fig. 5: An one-level mapping example to illustrate the ideas of the tables used in the paper, where it assumes that array elements are distributed over 4 processors with cyclic(4) distribution and the access stride is 5. istics of these tables and how it works in the proposed approach. The complexities in time and space will be analyzed in Section 4. For the sake of space limi- tation, the constructions of the tables are omitted in the paper. For further details, please refer to [18]. 3.2.1 C2P, P2C, and Offset Tables As well-known, the accessed elements on blocks have a repetitive pattern. By [22], all blocks can be classified into s classes according to the positions of the accessed elements on a block. Note that blocks of the same class have the same format. All blocks can be numbered in terms of class according to the rule: b mod s , where b is the block number of that block. A repetitive pattern contains blocks from class 0 to class s 1 and which is termed a class cycle in [22]. In addition, since s ? k, there is at most one accessed element on a block. Therefore, we can use a table to record the position of the only accessed element for every class. Different from [22], we assume that the accessed element on the block of class 0 is at the first position, that is, at position 0. The blocks with no accessed element are recorded by "-". With the table we can easily and efficiently get the position of an accessed element on a block if the class number of the block is given. Therefore, we can easily deduce S q from G since all blocks have been classified into classes. We denote the table recording the position of the accessed element on every class as C2P table. As an example, let us suppose that array elements are distributed over 4 processors with cyclic(4) distribution and the access stride is 5. The layout of array elements on processors is illustrated in Fig. 5. In this figure, the accessed elements are those elements with a white text on a black background. Obviously, all blocks are classified into 5 classes. Each class is colored by the gradations of gray color. The class number of a block is labeled at the bottom of the block. The blocks bounded by a dashed line indicate a repetitive pattern. The accessed elements on class are at positions 0, 1, 2, and 3, respec- tively. Therefore, the values of C2P(0), (1), (2), and (3) are 0, 1, 2, and 3, respectively. Moreover, there is no accessed element in class 4. So, C2P(4)="- ". Thus we can obtain the C2P table. Fig. 7(a) illustrates the C2P table for the example shown in Fig. 5. We can get the position of an accessed element on a block according to the class number of a block by using C2P table. In contrast to C2P table, a P2C table is to record the class number according to the position of accessed element on a block. Thus we can obtain the class number of a block according to the position of the accessed element on the block. Generally speaking, we can obtain the class number of a block according to the position of the accesses element on the block by using C2P table. However, it requires a search operation and, in some cases, we can not recognize the class number of a block according to the C2P table. For instance, when the number of classes is smaller than the block size, it is possible that, for some position, there is no class that its accessed element is at that position. It would cause confusion to recognize the class number by that posi- tion. Therefore, P2C table is necessary. P2C table can be constructed by C2P table. For example, considering C2P table shown in Fig. 7(a), one can scan the table to obtain P2C table. P2C table is illustrated in Fig. 7(b). Note that, for the above case that the number of classes is smaller than the block size, for the position that has no class number to correspond to, we assume the class number of that position to be the class number of the previous po- sition. For example, suppose that the distribution block size is 4 and the access stride is 6. All blocks can be classified into 3 classes. The accessed elements on class 0, 1, and 2 are at the positions of 0, 2, and -, respectively. As a result, positions at 0 and 2 are corresponding to the classes 0 and 1, re- spectively. We have P2C=(0, -, 1, -). Obviously, positions at 1 and 3 have no suitable class numbers to correspond to. With the above assumption, the class number corresponded by position 1 is 0, the same as that of the previous position. Similarly, the class number corresponded by position 3 is 1. Thus, we have P2C=(0, 0, 1, 1). Basically, C2P and P2C tables are in some sense like a hash table. The reason to make the above assumption for P2C table construction is to solve the offset prob- lem. The offset problem can be solved by the assumption in conjunction with another table Offset. Generally speaking, a global start G can be at any position in a block. However, as constructing C2P table, we have assumed that the accessed element on the block of class 0 should be at position 0, Further- more, as we construct P2C table, for the position that has no class number to correspond to, we assign the class number of the previous value to the current value. Nevertheless, according to C2P table, the class number has its real position to correspond to. Consequently, there is a difference between the real position and the assumed position if we make such an assumption. In order to make use of C2P table in every case, we use another table to record the difference in order to make up the shortcomings of C2P table. The table is denoted asOffset table in the paper. It has been discussed that when the number of classes is larger than the block size, each position has its suitable class number to correspond to. In such a case, Offset table is of no use. Fig. 5 is an example under such a condition and the Offset table is shown in Fig. 7(c). On the other hand, if the number of classes is smaller than the block size, Follow the example used in the explanation of P2C table, in which it assumes that the distribution block size is 4 and the access stride is 6. The C2P and P2C tables are (0, 2, -) and (0, 0, 1, 1), respectively. Since position 1 has no suitable class number to correspond to, we assign the class number corresponded by position 0 to position 1. Although, position 1 correspond to class 0, the real position of accessed element on the block of class 0 is at position 0 according to C2P ta- ble. Thus there is a 1 difference between the assumed value and the real value. As a result, Offset(1)=1. Similarly, Offset(3)=1. There is no problem on positions 0 and 2 since they have their suitable class numbers to correspond to. Consequently, Offset table is (0, 1, 0, 1). 3.2.2 NextActive and Jump Tables As previously described, a block contains at most one accessed element when the access stride is larger than the block size. Of course, a block may contain no accessed element at all in such a case. Thus, we name a block that has elements to be accessed as an active block; otherwise, it is termed an empty block. The tables NextAct and Jump that we would like to introduced are used for jumping over the empty (a) (b) Fig. (a) A one group ordered sequence. (b) A multiple groups ordered sequence. blocks to an active block. One important observation here is that, from processor's viewpoint, blocks on processors have a repetitive pattern in terms of classes. To explain concretely, let us take a look at the example shown in Fig. 5. The blocks on processor are in classes 0, 4, 3, 2, 1, and then repeat again from class 0. Similar situation also happens on processors . The sequence of class numbers on p 1 is 1, 0, 4, 3, 2, and that for p 2 is 2, 1, 0, 4, 3, and that for p 3 is 3, 2, 1, 0, 4. It is interesting that the sequence of class numbers on each processor is the same except the initial class number on each processor. That is, the sequence of class numbers on each processor can be viewed as the sequence 0, 4, and the initial class numbers for p 0 and p 3 are 0, 1, 2, and 3, respectively. We use the notation (0; 4; 3; 2; 1) to denote the ordered sequence. Clearly, all class numbers have appeared in the ordered sequence. Thus, we say that there is only one group in the ordered sequence. Fig. 6(a) illustrates the one group ordered sequence for this example. It should be addressed that it is possible that the sequence of class numbers on each processor may be different and there may be more than one group in an ordered sequence. However, groups are mutually disjoint and a processor can belong to one and only one group. We give an example to illustrate the phenomenon. Suppose that array elements are distributed over 2 processors with cyclic(3) distribution and the access stride is 12. There are 4 classes for this example. The sequence of class numbers on 2. and that on p 1 is 1, 3. The ordered sequence can be represented as (0; 2)(1; 3). Obviously, the ordered sequence contains two groups. One is (0; 2) and another is (1; 3). (0; 2) and (1; are mutually disjoint. Processor p 0 belongs to the group (0; belongs to the group (1; 3). Fig. 6(b) illustrates the multiple groups ordered sequence for (a) (c) Offset (d) NextAct Fig. 7: Tables used for starting elements findings. this example. It is important to have such a discovery since we can obtain the class number of the next block on a processor from current block if the class number of the current block is known. Based on the discovery, we use one table to record the class number of the next active block from current block on a processor and another to record how many empty blocks we need to skip over. They are named NextAct and Jump, respectively. The constructions of the two tables are based on the ordered sequence and C2P table. If current block is not an empty block, we need not to jump any block. Thus, the value in NextAct table for that block is recorded by its class number and that in Jump table is recorded by 0. Otherwise, we can traverse the ordered sequence to find an active block. Then the value in NextAct table for that block is recorded by the class number of the active block and that in Jump table is recorded by the number of blocks that we have traversed. If we can not find an active block, both the values in Nex- tAct and Jump tables are recorded by "-". Such as the example where array elements are distributed over 4 processors with cyclic(4) distribution and the access stride is 8, the NextAct and Jump tables are (0, -) and (0, -), respectively. Although a processor can belong to one and only one group, NextAct table is suitable for all processors since the construction of the table is based on the class number, not on group. The NextAct and Jump tables for the example shown in Fig. 5 are illustrated in Fig. 7(d) and (e), respectively. 3.3 The Algorithm With these tables we can evaluate the starting element from a given global start in O(1) time complex- ity. Fig. 8 illustrates the algorithm to evaluate the starting element from a given global start. We term the algorithm Start Computation algorithm. The basic concept of the Start Computation algorithm Algorithm: Start Computation algorithm for the case of s ? k. Input: G, a global start, p, the processor where the global start is allocated, q, the processor that we would like to find its starting element, where q 6= p k, the distribution block size, P , the number of processors, s, the access stride, C, the number of classes, where C2P, P2C, Offset, NextAct, and Jump tables. Output: S q , the starting element on processor q. Steps: 1. pos 2. dist 3. 4. pos 5. if pos 7. return no starting element on q 8. else pos Jump(c)\Lambdak 9. endif 10. endif 12. if q ! p then 13. 14. endif 16. return S q Fig. 8: Start Computation algorithm for the case of is that, from the viewpoint of the global start, we try to figure out the distance between the starting element and the global start. With the distance we can, therefore, get the local address of the starting element by adding the distance to the local address of the global start. The details of the algorithm is described as follows. Given G, the local address of a global start, and where G is allocated, Step 1 is to calculate the position on a block for the global start. The obtained value is stored in pos g . Step 2 is to measure the distance between processors p and q, which is then stored in dist. In Step 3, P2C(pos g ) can obtain the class number of the block which the global start is on. Thus, Step 3 can get the class number of the block on processor q, which may contain the starting element. The class number obtained in Step 3 is represented by c. According to C2P table, C2P(c) can get the position of the accessed element on the block of class c, if ever. Therefore, Step 4 can obtain the position on a block for the starting element on processor q. The obtained position is represented by pos s . If pos s does not equal "-", it means that there is an accessed element on the block. Of course, the accessed element is the starting element. We can go direct to Step 11 to evaluate the distance between the starting element and the global start. If q ! p, we still need to add a size of a block to the distance since the starting element must be at one more course than the global start. That is what Steps 12-14 has done. As a result, the local address of the starting element can be obtained, just as Step 15 shows. On the other hand, if pos means that there has no accessed element on the block. We can use NextAct table to obtain the class number of the next active block. If NextAct(c) = "-", it implies that there exists no active block on the processor. Certainly, there is no starting element on the pro- cessor. Otherwise, which means that we can find an active block on the processor, we can get the number of blocks required to jump from the current block to the next active block and the position of the accessed element on the active block from Jump and NextAct tables, respectively. Since pos s in Step 4 represents the position of the starting element on the block with the same course as the global start, hence, the distance caused by the number of blocks required to jump to an active block should be added to pos s in such a case. Thus, we have Step 8. Steps after 11 are the same as explained in the previous paragraph. Let us take Fig. 9 as an example, where it assumes that Given an global start 37, whose local address is 9 on processor we first to find the starting element for processor . The input of the Start Computation algorithm is . The tables used for the example are the same as shown in Fig. 7. Following the Steps from 1 to 4 in the algorithm we can obtain that pos 2. Since pos s does not equal "-", we go direct to Step 11 and we obtain that offset = 1. Due to the invalidation of the condition in Step 12, we go direct to Step 15 and we have S 2 which corresponds to the array element 42 in terms of global address. On the same input except we take the finding of the starting element on processor p 0 as Fig. 9: Layout of array elements on processors for the case of s 2 ? k, another MIV example, where another example. After executing the Step 4, we have pos 4, and pos Since pos s equals "-", which means that the block contains no accessed element, we go to Step 6. According to NextAct and Jump tables, there is an active block at one block after the empty block on processor p 0 . By Step 8, we have pos 7. After Step 11, we have offset = 6. As needs to add 4, the size of a block. It turns out that which corresponds to the array element 67 in terms of global address. Evidently, the time complexity of Start Computation algorithm is O(1). The complexity analyses of the tables used in the algorithm and the performance comparisons against the existing methods will be discussed in Section 4. Performance Analyses and Comparisons Performance analyses of the tables used in Start Computation algorithm are shown in Table 1. Performance comparisons of our method against the existing methods are shown in Table 2. For the sake of space limitation, please refer to [18] for more details. Conclusions In this paper, we have presented an efficient approach to the evaluation of the starting element for some processor from a given global start, which is a key step to solve the MIV address generation problem Table 1: Performance Analyses. Table Complexity Time Space Offset NextAct O(C) C Table 2: Performance Comparisons. [10]'s [17]'s Ours start comp. O(1) O(1) O(1) preprocess O(1) O(1) O(s 2 start comp. O(Pk) O(k) O(1) outer loop repetitive Pk Pk Pk iterations Total in data-parallel programs, assuming array is block- cyclically distributed and its subscript is affine. The approach is a table-based approach. The constructions of these tables require O(s plexity, where s 2 is the access stride of the inner loop. With these tables, the Start Computation algorithm can run in O(1) time. In addition, we have discovered that there exists a repetitive pattern for every iterations. Therefore, the MIV address generation problem can be solved in is the access stride of the outer loop. In the future, we would like to apply the address generation approach to evaluate communication sets. Furthermore, the address generation and communication sets evaluation for general affine subscripts are also under investigation. --R A linear algebra framework for static HPF code distribution. Programming in Vienna Fortran. Generating local addresses and communication sets for data parallel programs. Automatic Parallelization for Distributed-Memory Multiprocessing Systems Concrete Mathematics. On compiling array expressions for efficient execution on distributed-memory machines High Performance Fortran Forum. Efficient address generation for block-cyclic distri- butions A linear-time algorithm for computing the memory access sequence in data-parallel programs Compiling global name-space parallel loops for distributed execu- tion Local iteration set computation for block-cyclic distributions Computing the local iteration set of a block-cyclically distributed reference with affine subscripts Optimizing the representation of local iteration sets and access sequences for block-cyclic distributions Code generation for complex subscripts in data-parallel programs Generating communication for array state- ments: Design Efficient computation of address sequences in data parallel programs using closed forms for basis vectors. An Optimizing Fortran D Compiler for MIMD Distributed-Memory Machines Compiling array references with affine functions for data-parallel programs --TR Concrete mathematics: a foundation for computer science Compile-time generation of regular communications patterns Vienna FortranMYAMPERSANDmdash;a Fortran language extension for distributed memory multiprocessors The high performance Fortran handbook Generating communication for array statements Compilation techniques for block-cyclic distributions An optimizing Fortran D compiler for MIMD distributed-memory machines Generating local addresses and communication sets for data-parallel programs A linear-time algorithm for computing the memory access sequence in data-parallel programs Efficient address generation for block-cyclic distributions Compiling array expressions for efficient execution on distributed-memory machines Efficient computation of address sequences in data parallel programs using closed forms for basis vectors An Empirical Study of Fortran Programs for Parallelizing Compilers Compiling Global Name-Space Parallel Loops for Distributed Execution Code Generation for Complex Subscripts in Data-Parallel Programs

affine subscripts;data-parallel languages;data distribution;address generation;multiple induction variables MIVs;distributed-memory multicomputers;single program multiple data SPMD

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Solving Fundamental Problems on Sparse-Meshes.

AbstractA sparse-mesh, which has PUs on the diagonal of a two-dimensional grid only, is a cost effective distributed memory machine. Variants of this machine have been considered before, but none are as simple and pure as a sparse-mesh. Various fundamental problems (routing, sorting, list ranking) are analyzed, proving that sparse-meshes have great potential. It is shown that on a two-dimensional $n \times n$ sparse-mesh, which has $n$ PUs, for h-relations can be routed in $(h steps. The results are extended for higher dimensional sparse-meshes. On a $d$-dimensional $n \times \cdots \times n$ sparse-mesh, with h-relations are routed in $(6 \cdot (d - 1) / \epsilon - steps.

Introduction On ordinary two-dimensional meshes we must accept that, due to their small bisection width, for most problems the maximum achievable speed-up with n 2 processing units (PUs) is only \Theta(n). On the other hand, networks such as hypercubes impose increasing conditions on the interconnection modules with increasing network sizes. Cube-connected-cycles do not have this problem, but are harder to program due to their irregularity. Anyway, because of a basic theorem from VLSI lay-out [18], all planar architectures have an area that is quadratic in their bisection-width. But this in turn means that, except for several special problems with much locality (the Warshall algorithm most notably), we must accept that the hardware cost is quadratic in the speed-up that we may hope to obtain (later we will also deal with three-dimensional lay-outs). Once we have accepted this, we should go for the simplest and cheapest architecture achieving this, and a good candidate is the sparse-mesh considered in this paper. Network. A sparse-mesh with n PUs consists of a two-dimensional n \Theta n grid of buses, with PUs connected to them at the diagonal. The horizontal buses are called row-buses, the vertical buses are called column-buses. PU i , send data along the i-th row-bus and receive data from the i-th column-bus. In one step, PU i can send one packet of standard length to an arbitrary PU j . The packet first travels along the i-th row-bus to position (i; j) and then turns into the j-th column-bus. Bus conflicts are forbidden: the algorithm must guarantee that in every step a bus is used for the transfer of only one packet. See Figure 1 for an illustration. In comparison to a mesh, we have reduced the number of PUs from n 2 to saving substantially on the hardware cost. In comparison with other Max-Planck-Institut f?r Informatik, Im Stadtwald, Saarbr-ucken, Germany, jopsi@mpi- sb.mpg.de, http://www.mpi-sb.mpg.de/-jopsi/. Figure 1: A sparse-mesh with 8 PUs and an 8 \Theta 8 network of buses. The large circles indicate the PUs with their indices, the smaller circles the connections between the buses. networks, we have a very simple interconnection network that can be produced easily and is scalable without problem. The sparse-mesh is very similar to the Parallel Alternating Direction Machine, PADAM, considered in [2, 3] and the coated-mesh considered in [10]. Though similar in inspiration, the sparse-mesh is simpler. In Section 6 the network is generalized for higher dimensions. Problems. Parallel computation is possible only provided that the PUs can exchange data. Among the many communication patterns, patterns in which each PU sends and receives at most h packets, h-relations, have attracted most attention. An h-relation is balanced, if every PU sends h=n packets to each PU. Lemma 1 On a sparse-mesh with n PUs, balanced h-relations can be performed in h steps. Unfortunately, not all h-relations are balanced, and thus there is a need for algorithms that route arbitrary h-relations efficiently. Also for cases that the PUs have to route less than n packets, the above algorithm does not work. Offline, all h-relations can be routed in h steps. The algorithm proceeds as follows: Algorithm offline route 1. Construct a bipartite graph with n vertices on both sides. Add an edge from Node i on the left to Node j on the right for each packet going from PU i to PU j . 2. Color this h-regular bipartite graph with h colors. 3. Perform h routing steps. Route all packets whose edges got Color t, 0 - h, in Step t. This coloring idea is standard since [1]. Its feasibility is guaranteed by Hall's theorem. As clearly at least h steps are required for routing an h-relation, the offline algorithm is optimal. This shows that the h-relation routing problem is equivalent to the problem of constructing a conflict-free bus-allocation schedule. In [3] routing h-relations is considered for the PADAM. This network is so similar to the sparse-mesh, that most results carry over. It was shown that for h - n, h-relations can be routed in O(h) time. Further it was shown that 1-optimality can be achieved for log log n). Here and in the remainder we call a routing algorithm c-optimal, if it routes all h-relations in at most c \Delta Other fundamental problems that should be solved explicitly on any network, are sorting and list ranking. With h keys/nodes per PU, their communication complexity is comparable to that of an h-relation. In [3], the list-ranking problem was solved asymptotically optimally for log log n). New Results. In this paper we address the problems of routing, sorting and list ranking. Applying randomization, it is rather easy to route n ffl -relations, (2=ffl)-optimally. An intricate deterministic algorithm is even (1=ffl)- optimal. This might be the best achievable. We present a deterministic sampling technique, by which all routing algorithms can be turned into sorting algorithms with essentially the same time consumption. In addition to this, we give an algorithm for ranking randomized lists, that runs in 6=ffl steps for lists with nodes per PU. Our generalization of the sparse-mesh to higher- dimensions is based on the generalized diagonals that were introduced in [6]. At least for three dimensions this makes practical sense: n 2 PUs are interconnected by a cubic amount of hardware, which means an asymptotically better ratio than for two-dimensional sparse-meshes. On higher dimensional sparse-meshes everything becomes much harder, because it is no longer true that all one-relations can be routed in a single step. We consider our deterministic routing algorithm for higher dimensional sparse-meshes to be the most interesting of the paper. All results constitute considerable improvements over those in [3]. The results of this paper demonstrate that networks like the sparse-mesh are versatile, not only in theory, but even with a great practical potential: we show that for realistic sizes of the network, problems with limited parallel slackness can be solved efficiently. All proofs are omitted due to a lack of space. Randomized Routing Using the idea from [19] it is easy to obtain a randomized 2-optimal algorithm for large h. The algorithm consists of two randomizations, routings in which the destinations are randomly distributed. In Round 1, all packets are routed to randomly chosen intermediate destinations; in Round 2, all packets are routed to their actual destinations. Lemma 2 On a sparse-mesh with n PUs, h-relations can be routed in 2 \Delta h+o(h) steps, for all log n), with high probability. We show how to route randomizations for n). First the PUs are divided into subsets of PUs, S then we do the following: Algorithm random route 1. Perform In Superstep t, its packets with destination in S (j+t) mod p n to PU i in S (j+t) mod 2. Perform In Superstep t, n, sends all its packets with destination in PU (i+t) mod n in S j to their destinations. This approach can easily be generalized. For log n), the algorithm consists of 1=ffl routing rounds. In Round r, 1 - r - 1=ffl, packets are routed to the subsets, consisting of n 1\Gammaffl\Deltar PUs, in which their destinations lie. Theorem 1 On a sparse-mesh with n PUs, random route routes randomizations with log n) in (h probability. Arbitrary h-relations can be routed in twice as many steps. List ranking is the problem of determining the rank for every node of a set of to the final node of its list. h denotes the number of nodes per PU. The edges of a regular bipartite graph of degree m can be colored by splitting the graph in two subgraphs log m times, each with half the previous degree [4]. Lev, Pippenger and Valiant [11] have shown that each halving step can be performed by solving two problems that are very similar to list ranking (determining the distance to the node with minimal index on a circle). Thus, if list-ranking is solved in time T , then bipartite graphs of degree m can be colored in time In [3] list ranking on PADAMs is performed by simulating a work- optimal PRAM algorithm. For coloring the eges of a bipartite regular graph, the above idea was used: Lemma 3 [3] On a sparse-mesh with n PUs, list ranking can be solved in O(h) steps, for all h - n \Delta log log n. A bipartite regular graph with h \Delta n edges, can be colored in O(log(h \Delta n) \Delta h) steps. The algorithms are asymptotically optimal, but the hidden constants are quite bad, and the range of applicability is limited to large h. In the following we present a really good and versatile randomized list-ranking algorithm. It is based on the repeated-halving algorithm from [14]. This algorithm has the unique property, that not only the number of participating nodes is reduced in every round, but that the size of the processor network is reduced as well. Generally, the value of this property is limited, but on the sparse-mesh, where we need a certain minimal h for efficient routing, this is very nice. 3.1 Repeated-Halving It is assumed that the nodes of the lists are randomly distributed over the PUs. If this is not the case, they should be randomized first. The algorithm consists of log n reduction steps. In each step, first the set of PUs is divided in two halves, . The nodes in S 0 with current successor in S 1 and vice-versa, are called masters, the other nodes are non-masters. The current successor of a node is stored in its crs field. Each PU holds nodes. Algorithm reduce 1. Each non-master p follows the links until a master or a final node is reached and sets crs(p) to this node. 2. Each master p asks 3. Each master p asks In a full algorithm one must also keep track of the distances. After Step 2, each master in S i , points to the subsequent master in S (i+1) mod 2 . Thus, after Step 3, each master in S i , points to the subsequent master in S i itself. Now recursion can be applied on the masters. This does not involve communication between S 0 and S 1 anymore. Details are provided in [14]. The expected number of masters in each subset equals h \Delta n=4. Thus, after log n rounds, we have n subproblems of expected size h=n, that can be solved internally. Hereafter, the reduction must be reversed: Algorithm expand 1. Each non-master p asks For Step 1 of reduce we repeat pointer-jumping rounds as long as necessary. In every such round, each node p who has not yet reached a master or a final node, asks and the distance thereto. Normally pointer- jumping is inefficient, but in this case, because the expected length of the lists is extremely short, the number of nodes that participates decreases rapidly with each performed round [16]. Theorem 2 On a sparse-mesh with n PUs, list ranking can be solved in 14 \Delta h+ o(h) steps, for all log n), with high probability. For smaller h, the average number of participating nodes per PU decreases to less than 1. This is not really a problem: for such a case, the maximum number of participating nodes in any PU can easily be estimated on O(log n), and the routing operations can be performed in O(log 2 n). In comparison to the the total routing time, the cost of these later reduction rounds is negligible. Using Theorem 1, this gives Corollary 1 On a sparse-mesh with n PUs, list ranking can be solved in 14=ffl \Delta log n), with high probability. 3.2 Sparse-Ruling-Sets The performance of the previous algorithm can be boosted by first applying the highly efficient sparse-ruling-sets algorithm from [13] to reduce the number of nodes by a factor of !(1). We summarize the main ideas. Algorithm sparse ruling sets 1. In each PU randomly select h 0 nodes as rulers. 2. The rulers initiate waves that run along their list until they reach the next ruler or a final element. If a node p is reached by a wave from a ruler p 0 , then 3. The rulers send their index to the ruler whose wave reached them. By a wave, we mean a series of sending operations in which a packet that is destined for a node p is forwarded to crs(p). Hereafter, we can apply the previous list-ranking algorithm for ranking the rulers. Finally each non-ruler p asks mst(p) for the index of the last element of its list and the distance thereof. Details can be found in [13], particularly the initial nodes must be handled carefully. Theorem 3 On a sparse-mesh with n PUs, list ranking can be solved in 6=ffl \Delta log n), with high probability. Under the same conditions, a bipartite regular graph with h \Delta n edges, can be colored in O(log(h steps. 4 Faster Routing Our goal is to construct a deterministic O(1=ffl)-optimal algorithm for routing -relations. As in [3], we apply the idea from [17] for turning offline routing algorithms into online algorithms by solving Step 2 of offline route after sufficient reduction of the graph online. However, here we obtain much more interesting results. log n), and define f by log n) 1=2 . As in random route, we are going to perform 1=ffl rounds. In Round r, 1 - r - 1=ffl, all packets are routed to the subsets, consisting of n 1\Gammaffl\Deltar PUs each, in which their destinations lie. We describe Round 1. First the PUs are divided in n ffl sub- each consisting of n 1\Gammaffl PUs. Then, we perform the following steps: Algorithm determine destination 1. Each PU i , its packets on the indices of their destination subsets. The packets going to the same S j , are filled into superpackets of size f \Delta log n, leaving one partially filled or empty superpacket for each j. The superpackets p, going to the same S j , are numbered consecutively and starting from 0 with numbers a p . ff ij denotes the total number of superpackets going from S i to S j . 2. For each j, 0 the PUs perform a parallel prefix on the ff ij , to make the numbers A available in PU i , 3. For each superpacket p in PU i , dest For each superpacket p with destination in S j , dest p gives the index of the PU in S j , to which p is going to be routed first. Because the PUs in S j are the destination of h \Delta n 1\Gammaffl =(f \Delta log n)+n superpackets, the bipartite graph with n nodes on both sides and one edge for every superpacket has degree h=(f \Delta log n) Its edges can be colored in o(h) time. If the edge corresponding to a superpacket gets Color t, 1 - t ! h=(f \Delta log n)+n ffl , then p is going to be routed in superstep t. Lemma 4 On a sparse-mesh with n PUs, and log n) all packets can be routed to their destination subsets of size n 1\Gammaffl in h steps. Repeating the above steps 1=ffl times, the packets eventually reach their destinations Theorem 4 On a sparse-mesh with n PUs, h-relations with can be routed in (h steps. Our algorithm is not entirely deterministic: the underlying list-ranking algorithm is randomized. It is likely that deterministic list-ranking for a problem with nodes per PU can be performed in O(h=ffl) time. But, for our main claim, that n ffl -relations can be routed (1=ffl)-optimally, we do not need this: if n) the size of the superpackets can be taken f \Delta log 2 n, and we can simply apply pointer-jumping for the list-ranking. 5 Faster Sorting On meshes the best deterministic routing algorithm is more or less a sorting algorithm [9, 8]. For sparse-meshes the situation is different: the routing algorithm presented in Section 4 is in no way a sorting algorithm. However, it can be enhanced to sort in essentially the same time. We first consider log n). This case also gives the final round in the sorting algorithm for smaller h hereafter. Define log(n). In [15] it is shown how to deterministically select a high-quality sample. The basic steps are: Algorithm refined sampling 1. Each PU internally sorts all its keys and selects those with ranks j \Delta h=m, m, to its sample. 2. Perform log n merge and reduce rounds: in Round r, 0 - r ! log n, the samples, each of size m, in two subsets of 2 r PUs are merged, and only the keys with even ranks are retained. 3. Broadcast the selected sample M of size m to all PUs. Lemma 5 On a sparse-mesh with n PUs, refined sampling selects a sample of size m in o(h) steps. The global rank, Rank p , of an element p, with rank rank p among the elements of M satisfies Thus, by merging the sample and its packets, a PU can estimate for each of its packets p its global rank Rank p with an error bounded by O(h This implies that if we set dest for each packet p, where rank p is defined as above, that then at most h packets have the same dest-value. Thus all packets can be routed to the PU given by their dest-values in h steps. Furthermore, after each PU has sorted all the packets it received, the packets can be rearranged in o(h) steps so that each PU holds exactly h packets. Lemma 6 On a sparse-mesh with n PUs, the sorting algorithm based on refined sampling sorts an h-relation in h For log n), we stick close to the pattern of the routing algorithm from Section 4: we perform 1=ffl rounds of bucket-sort with buckets of decreasing sizes. . For each round of the bucket sorting, we run refined sampling with log(n). This sample selection takes steps. In Round r, 1 - r - 1=ffl, the sample is used to guess in which subset of size n 1\Gammar=ffl each packet belongs. Then they are routed to these subsets with the algorithm of Section 4. Theorem 5 On a sparse-mesh with n PUs, the sorting algorithm based on refined sampling sorts an h-relation in (h+o(h))=ffl steps, for all The remark at the end of Section 4 can be taken over here: n) is required for making the algorithm deterministic, but this has no impact on the claim that (1=ffl)-optimality can be achieved for 6 Higher Dimensional Sparse-Meshes One of the shortcomings of the PADAM [2, 3] is that it has no natural generalization for higher-dimensions. The sparse-mesh can be generalized easily. subset of a d-dimensional n \Theta \Delta \Delta \Delta \Theta n grid is a diagonal, if the projection of all its points on any of the (d \Gamma 1)-dimensional coordinate planes is a bijection. For this we need the following definition (a simplification of the definition in [6]): Diag Lemma 7 [6] In a d-dimensional n \Theta \Delta \Delta \Delta \Theta n grid, Diag d is a diagonal. Figure 2: A three-dimensional 4 \Theta 4 \Theta 4 PUs, such that if they were occupied by towers in a three-dimensional chess game, none of them could capture another one. The d-dimensional sparse-mesh, consists of the n d\Gamma1 PUs on Diag d , inter-connected by d \Delta n d\Gamma1 buses of length n. The PU at position denoted PU x0 ;:::;x . The routing is dimension-ordered. Thus, after t substeps, d, a packet traveling from PU x0 ;:::;x to PU x 0;:::;x 0 has reached position The scheduling should exclude bus conflicts. Diag 3 is illustrated in Figure 2. 6.1 Basic Results Now it is even harder to find a conflict-free allocation of the buses. Different from before, it is no longer automatically true, that every one-relation can be routed offline in one step (but see Lemma 10!). For example, in a four-dimensional sparse-mesh, under the permutation at the positions (0; 0; x pass through position (0; 0; 0; 0). However, for the most common routing pattern, balanced h-relations, this is no problem, because it can be written as the composition of n shifts. A shift is a routing pattern under which, for all x the packets in have to be routed to PU (x0+s0 ) mod n;:::;(x given s Lemma 8 A shift in which each PU contributes one packets can be routed in one step. Lemma 8 implies that the randomized routing algorithm from Section 2 carries on with minor modifications. One should first solve the case of large log n) and then extend the algorithms to smaller Theorem 6 On a d-dimensional sparse-mesh with n d\Gamma1 PUs, the generalization of random route routes randomizations with log n) in (d \Gamma 1) steps, with high probability. Arbitrary h-relations can be routed in twice as many steps. The algorithm for ranking randomized lists of Section 3 has the complexity of a few routing operations, and this remains true. The same holds for our technique of Section 5 for enhancing a routing algorithm into a sorting algorithm with the same complexity. The only algorithm that does not generalize is the deterministic algorithms of Section 4, because it is not based on balanced h-relations. 6.2 Deterministic Routing In this section we analyze the possibilities of deterministic routing on higher-dimensional sparse-meshes. Actually, for the sake of a simple notation, we describe our algorithms for two-dimensional sparse-meshes, but we will take care that they consist of a composition of shifts. For log n), we can match the randomized result. The algorithm is a deterministic version of the randomized one: it consists of two phases. In the first phase the packets are smoothed-out, in the second they are routed to their destinations. Smoothing-out means rearranging the packets so that every PU holds approximately the same number of packets with destinations in each of the PUs. Let Algorithm deterministic route 1. Each PU i , its packets on the indices of their destination PU. The packets going to the same PU j , superpackets of size f \Delta log n, leaving one partially filled or empty superpacket for each j. 2. Construct a bipartite graph with n nodes on both sides, and an edge from Node i on the left to Node j on the right, for every superpacket going from PU i to PU j . Color this graph with (f colors. A superpacket p with Color dest 3. Perform shifts. In Shift t, routes the dest to PU (i+t) mod n . 4. Perform shifts. In Shift t, routes the with destination in PU (i+t) mod n to this PU. The algorithm works, because the coloring assures that for each source and destination PU there are exactly f +1 packets with the same dest-value. This implies that both routings are balanced. Theorem 7 On a d-dimensional sparse-mesh with n d\Gamma1 PUs deterministic route routes h-relations with steps. For smaller h, ideas from deterministic route are combined with ideas from random route: the algorithm consists of rounds in which the packets are smoothed-out and then routed to the subsets in which their destinations lie. The fact that this second phase must be balanced, implies that the smoothing must be almost perfect. Another essential observation is that if packets are redistributed among subsets of PUs, and if this routing has to be balanced, that then the number of subsets should not exceed h. Let log n), and set . The PUs are divided in n ffl subsets S i of n 1\Gammaffl PUs each. Then all packets are routed to the subset in which their destinations lie, by performing Algorithm deterministic route 1. Each PU i , its packets on the indices of their destination PU. The packets going to the same S j , are filled into superpackets of size f \Delta log n, leaving one partially filled or empty superpacket for each j. 2. Construct a bipartite graph with n ffl nodes on both sides, and an edge from Node i on the left to Node j on the right, for every superpacket going from S i to S j . Color this graph with (f colors. For a superpacket p with Color 3. In each S i , the rank, rank p , of each packet p with dest among the packets p 0 with dest 4. Rearrange the packets within the S i , , such that thereafter a packet p with rank stands in PU j mod n 1\Gammaffl of S i . 5. Perform shifts. In Shift t, routes the f dest to PU i in S (j+t) mod n ffl . 6. In each S i , the rank, rank p , of each packet p with destination in S j among the packets with destination in this subset. 7. Rearrange the packets within the S i , , such that thereafter a packet p with rank stands in PU j mod n 1\Gammaffl of S i . 8. Perform shifts. In Shift t, routes the f with destination in S (j+t) mod n ffl to PU i in this subset. The ranks can be computed as in Section 4 in O(n ffl \Delta log steps. The coloring guarantees that Lemma 9 After Step 4, for each j, 0 every PU holds exactly f superpackets p with dest holds exactly f with destination in S j . The rearrangements within the subsets are routings with a large h as described by Theorem 7 (the superpackets do not need to be filled into supersuperpackets!), and thus they can be performed without further recursion. Theorem 8 On a d-dimensional sparse-mesh with n d\Gamma1 PUs deterministic route routes h-relations with log n) in (6 steps. From an algorithmic point of view deterministic route is the climax of the paper: all developed techniques are combined in a non-trivial way, to obtain a fairly strong result. As in Section 4, n) is required for also making the coloring deterministic. 6.3 Three-Dimensional Sparse-Meshes Three-dimensional sparse-meshes are practically the most interesting generaliza- tion. It is a lucky circumstance that for them even the results from Section 4 carry on, because of the following On a three-dimensional sparse-mesh, each one-relation can be routed in a single step. Proof: The permutation is a composition of OE corrects the ith coordinate. OE 0 maps a position Diag 3 to a position The injectivity of the projection on the plane x carries over to OE 0 . Thus, after OE 0 , no two packets stand in the same position. Analogously, it follows that the inverse of OE 2 , is an injection. Thus, not even at the beginning of OE 2 , that is after OE 1 , two packets may have been sharing a position. leads to the following analogue of Theorem 4: Theorem 9 On a three-dimensional n \Theta n \Theta n sparse-mesh, h-relations with log n) can be routed in (2 steps. The coated-mesh [10] can be trivially generalized for higher dimensions. How- ever, whereas routing on a two-dimensional coated-mesh is almost as easy as on a sparse-mesh, there is no analogue of Lemma 10 for three-dimensional coated- meshes. 7 Conclusion Our analysis reveals that the sparse-mesh has a very different character than the mesh. Whereas on a mesh several approaches essentially give the same result, we see that on a sparse-mesh, there is a great performance difference. For column-sort performs poorly, because the operations in subnetworks are costly, while on a mesh they are for free. On a mesh also the randomized algorithm inspired by [19] performs optimal [12, 7], because there the worst-case time consumption is determined by the time the packets need to cross the bisection. For the sparse-mesh this argument does not apply, and our deterministic routing algorithm is twice as fast. --R 'A Unified Framework for Off-Line Permutation Routing in Parallel Networks,' Mathematical Systems Theory 'On Edge Coloring Bipartite Graphs,' SIAM Journal on Computing An Introduction to Parallel Algorithms 'Randomized Multipacket Routing and Sorting on Meshes,' Algorithmica 'Work-Optimal Simulation of PRAM Models on Meshes,' Nordic Journal of Computing 'A Fast Parallel Algorithm for Routing in Permutation Networks,' IEEE Transactions on Computers 'k-k Routing, k-k Sorting, and Cut-Through Routing on the Mesh,' Journal of Algorithms 'From Parallel to External List Ranking,' Techn. Permutation Routing and Sorting on Meshes with Row and Column Buses --TR --CTR Martti Forsell, A parallel computer as a NOC region, Networks on chip, Kluwer Academic Publishers, Hingham, MA,

routing;theory of parallel computation;algorithms;meshes;sorting;networks;list-ranking

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Data prefetch mechanisms.

The expanding gap between microprocessor and DRAM performance has necessitated the use of increasingly aggressive techniques designed to reduce or hide the latency of main memory access. Although large cache hierarchies have proven to be effective in reducing this latency for the most frequently used data, it is still not uncommon for many programs to spend more than half their run times stalled on memory requests. Data prefetching has been proposed as a technique for hiding the access latency of data referencing patterns that defeat caching strategies. Rather than waiting for a cache miss to initiate a memory fetch, data prefetching anticipates such misses and issues a fetch to the memory system in advance of the actual memory reference. To be effective, prefetching must be implemented in such a way that prefetches are timely, useful, and introduce little overhead. Secondary effects such as cache pollution and increased memory bandwidth requirements must also be taken into consideration. Despite these obstacles, prefetching has the potential to significantly improve overall program execution time by overlapping computation with memory accesses. Prefetching strategies are diverse, and no single strategy has yet been proposed that provides optimal performance. The following survey examines several alternative approaches, and discusses the design tradeoffs involved when implementing a data prefetch strategy.

Introduction By any metric, microprocessor performance has increased at a dramatic rate over the past decade. This trend has been sustained by continued architectural innovations and advances in microprocessor fabrication technology. In contrast, main memory dynamic RAM (DRAM) performance has increased at a much more leisurely rate, as shown in Figure 1. This expanding gap between microprocessor and DRAM performance has necessitated the use of increasingly aggressive techniques designed to reduce or hide the large latency of memory accesses [16]. Chief among the latency reducing techniques is the use of cache memory hierarchies [34]. The static RAM (SRAM) memories used in caches have managed to keep pace with processor memory request rates but continue to be too expensive for a main store technology. Although the use of large cache hierarchies has proven to be effective in reducing the average memory access penalty for programs that show a high degree of locality in their addressing patterns, it is still not uncommon for scientific and other data-intensive programs to spend more than half their run times stalled on memory requests [25]. The large, dense matrix operations that form the basis of many such applications typically exhibit little locality and therefore can defeat caching strategies. The poor cache utilization of these applications is partially a result of the "on demand" memory fetch policy of most caches. This policy fetches data into the cache from main memory only after the processor has requested a word and found it absent from the cache. The situation is illustrated in Figure 2a where computation, including memory references satisfied within the cache hierarchy, are represented by the upper time line while main memory access time is represented by the lower time line. In this figure, the data blocks associated with memory references r1, r2, and r3 are not found in the cache hierarchy and must therefore be fetched from main memory. Assuming the referenced data word is needed immediately, the processor will be stalled while it waits for the corresponding cache block to be fetched. Once the data returns from main memory it is cached and forwarded to the processor where computation may again proceed. Year Performance101000 System Figure 1. System and DRAM performance since 1988. System performance is measured by SPECfp92 and DRAM performance by row access times. All values are normalized to their 1988 equivalents (source: Internet SPECtable, ftp://ftp.cs.toronto.edu/pub/jdd/spectable). Note that this fetch policy will always result in a cache miss for the first access to a cache block since only previously accessed data are stored in the cache. Such cache misses are known as cold start or compulsory misses. Also, if the referenced data is part of a large array operation, it is likely that the data will be replaced after its use to make room for new array elements being streamed into the cache. When the same data block is needed later, the processor must again bring it in from main memory incurring the full main memory access latency. This is called a capacity miss. Many of these cache misses can be avoided if we augment the demand fetch policy of the cache with the addition of a data prefetch operation. Rather than waiting for a cache miss to perform a memory fetch, data prefetching anticipates such misses and issues a fetch to the memory system in advance of the actual memory reference. This prefetch proceeds in parallel with processor computation, allowing the memory system time to transfer the desired data from main memory to the cache. Ideally, the prefetch will complete just in time for the processor to access the needed data in the cache without stalling the processor. An increasingly common mechanism for initiating a data prefetch is an explicit fetch instruction issued by the processor. At a minimum, a fetch specifies the address of a data word to be brought into cache space. When the fetch instruction is executed, this address is simply passed on to the memory system without forcing the processor to wait for a response. The cache responds to the fetch in a manner similar to an ordinary load instruction with the exception that the AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA3010(a) (b) (c) Time AAAA AAAA AAAA AAAA AAAA AAAA Computation Memory Acesss Cache Hit Cache Miss Prefetch Figure 2. Execution diagram assuming a) no prefetching, b) perfect prefetching and c) degraded prefetching. referenced word is not forwarded to the processor after it has been cached. Figure 2b shows how prefetching can be used to improve the execution time of the demand fetch case given in Figure 2a. Here, the latency of main memory accesses is hidden by overlapping computation with memory accesses resulting in a reduction in overall run time. This figure represents the ideal case when prefetched data arrives just as it is requested by the processor. A less optimistic situation is depicted in Figure 2c. In this figure, the prefetches for references r1 and r2 are issued too late to avoid processor stalls although the data for r2 is fetched early enough to realize some benefit. Note that the data for r3 arrives early enough to hide all of the memory latency but must be held in the processor cache for some period of time before it is used by the processor. During this time, the prefetched data are exposed to the cache replacement policy and may be evicted from the cache before use. When this occurs, the prefetch is said to be useless because no performance benefit is derived from fetching the block early. A prematurely prefetched block may also displace data in the cache that is currently in use by the processor, resulting in what is known as cache pollution. Note that this effect should be distinguished from normal cache replacement misses. A prefetch that causes a miss in the cache that would not have occurred if prefetching was not in use is defined as cache pollution. If, however, a prefetched block displaces a cache block which is referenced after the prefetched block has been used, this is an ordinary replacement miss since the resulting cache miss would have occurred with or without prefetching. A more subtle side effect of prefetching occurs in the memory system. Note that in Figure 2a the three memory requests occur within the first 31 time units of program startup whereas in Figure 2b, these requests are compressed into a period of 19 time units. By removing processor stall cycles, prefetching effectively increases the frequency of memory requests issued by the processor. Memory systems must be designed to match this higher bandwidth to avoid becoming saturated and nullifying the benefits of prefetching. This is can be particularly true for multiprocessors where bus utilization is typically higher than single processor systems. It is also interesting to note that software prefetching can achieve a reduction in run time despite adding instructions into the execution stream. In Figure 3, the memory effects from Figure 2 are ignored and only the computational components of the run time are shown. Here, it can be seen that the three prefetch instructions actually increase the amount of work done by the processor. Several hardware-based prefetching techniques have also been proposed which do not require the use of explicit fetch instructions. These techniques employ special hardware which monitors the processor in an attempt to infer prefetching opportunities. Although hardware prefetching incurs no instruction overhead, it often generates more unnecessary prefetches than software prefetching. Unnecessary prefetches are more common in hardware schemes because they speculate on future memory accesses without the benefit of compile-time information. If this speculation is incorrect, cache blocks that are not actually needed will be brought into the cache. Although unnecessary prefetches do not affect correct program behavior, they can result in cache pollution and will AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA Prefetch Overhead Prefetching Figure 3. Software prefetching overhead. consume memory bandwidth. To be effective, data prefetching must be implemented in such a way that prefetches are timely, useful, and introduce little overhead. Secondary effects in the memory system must also be taken into consideration when designing a system that employs a prefetch strategy. Despite these obstacles, data prefetching has the potential to significantly improve overall program execution time by overlapping computation with memory accesses. Prefetching strategies are diverse and no single strategy has yet been proposed which provides optimal performance. In the following sections, alternative approaches to prefetching will be examined by comparing their relative strengths and weaknesses. 2. Background Prefetching, in some form, has existed since the mid-sixties. Early studies [1] of cache design recognized the benefits of fetching multiple words from main memory into the cache. In effect, such block memory transfers prefetch the words surrounding the current reference in hope of taking advantage of the spatial locality of memory references. Hardware prefetching of separate cache blocks was later implemented in the IBM 370/168 and Amdahl 470V [33]. Software techniques are more recent. Smith first alluded to this idea in his survey of cache memories [34] but at that time doubted its usefulness. Later, Porterfield [29] proposed the idea of a "cache load instruction" with several RISC implementations following shortly thereafter. Prefetching is not restricted to fetching data from main memory into a processor cache. Rather, it is a generally applicable technique for moving memory objects up in the memory hierarchy before they are actually needed by the processor. Prefetching mechanisms for instructions and file systems are commonly used to prevent processor stalls, for example [38,28]. For the sake of brevity, only techniques that apply to data objects residing in memory will be considered here. Non-blocking load instructions share many similarities with data prefetching. Like prefetches, these instructions are issued in advance of the data's actual use to take advantage of the parallelism between the processor and memory subsystem. Rather than loading data into the cache, however, the specified word is placed directly into a processor register. Non-blocking loads are an example of a binding prefetch, so named because the value of the prefetched variable is bound to a named location (a processor register, in this case) at the time the prefetch is issued. Although non-blocking loads will not be discussed further here, other forms of binding prefetches will be examined. Data prefetching has received considerable attention in the literature as a potential means of boosting performance in multiprocessor systems. This interest stems from a desire to reduce the particularly high memory latencies often found in such systems. Memory delays tend to be high in multiprocessors due to added contention for shared resources such as a shared bus and memory modules in a symmetric multiprocessor. Memory delays are even more pronounced in distributed-memory multiprocessors where memory requests may need to be satisfied across an interconnection network. By masking some or all of these significant memory latencies, prefetching can be an effective means of speeding up multiprocessor applications. Due to this emphasis on prefetching in multiprocessor systems, many of the prefetching mechanisms discussed below have been studied either largely or exclusively in this context. Because several of these mechanisms may also be effective in single processor systems, multiprocessor prefetching is treated as a separate topic only when the prefetch mechanism is inherent to such systems. 3. Software Data Prefetching Most contemporary microprocessors support some form of fetch instruction which can be used to implement prefetching [3,31,37]. The implementation of a fetch can be as simple as a load into a processor register that has been hardwired to zero. Slightly more sophisticated implementations provide hints to the memory system as to how the prefetched block will be used. Such information may be useful in multiprocessors where data can be prefetched in different sharing states, for example. Although particular implementations will vary, all fetch instructions share some common characteristics. Fetches are non-blocking memory operations and therefore require a lockup-free cache [21] that allows prefetches to bypass other outstanding memory operations in the cache. Prefetches are typically implemented in such a way that fetch instructions cannot cause exceptions. Exceptions are suppressed for prefetches to insure that they remain an optional optimization feature that does not affect program correctness or initiate large and potentially unnecessary overhead, such as page faults or other memory exceptions. The hardware required to implement software prefetching is modest compared to other prefetching strategies. Most of the complexity of this approach lies in the judicious placement of fetch instructions within the target application. The task of choosing where in the program to place a fetch instruction relative to the matching load or store instruction is known as prefetch scheduling. In practice, it is not possible to precisely predict when to schedule a prefetch so that data arrives in the cache at the moment it will be requested by the processor, as was the case in Figure 2b. The execution time between the prefetch and the matching memory reference may vary, as will memory latencies. These uncertainties are not predictable at compile time and therefore require careful consideration when scheduling prefetch instructions in a program. Fetch instructions may be added by the programmer or by the compiler during an optimization pass. Unlike many optimizations which occur too frequently in a program or are too tedious to implement by hand, prefetch scheduling can often be done effectively by the programmer. Studies have indicated that adding just a few prefetch directives to a program can substantially improve performance [24]. However, if programming effort is to be kept at a minimum, or if the program contains many prefetching opportunities, compiler support may be required. Whether hand-coded or automated by a compiler, prefetching is most often used within loops responsible for large array calculations. Such loops provide excellent prefetching opportunities because they are common in scientific codes, exhibit poor cache utilization and often have predictable array referencing patterns. By establishing these patterns at compile-time, fetch instructions can be placed inside loop bodies so that data for a future loop iteration can be prefetched during the current iteration. As an example of how loop-based prefetching may be used, consider the code segment shown in Figure 4a. This loop calculates the inner product of two vectors, a and b, in a manner similar to the innermost loop of a matrix multiplication calculation. If we assume a four-word cache block, this code segment will cause a cache miss every fourth iteration. We can attempt to avoid these cache misses by adding the prefetch directives shown in Figure 4b. Note that this figure is a source code representation of the assembly code that would be generated by the compiler. This simple approach to prefetching suffers from several problems. First, we need not prefetch every iteration of this loop since each fetch actually brings four words (one cache block) into the cache. Although the extra prefetch operations are not illegal, they are unnecessary and will degrade performance. Assuming a and b are cache block aligned, prefetching should be done only on every fourth iteration. One solution to this problem is to surround the fetch directives with an if condition that tests when i modulo true. The overhead of such an explicit prefetch predicate, however, would likely offset the benefits of prefetching and therefore should be avoided. A better solution is to unroll the loop by a factor of r where r is equal to the number of words to be prefetched per cache block. As shown in Figure 4c, unrolling a loop involves replicating the loop body r times and increasing the loop stride to r. Note that the fetch for for for for (a) (b) (c) (d) Figure 4. Inner product calculation using a) no prefetching, b) simple prefetching, c) prefetching with loop unrolling and d) software pipelining. directives are not replicated and the index value used to calculate the prefetch address is changed from i+1 to i+r. The code segment given in Figure 4c removes most cache misses and unnecessary prefetches but further improvements are possible. Note that cache misses will occur during the first iteration of the loop since prefetches are never issued for the initial iteration. Unnecessary prefetches will occur in the last iteration of the unrolled loop where the fetch commands attempt to access data past the loop index boundary. Both of the above problems can be remedied by using software pipelining techniques as shown in Figure 4d. In this figure, we have extracted select code segments out of the loop body and placed them on either side of the original loop. Fetch statements have been prepended to the main loop to prefetch data for the first iteration of the main loop, including ip. This segment of code is referred to as the loop prolog. An epilog is added to the end of the main loop to execute the final inner product computations without initiating any unnecessary prefetch instructions. The code given in Figure 4 is said to cover all loop references because each reference is preceded by a matching prefetch. However, one final refinement may be necessary to make these prefetches effective. The examples in Figure 4 have been written with the implicit assumption that prefetching one iteration ahead of the data's actual use is sufficient to hide the latency of main memory accesses. This may not be the case. Although early studies [4] were based on this assumption, Klaiber and Levy [20] recognized that this was not a sufficiently general solution. When loops contain small computational bodies, it may be necessary to initiate prefetches d iterations before the data is referenced. Here, d is known as the prefetch distance and is expressed in units of loop iterations. Mowry, et. al. [25] later simplified the computation of d to l s where l is the average memory latency, measured in processor cycles, and s is the estimated cycle time of the shortest possible execution path through one loop iteration, including the prefetch overhead. By choosing the shortest execution path through one loop iteration and using the ceiling operator, this calculation is designed to err on the conservative side and thus increase the likelihood that prefetched data will be cached before it is requested by the processor. Returning to the main loop in Figure 4d, let us assume an average miss latency of 100 processor cycles and a loop iteration time of 45 cycles so that d = 3. Figure 5 shows the final version of the inner product loop which has been altered to handle a prefetch distance of three. Note that the prolog has been expanded to include a loop which prefetches several cache blocks for the initial three iterations of the main loop. Also, the main loop has been shortened to stop prefetching three iterations before the end of the computation. No changes are necessary for the epilog which carries out the remaining loop iterations with no prefetching. The loop transformations outlined above are fairly mechanical and, with some refinements, can be applied recursively to nested loops. Sophisticated compiler algorithms based on this approach have been developed to automatically add fetch instructions during an optimization pass of a compiler [25], with varying degrees of success. Bernstein, et al. [3] measured the run-times of twelve scientific benchmarks both with and without the use of prefetching on a PowerPC 601- based system. Prefetching typically improved run-times by less than 12% although one benchmark ran 22% faster and three others actually ran slightly slower due to prefetch instruction overhead. Santhanam, et al. [31] found that six of the ten SPECfp95 benchmark programs ran between 26% and 98% faster on a PA8000-based system when prefetching was enabled. Three of the four remaining SPECfp95 programs showed less than a 7% improvement in run-time and one program was slowed down by 12%. Because a compiler must be able to reliably predict memory access patterns, prefetching is normally restricted to loops containing array accesses whose indices are linear functions of the loop indices. Such loops are relatively common in scientific codes but far less so in general applications. Attempts at establishing similar software prefetching strategies for these applications are hampered by their irregular referencing patterns [9,22,23]. Given the complex control structures typical of general applications, there is often a limited window in which to reliably predict when a particular datum will be accessed. Moreover, once a cache block has been accessed, there is less of a chance that several successive cache blocks will also be requested when data structures such as graphs and linked lists are used. Finally, the comparatively high temporal locality of many general applications often result in high cache utilization thereby diminishing the benefit of prefetching. Even when restricted to well-conformed looping structures, the use of explicit fetch instructions exacts a performance penalty that must be considered when using software prefetching. Fetch instructions add processor overhead not only because they require extra execution cycles but also because the fetch source addresses must be calculated and stored in the processor. Ideally, this prefetch address should be retained so that it need not be recalculated for the matching load or store instruction. By allocating and retaining register space for the prefetch addresses, however, the compiler will have less register space to allocate to other active variables. The addition of fetch instructions is therefore said to increase register pressure which, in turn, may result in additional spill code to manage variables "spilled" out to main memory due to insufficient register space. The problem is exacerbated when the prefetch distance is greater than one since this implies either maintaining d address registers to hold multiple prefetch addresses or storing these addresses in memory if the required number of address registers are not available. Comparing the transformed loop in Figure 5 to the original loop, it can be seen that software prefetching also results in significant code expansion which, in turn, may degrade instruction cache performance. Finally, because software prefetching is done statically, it is unable to detect when a prefetched block has been prematurely evicted and needs to be re-fetched. for for prolog -prefetching only main loop -prefetching and computation computation only Figure 5. Final inner product loop transformation. 4. Hardware Data Prefetching Several hardware prefetching schemes have been proposed which add prefetching capabilities to a system without the need for programmer or compiler intervention. No changes to existing executables are necessary so instruction overhead is completely eliminated. Hardware prefetching also can take advantage of run-time information to potentially make prefetching more effective. 4.1 Sequential prefetching Most (but not all) prefetching schemes are designed to fetch data from main memory into the processor cache in units of cache blocks. It should be noted, however, that multiple word cache blocks are themselves a form of data prefetching. By grouping consecutive memory words into single units, caches exploit the principle of spatial locality to implicitly prefetch data that is likely to be referenced in the near future. The degree to which large cache blocks can be effective in prefetching data is limited by the ensuing cache pollution effects. That is, as the cache block size increases, so does the amount of potentially useful data displaced from the cache to make room for the new block. In shared-memory multiprocessors with private caches, large cache blocks may also cause false sharing which occurs when two or more processors wish to access different words within the same cache block and at least one of the accesses is a store. Although the accesses are logically applied to separate words, the cache hardware is unable to make this distinction since it operates only on whole cache blocks. The accesses are therefore treated as operations applied to a single object and cache coherence traffic is generated to ensure that the changes made to a block by a store operation are seen by all processors caching the block. In the case of false sharing, this traffic is unnecessary since only the processor executing the store references the word being written. Increasing the cache block size increases the likelihood of two processors sharing data from the same block and hence false sharing is more likely to arise. Sequential prefetching can take advantage of spatial locality without introducing some of the problems associated with large cache blocks. The simplest sequential prefetching schemes are variations upon the one block lookahead (OBL) approach which initiates a prefetch for block b+1 when block b is accessed. This differs from simply doubling the block size in that the prefetched blocks are treated separately with regard to the cache replacement and coherence policies. For example, a large block may contain one word which is frequently referenced and several other words which are not in use. Assuming an LRU replacement policy, the entire block will be retained even though only a portion of the block's data is actually in use. If this large block were replaced with two smaller blocks, one of them could be evicted to make room for more active data. Similarly, the use of smaller cache blocks reduces the probability that false sharing will occur. OBL implementations differ depending on what type of access to block b initiates the prefetch of b+1. Smith [34] summarizes several of these approaches of which the prefetch-on-miss and tagged prefetch algorithms will be discussed here. The prefetch-on-miss algorithm simply initiates a prefetch for block b+1 whenever an access for block b results in a cache miss. If b+1 is already cached, no memory access is initiated. The tagged prefetch algorithm associates a tag bit with every memory block. This bit is used to detect when a block is demand-fetched or a prefetched block is referenced for the first time. In either of these cases, the next sequential block is fetched. Smith found that tagged prefetching reduced cache miss ratios in a unified (both instruction and data) cache by between 50% and 90% for a set of trace-driven simulations. Prefetch-on-miss was less than half as effective as tagged prefetching in reducing miss ratios. The reason prefetch-on- miss is less effective is illustrated in Figure 6 where the behavior of each algorithm when accessing three contiguous blocks is shown. Here, it can be seen that a strictly sequential access pattern will result in a cache miss for every other cache block when the prefetch-on-miss algorithm is used but this same access pattern results in only one cache miss when employing a tagged prefetch algorithm. The HP PA7200 [5] serves as an example of a contemporary microprocessor that uses OBL prefetch hardware. The PA7200 implements a tagged prefetch scheme using either a directed or an undirected mode. In the undirected mode, the next sequential line is prefetched. In the directed mode, the prefetch direction (forward or backward) and distance can be determined by the pre/post-increment amount encoded in the load or store instructions. That is, when the contents of an address register are auto-incremented, the cache block associated with a new address is prefetched. Compared to a base case with no prefetching, the PA7200 achieved run-time improvements in the range of 0% to 80% for 10 SPECfp95 benchmark programs [35]. Although performance was found to be application-dependent, all but two of the programs ran more than 20% faster when prefetching was enabled. Note that one shortcoming of the OBL schemes is that the prefetch may not be initiated far enough in advance of the actual use to avoid a processor memory stall. A sequential access stream resulting from a tight loop, for example, may not allow sufficient lead time between the use of block b and the request for block b+1. To solve this problem, it is possible to increase the number of blocks prefetched after a demand fetch from one to K, where K is known as the degree of prefetching. Prefetching K > 1 subsequent blocks aids the memory system in staying ahead of demand-fetched prefetched demand-fetched prefetched demand-fetched prefetched demand-fetched prefetched demand-fetched prefetched demand-fetched prefetched prefetched0 demand-fetched prefetched prefetched prefetched demand-fetched prefetched demand-fetched prefetched prefetched demand-fetched prefetched prefetched prefetched1 prefetched prefetched prefetched (c) (a) (b) Figure 6. Three forms of sequential prefetching: a) Prefetch on miss, b) tagged prefetch and c) sequential prefetching with 2. rapid processor requests for sequential data blocks. As each prefetched block, b, is accessed for the first time, the cache is interrogated to check if blocks b+1, . b+K are present in the cache and, if not, the missing blocks are fetched from memory. Note that when scheme is identical to tagged OBL prefetching. Although increasing the degree of prefetching reduces miss rates in sections of code that show a high degree of spatial locality, additional traffic and cache pollution are generated by sequential prefetching during program phases that show little spatial locality. Przybylski [30] found that this overhead tends to make sequential prefetching unfeasible for values of K larger than one. Dahlgren and Stenstr-m [11] proposed an adaptive sequential prefetching policy that allows the value of K to vary during program execution in such a way that K is matched to the degree of spatial locality exhibited by the program at a particular point in time. To do this, a prefetch efficiency metric is periodically calculated by the cache as an indication of the current spatial locality characteristics of the program. Prefetch efficiency is defined to be the ratio of useful prefetches to total prefetches where a useful prefetch occurs whenever a prefetched block results in a cache hit. The value of K is initialized to one, incremented whenever the prefetch efficiency exceeds a predetermined upper threshold and decremented whenever the efficiency drops below a lower threshold as shown in Figure 7. Note that if K is reduced to zero, prefetching is effectively disabled. At this point, the prefetch hardware begins to monitor how often a cache miss to block b occurs while block b-1 is cached and restarts prefetching if the respective ratio of these two numbers exceeds the lower threshold of the prefetch efficiency. Simulations of a shared memory multiprocessor found that adaptive prefetching could achieve appreciable reductions in cache miss ratios over tagged prefetching. However, simulated run-time comparisons showed only slight differences between the two schemes. The lower miss ratio of adaptive sequential prefetching was found to be partially nullified by the associated overhead of increased memory traffic and contention. Jouppi [19] proposed an approach where K prefetched blocks are brought into a FIFO stream buffer before being brought into the cache. As each buffer entry is referenced, it is brought into the cache while the remaining blocks are moved up in the queue and a new block is prefetched into the tail position. Note that since prefetched data are not placed directly into the cache, this scheme avoids any cache pollution. However, if a miss occurs in the cache and the desired block is also not found at the head of the stream buffer, the buffer is flushed. Therefore, prefetched blocks must be accessed in the order they are brought into the buffer for stream buffers to provide a performance benefit. K- K++ time upper threshold lower threshold prefetch efficiency Figure 7. Sequential adaptive prefetching Palacharla and Kessler [27] studied stream buffers as a replacement for a secondary cache. When a primary cache miss occurs, one of several stream buffers is allocated to service the new reference stream. Stream buffers are allocated in LRU order and a newly allocated buffer immediately fetches the next K blocks following the missed block into the buffer. Palacharla and Kessler found that eight stream buffers and performance in their simulation study. With these parameters, stream buffer hit rates (the percentage of primary cache misses that are satisfied by the stream buffers) typically fell between 50% and 90%. However, Memory bandwidth requirements were found to increase sharply as a result of the large number of unnecessary prefetches generated by the stream buffers. To help mitigate this effect, a small history buffer is used to record the most recent primary cache misses. When this history buffer indicates that misses have occurred for both block b and block b + 1, a stream is allocated and blocks b are prefetched into the buffer. Using this more selective stream allocation policy, bandwidth requirements were reduced at the expense of some slightly reduced stream buffer hit rates. The stream buffers described by Palacharla and Kessler were found to provide an economical alternative to large secondary caches and were eventually incorporated into the Cray T3E multiprocessor [26]. In general, sequential prefetching techniques require no changes to existing executables and can be implemented with relatively simple hardware. Compared to software prefetching, sequential hardware prefetching performs poorly when non-sequential memory access patterns are encountered, however. Scalar references or array accesses with large strides can result in unnecessary prefetches because these types of access patterns do not exhibit the spatial locality upon which sequential prefetching is based. To enable prefetching of strided and other irregular data access patterns, several more elaborate hardware prefetching techniques have been proposed. 4.2 Prefetching with arbitrary strides Several techniques have been proposed which employ special logic to monitor the processor's address referencing pattern to detect constant stride array references originating from looping structures [2,13,32]. This is accomplished by comparing successive addresses used by load or store instructions. Chen and Baer's scheme [7] is perhaps the most aggressive proposed thus far. To illustrate its design, assume a memory instruction, m i , references addresses a 1 , a 2 and a 3 during three successive loop iterations. Prefetching for m i will be initiated if a a where D is now assumed to be the stride of a series of array accesses. The first prefetch address will then be A a 3 is the predicted value of the observed address, a 3 . Prefetching continues in this way until the equality A a no longer holds true. Note that this approach requires the previous address used by a memory instruction to be stored along with the last detected stride, if any. Recording the reference histories of every memory instruction in the program is clearly impossible. Instead, a separate cache called the reference prediction table (RPT) holds this information for only the most recently used memory instructions. The organization of the RPT is given in Figure 8. Table entries contain the address of the memory instruction, the previous address accessed by this instruction, a stride value for those entries which have established a stride and a state field which records the entry's current state. The state diagram for RPT entries is given in Figure 9. The RPT is indexed by the CPU's program counter (PC). When memory instruction m i is executed for the first time, an entry for it is made in the RPT with the state set to initial signifying that no prefetching is yet initiated for this instruction. If m i is executed again before its RPT entry has been evicted, a stride value is calculated by subtracting the previous address stored in the RPT from the current effective address. To illustrate the functionality of the RPT, consider the matrix multiply code and associated RPT entries given in Figure 10. In this example, only the load instructions for arrays a, b and c are considered and it is assumed that the arrays begin at addresses 10000, 20000 and 30000, respectively. For simplicity, one word cache blocks are also assumed. After the first iteration of the innermost loop, the state of the RPT is as given in Figure 10b where instruction addresses are represented by their pseudo-code mnemonics. Since the RPT does not yet contain entries for these instructions, the stride fields are initialized to zero and each entry is placed in an initial state. All three references result in a cache miss. After the second iteration, strides are computed as shown in Figure 10c. The entries for the array references to b and c are placed in a transient state because the newly computed strides do not match the previous stride. This state indicates that an instruction's referencing pattern may be in transition and a tentative prefetch is issued for the block at address effective address is not already cached. The RPT entry for the reference to array a is placed in a steady state because the previous and current strides match. Since this entry's stride is zero, no prefetching will be issued for this instruction. Although the reference to array a hits in the cache due a demand fetch in the previous iteration, the references to arrays b and c once again result in a cache miss. During the third iteration, the entries for array references b and c move to the steady state when the tentative strides computed in the previous iteration are confirmed. The prefetches issued during the second iteration result in cache hits for the b and c references, provided that a prefetch distance instruction tag previous address stride state PC effective address prefetch address Figure 8. The organization of the reference prediction table. of one is sufficient. From the above discussion, it can be seen that the RPT improves upon sequential policies by correctly handling strided array references. However, as described above, the RPT still limits the prefetch distance to one loop iteration. To remedy this shortcoming, a distance field may be added to the RPT which specifies the prefetch distance explicitly. Prefetch addresses would then be calculated as effective address The addition of the distance field requires some method of establishing its value for a given RPT entry. To calculate an appropriate value, Chen and Baer decouple the maintenance of the RPT from its use as a prefetch engine. The RPT entries are maintained under the direction of the PC as described above but prefetches are initiated separately by a pseudo program counter, called the lookahead program counter (LA-PC) which is allowed to precede the PC. The difference between the PC and LA-PC is then the prefetch distance, d. Several implementation issues arise with the addition of the lookahead program counter and the interested reader is referred to [2] for a complete description. In [8], Chen and Baer compared RPT prefetching to Mowry's software prefetching scheme [25] and found that neither method showed consistently better performance on a simulated shared memory multiprocessor. Instead, it was found that performance depended on the individual program characteristics of the four benchmark programs upon which the study was based. Software prefetching was found to be more effective with certain irregular access patterns for which an indirect reference is used to calculate a prefetch address. The RPT may not be able to establish an access pattern for an instruction which uses an indirect address because the instruction may generate effective addresses which are not separated by a constant stride. Also, the RPT is less efficient at the beginning and end of a loop. Prefetches are issued by the RPT only after an access pattern has been established. This means that no prefetches will be issued for array data for at least the first two iterations. Chen and Baer also noted that it may take several iterations for the RPT to achieve a prefetch distance that completely masks memory latency when the LA-PC was used. Finally, the RPT will always prefetch past array bounds because an incorrect prediction is necessary to stop subsequent prefetching. However, during loop steady state, the RPT was able to dynamically adjust its prefetch distance to achieve a better overlap with memory latency than the software scheme for some array access patterns. Also, software prefetching incurred instruction overhead resulting from prefetch address calculation, fetch instruction execution and spill code. initial steady transient no prediction Correct stride prediction Incorrect stride prediction Incorrect prediction with stride update initial Start state. No prefetching. transient Stride in transition. Tentative prefetch. steady Constant Stride. Prefetch if stride - 0. no prediction prefetching. Figure 9. State transition graph for reference prediction table entries. Dahlgren and Stenstr-m [10] compared tagged and RPT prefetching in the context of a distributed shared memory multiprocessor. By examining the simulated run-time behavior of six benchmark programs, it was concluded that RPT prefetching showed limited performance benefits over tagged prefetching, which tends to perform as well or better for the most common memory access patterns. Dahlgren showed that most array strides were less than the block size and therefore were captured by the tagged prefetch policy. In addition, it was found that some scalar references showed a limited amount of spatial locality that could captured by the tagged prefetch policy but not by the RPT mechanism. If memory bandwidth is limited, however, it was conjectured that the more conservative RPT prefetching mechanism may be preferable since it tends to produce fewer useless prefetches. As with software prefetching, the majority of hardware prefetching mechanisms focus on very regular array referencing patterns. There are some notable exceptions, however. Harrison and Mehrotra [17] have proposed extensions to the RPT mechanism which allow for the prefetching of data objects connected via pointers. This approach adds fields to the RPT which enable the detection of indirect reference strides arising from structures such as linked lists and sparse matrices. Joseph and Grunwald [18] have studied the use of a Markov predictor to drive a data prefetcher. By dynamically recording sequences of cache miss references in a hardware table, the prefetcher attempts to predict when a previous pattern of misses has begun to repeat itself. When float a[100][100], b[100][100], c[100][100]; for for for (a) Tag Previous Address Stride State ld b[i][k] 20,000 0 initial ld c[k][j] 30,000 0 initial ld a[i][j] 10,000 0 initial (b) Tag Previous Address Stride State ld b[i][k] 20,004 4 transient ld c[k][j] 30,400 400 transient ld a[i][j] 10,000 0 steady (c) Tag Previous Address Stride State ld b[i][k] 20,008 4 steady ld c[k][j] 30,800 400 steady ld a[i][j] 10,000 0 steady (d) Figure 10. The RPT during execution of matrix multiply. the current cache miss address is found in the table, prefetches for likely subsequent misses are issued to a prefetch request queue. To prevent cache pollution and wasted memory bandwidth, prefetch requests may be displaced from this queue by requests that belong to reference sequences with higher a probability of occurring. 5. Integrating Hardware and Software Prefetching Software prefetching relies exclusively on compile-time analysis to schedule fetch instructions within the user program. In contrast, the hardware techniques discussed thus far infer prefetching opportunities at run-time without any compiler or processor support. Noting that each of these approaches has its advantages, some researchers have proposed mechanisms that combine elements of both software and hardware prefetching. Gornish and Veidenbaum [15] describe a variation on tagged hardware prefetching in which the degree of prefetching (K) for a particular reference stream is calculated at compile time and passed on to the prefetch hardware. To implement this scheme, a prefetching degree (PD) field is associated with every cache entry. A special fetch instruction is provided that prefetches the specified block into the cache and then sets the tag bit and the value of the PD field of the cache entry holding the prefetched block. The first K blocks of a sequential reference stream are prefetched using this instruction. When a tagged block, b, is demand fetched, the value in its PD field, K b , is added to the block address to calculate a prefetch address. The PD field of the newly prefetched block is then set to K b and the tag bit is set. This insures that the appropriate value of K is propagated through the reference stream. Prefetching for non-sequential reference patterns is handled by ordinary fetch instructions. Zheng and Torrellas [39] suggest an integrated technique that enables prefetching for irregular data structures. This is accomplished by tagging memory locations in such a way that a reference to one element of a data object initiates a prefetch of either other elements within the referenced object or objects pointed to by the referenced object. Both array elements and data structures connected via pointers can therefore be prefetched. This approach relies on the compiler to initialize the tags in memory, but the actual prefetching is handled by hardware within the memory system. The use of a programmable prefetch engine has been proposed by Chen [6] as an extension to the reference prediction table described in Section 4.2. Chen's prefetch engine differs from the RPT in that the tag, address and stride information are supplied by the program rather than being dynamically established in hardware. Entries are inserted into the engine by the program before entering looping structures that can benefit from prefetching. Once programmed, the prefetch engine functions much like the RPT with prefetches being initiated when the processor's program counter matches one of the tag fields in the prefetch engine. VanderWiel and Lilja [36] propose a prefetch engine that is external to the processor. The engine is a general processor that executes its own program to prefetch data for the CPU. Through a shared second-level cache, a producer-consumer relationship is established between the two processors in which the engine prefetches new data blocks into the cache only after previously prefetched data have been accessed by the compute processor. The processor also partially directs the actions of the prefetch engine by writing control information to memory-mapped registers within the prefetch engine's support logic. These integrated techniques are designed to take advantage of compile-time program information without introducing as much instruction overhead as pure software prefetching. Much of the speculation performed by pure hardware prefetching is also eliminated, resulting in fewer unnecessary prefetches. Although no commercial systems yet support this model of prefetching, the simulation studies used to evaluate the above techniques indicate that performance can be enhanced over pure software or hardware prefetch mechanisms. 6. Prefetching in Multiprocessors In addition to the prefetch mechanisms above, several multiprocessor-specific prefetching techniques have been proposed. Prefetching in these systems differs from uniprocessors for at least three reasons. First, multiprocessor applications are typically written using different programming paradigms than uniprocessors. These paradigms can provide additional array referencing information which enable more accurate prefetch mechanisms. Second, multiprocessor systems frequently contain additional memory hierarchies which provide different sources and destinations for prefetching. Finally, the performance implications of data prefetching can take on added significance in multiprocessors because these systems tend to have higher memory latencies and more sensitive memory interconnects. Fu and Patel [12] examined how data prefetching might improve the performance of vectorized multiprocessor applications. This study assumes vector operations are explicitly specified by the programmer and supported by the instruction set. Because the vectorized programs describe computations in terms of a series of vector and matrix operations, no compiler analysis or stride detection hardware is required to establish memory access patterns. Instead, the stride information encoded in vector references is made available to the processor caches and associated prefetch hardware. Two prefetching policies were studied. The first is a variation upon the prefetch-on-miss policy in which K consecutive blocks following a cache miss are fetched into the processor cache. This implementation of prefetch-on-miss differs from that presented earlier in that prefetches are issued only for scalars and vector references with a stride less than or equal to the cache block size. The second prefetch policy, which will be referred to as vector prefetching here, is similar to the first policy with the exception that prefetches for vector references with large strides are also issued. If the vector reference for block b misses in the cache, then blocks b, b are fetched. Fu and Patel found both prefetch policies improve performance over the no prefetch case on an Alliant FX/8 simulator. Speedups were more pronounced when smaller cache blocks were assumed since small block sizes limit the amount of spatial locality a non-prefetching cache can capture while prefetching caches can offset this disadvantage by simply prefetching more blocks. In contrast to other studies, Fu and Patel found both sequential prefetching policies were effective for values of K up to 32. This is in apparent conflict with earlier studies which found sequential prefetching to degrade performance for K > 1. Much of this discrepancy may be explained by noting how vector instructions are exploited by the prefetching scheme used by Fu and Patel. In the case of prefetch-on-miss, prefetching is suppressed when a large stride is specified by the instruction. This avoids useless prefetches which degraded the performance of the original policy. Although vector prefetching does issue prefetches for large stride referencing patterns, it is a more precise mechanism than other sequential schemes since it is able to take advantage of stride information provided by the program. Comparing the two schemes, it was found that applications with large strides benefited the most from vector prefetching, as expected. For programs in which scalar and unit-stride references dominate, the prefetch-on-miss policy tended to perform slightly better. For these programs, the lower miss ratios resulting from the vector prefetching policy were offset by the corresponding increase in bus traffic. Gornish, et. al. [14] examined prefetching in a distributed memory multiprocessor where global and local memory are connected through a multistage interconnection network. Data are prefetched from global to local memory in large, asynchronous block transfers to achieve higher network bandwidth than would be possible with word-at-a-time transfers. Since large amounts of data are prefetched, the data are placed in local memory rather than the processor cache to avoid excessive cache pollution. Some form of software-controlled caching is assumed to be responsible for translating global array addresses to local addresses after the data been placed in local memory. As with software prefetching in single-processor systems, loop transformations are performed by the compiler to insert prefetch operations into the user code. However, rather than inserting fetch instructions for individual words within the loop body, entire blocks of memory are prefetched before the loop is entered. Figure 11 shows how this block prefetching may be used with a vector-matrix product calculation. In Figure 11b, the iterations of the original loop (Figure 11a) have been partitioned among NPROC processors of the multiprocessor system so that each processor iterates over 1 NPROC th of a and c. Also note that the array c is prefetched a row at a time. Although it is possible to pull out the prefetch for c so that the entire array is fetched into local memory before entering the outermost loop, it is assumed here that c is very large and a prefetch of the entire array would occupy more local memory than is available. The block fetches given in Figure 11b will add processor overhead to the original computation in a manner similar to the software prefetching scheme described earlier. Although the block-oriented prefetch operations require size and stride information, significantly less overhead will be incurred than with the word-oriented scheme since fewer prefetch operations will be needed. Assuming equal problem sizes and ignoring prefetches for a, the loop given Figure 11 will generate N+1 block prefetches as compared to the 12 c hprefetches that would result from applying a word-oriented prefetching scheme. Although a single bulk data transfer is more efficient than dividing the transfer into several smaller messages, the former approach will tend to increase network congestion when several such messages are being transferred at once. Combined with the increased request rate prefetching induces, this network contention can lead to significantly higher average memory latencies. For a (b) (a) Figure 11. Block prefetching for a vector-matrix product calculation. set of six numerical benchmark programs, Gornish noted that prefetching increased average memory latency by a factor of between 5.3 and 12.7 over the no prefetch case. An implication of prefetching into the local memory rather than the cache is that the array a in Figure 11 cannot be prefetched. In general, this scheme requires that all data must be read-only between prefetch and use because no coherence mechanism is provided which allows writes by one processor to be seen by the other processors. Data transfers are also restricted by control dependencies within the loop bodies. If an array reference is predicated by a conditional statement, no prefetching is initiated for the array. This is done for two reasons. First, the conditional may only test true for a subset of the array references and initiating a prefetch of the entire array would result in the unnecessary transfer of a potentially large amount of data. Second, the conditional may guard against referencing non-existent data and initiating a prefetch for such data could result in unpredictable behavior. Honoring the above data and control dependencies limits the amount of data which can be prefetched. On average, 42% of loop memory references for the six benchmark programs used by Gornish could not be prefetched due to these constraints. Together with the increased average memory latencies, the suppression of these prefetches limited the speedup due to prefetching to less than 1.1 for five of the six benchmark programs. Mowry and Gupta [24] studied the effectiveness of software prefetching for the DASH DSM multiprocessor architecture. In this study, two alternative designs were considered. The first places prefetched data in a remote access cache (RAC) which lies between the interconnection network and the processor cache hierarchy of each node in the system. The second design alternative simply prefetched data from remote memory directly into the primary processor cache. In both cases, the unit of transfer was a cache block. The use of a separate prefetch cache such as the RAC is motivated by a desire to reduce contention for the primary data cache. By separating prefetched data from demand-fetched data, a prefetch cache avoids polluting the processor cache and provides more overall cache space. This approach also avoids processor stalls that can result from waiting for prefetched data to be placed in the cache. However, in the case of a remote access cache, only remote memory operations benefit from prefetching since the RAC is placed on the system bus and access times are approximately equal to those of main memory. Simulation runs of three scientific benchmarks found that prefetching directly into the primary cache offered the most benefit with an average speedup of 1.94 compared to an average of 1.70 when the RAC was used. Despite significantly increasing cache contention and reducing overall cache space, prefetching into the primary cache resulted in higher cache hit rates, which proved to be the dominant performance factor. As with software prefetching in single processor systems, the benefit of prefetching was application-specific. Speedups for two array-based programs achieved speedups over the non-prefetch case of 2.53 and 1.99 while the third, less regular, program showed a speedup of 1.30. 7. Conclusions Prefetching schemes are diverse. To help categorize a particular approach it is useful to answer three basic questions concerning the prefetching mechanism: 1) When are prefetches initiated, 2) where are prefetched data placed, and 3) what is prefetched? When Prefetches can be initiated either by an explicit fetch operation within a program, by logic that monitors the processor's referencing pattern to infer prefetching, or by a combination of these approaches. However they are initiated, prefetches must be issued in a timely manner. If a prefetch is issued too early there is a chance that the prefetched data will displace other useful data from the higher levels of the memory hierarchy or be displaced itself before use. If the prefetch is issued too late, it may not arrive before the actual memory reference and thereby introduce processor stall cycles. Prefetching mechanisms also differ in their precision. Software prefetching issues fetches only for data that is likely to be used while hardware schemes tend data in a more speculative manner. Where The decision of where to place prefetched data in the memory hierarchy is a fundamental design decision. Clearly, data must be moved into a higher level of the memory hierarchy to provide a performance benefit. The majority of schemes place prefetched data in some type of cache memory. Other schemes place prefetched data in dedicated buffers to protect the data from premature cache evictions and prevent cache pollution. When prefetched data are placed into named locations, such as processor registers or memory, the prefetch is said to be binding and additional constraints must be imposed on the use of the data. Finally, multiprocessor systems can introduce additional levels into the memory hierarchy which must be taken into consideration. What Data can be prefetched in units of single words, cache blocks, contiguous blocks of memory or program data objects. Often, the amount of data fetched is determined by the organization of the underlying cache and memory system. Cache blocks may be the most appropriate size for uniprocessors and SMPs while larger memory blocks may be used to amortize the cost of initiating a data transfer across an interconnection network of a large, distributed memory multiprocessor. These three questions are not independent of each other. For example, if the prefetch destination is a small processor cache, data must be prefetched in a way that minimizes the possibility of polluting the cache. This means that precise prefetches will need to be scheduled shortly before the actual use and the prefetch unit must be kept small. If the prefetch destination is large, the timing and size constraints can be relaxed. Once a prefetch mechanism has been specified, it is natural to wish to compare it with other schemes. Unfortunately, a comparative evaluation of the various proposed prefetching techniques is hindered by widely varying architectural assumptions and testing procedures. However, some general observations can be made. The majority of prefetching schemes and studies concentrate on numerical, array-based applications. These programs tend to generate memory access patterns that, although comparatively predictable, do not yield high cache utilization and therefore benefit more from prefetching than general applications. As a result, automatic techniques which are effective for general programs remain largely unstudied. To be effective, a prefetch mechanism must perform well for the most common types of memory referencing patterns. Scalar and unit-stride array references typically dominate in most applications and prefetching mechanisms should capture this type of access pattern. Sequential prefetching techniques concentrate exclusively on these access patterns. Although comparatively infrequent, large stride array referencing patterns can result in very poor cache utilization. RPT mechanisms sacrifice some scalar performance in order to cover strided referencing patterns. Software prefetching handles both types of referencing patterns but introduces instruction overhead. Integrated schemes attempt to reduce instruction overhead while still offering better prefetch coverage than pure hardware techniques. Finally, memory systems must be designed to match the added demands prefetching imposes. Despite a reduction in overall execution time, prefetch mechanisms tend to increase average memory latency. This is a result of effectively increasing the memory reference request rate of the processor thereby introducing congestion within the memory system. This particularly can be a problem in multiprocessor systems where buses and interconnect networks are shared by several processors. Despite these application and system constraints, data prefetching techniques have produced significant performance improvements on commercial systems. Efforts to improve and extend these known techniques to more diverse architectures and applications is an active and promising area of research. The need for new prefetching techniques is likely to continue to be motivated by increasing memory access penalties arising from both the widening gap between microprocessor and memory performance and the use of more complex memory hierarchies. 8. --R "Performance Evaluation of Computing Systems with Memory Hierarchies," "An Effective On-chip Preloading Scheme to Reduce Data Access Penalty," "Compiler Techniques for Data Prefetching on the PowerPC," "Software Prefetching," "Design of the HP PA 7200 CPU," "An Effective Programmable Prefetch Engine for On-chip Caches," "Effective Hardware-Based Data Prefetching for High Performance Processors," "A Performance Study of Software and Hardware Data Prefetching Schemes," "Data Access Microarchitectures for Superscalar Processors with Compiler-Assisted Data prefetching," "Effectiveness of Hardware-based Stride and Sequential Prefetching in Shared-memory Multiprocessors," "Fixed and Adaptive Sequential Prefetching in Shared-memory Multiprocessors," "Data Prefetching in Multiprocessor Vector Cache Memories," "Stride Directed Prefetching in Scalar Processors," "Compiler-directed Data Prefetching in Multiprocessors with Memory Hierarchies," "An Integrated Hardware/Software Scheme for Shared-Memory Multiprocessors," "Comparative Evaluation of Latency Reducing and Tolerating Techniques," "A Data Prefetch Mechanism for Accelerating General Computation," "Prefetching using Markov Predictors," "Improving Direct-mapped Cache Performance by the Addition of a Small Fully-associative Cache and Prefetch Buffers," "An Architecture for Software-Controlled Data Prefetching," "Lockup-free Instruction Fetch/prefetch Cache Organization," "SPAID: Software Prefetching in Pointer and Call-Intensive Environments," "Compiler-based Prefetching for Recursive Data Structures," "Tolerating Latency through Software-controlled Prefetching in Shared-memory Multiprocessors," "Design and Evaluation of a Compiler Algorithm for Prefetching," The Cray T3E Architecture Overview "Evaluating Stream Buffers as a Secondary Cache Replacement," "Exposing I/O concurrency with informed prefetching," Software Methods for Improvement of Cache Performance on Supercomputer Applications. "The Performance Impact of Block Sizes and Fetch Strategies," "Data Prefetching on the HP PA-8000," "Prefetch Unit for Vector Operations on Scalar Computers," "Sequential Program Prefetching in Memory Hierarchies," "Cache Memories," "When Caches are not Enough : Data Prefetching Techniques," "Hiding Memory Latency with a Data Prefetch Engine," "The MIPS R10000 Superscalar Microprocessor," "An intelligent I-cache prefetch mechanism," "Speeding up Irregular Applications in Shared-Memory Multiprocessors: Memory Binding and Group Prefetching," --TR Software prefetching Tolerating latency through software-controlled prefetching in shared-memory multiprocessors An architecture for software-controlled data prefetching Data prefetching in multiprocessor vector cache memories Data access microarchitectures for superscalar processors with compiler-assisted data prefetching An effective on-chip preloading scheme to reduce data access penalty Prefetch unit for vector operations on scalar computers (abstract) Design and evaluation of a compiler algorithm for prefetching directed prefetching in scalar processors Cache coherence in large-scale shared-memory multiprocessors Evaluating stream buffers as a secondary cache replacement A performance study of software and hardware data prefetching schemes Speeding up irregular applications in shared-memory multiprocessors Compiler techniques for data prefetching on the PowerPC An effective programmable prefetch engine for on-chip caches Compiler-based prefetching for recursive data structures Compiler-directed data prefetching in multiprocessors with memory hierarchies Prefetching using Markov predictors Data prefetching on the HP PA-8000 Dependence based prefetching for linked data structures The performance impact of block sizes and fetch strategies Improving direct-mapped cache performance by the addition of a small fully-associative cache and prefetch buffers Cache Memories Exposing I/O concurrency with informed prefetching When Caches Aren''t Enough The MIPS R10000 Superscalar Microprocessor Limited Bandwidth to Affect Processor Design Effective Hardware-Based Data Prefetching for High-Performance Processors Branch-Directed and Stride-Based Data Cache Prefetching Lockup-free instruction fetch/prefetch cache organization A study of branch prediction strategies Effectiveness of hardware-based stride and sequential prefetching in shared-memory multiprocessors Distributed Prefetch-buffer/Cache Design for High Performance Memory Systems Software methods for improvement of cache performance on supercomputer applications --CTR Nathalie Drach , Jean-Luc Bchennec , Olivier Temam, Increasing hardware data prefetching performance using the second-level cache, Journal of Systems Architecture: the EUROMICRO Journal, v.48 n.4-5, p.137-149, December 2002 Alexander Gendler , Avi Mendelson , Yitzhak Birk, A PAB-based multi-prefetcher mechanism, International Journal of Parallel Programming, v.34 n.2, p.171-188, April 2006 Addressing mode driven low power data caches for embedded processors, Proceedings of the 3rd workshop on Memory performance issues: in conjunction with the 31st international symposium on computer architecture, p.129-135, June 20-20, 2004, Munich, Germany Binny S. Gill , Dharmendra S. Modha, SARC: sequential prefetching in adaptive replacement cache, Proceedings of the USENIX Annual Technical Conference 2005 on USENIX Annual Technical Conference, p.33-33, April 10-15, 2005, Anaheim, CA Ismail Kadayif , Mahmut Kandemir , Guilin Chen, Studying interactions between prefetching and cache line turnoff, Proceedings of the 2005 conference on Asia South Pacific design automation, January 18-21, 2005, Shanghai, China Ismail Kadayif , Mahmut Kandemir , Feihui Li, Prefetching-aware cache line turnoff for saving leakage energy, Proceedings of the 2006 conference on Asia South Pacific design automation, January 24-27, 2006, Yokohama, Japan Aviral Shrivastava , Eugene Earlie , Nikil Dutt , Alex Nicolau, Aggregating processor free time for energy reduction, Proceedings of the 3rd IEEE/ACM/IFIP international conference on Hardware/software codesign and system synthesis, September 19-21, 2005, Jersey City, NJ, USA Vlad-Mihai Panait , Amit Sasturkar , Weng-Fai Wong, Static Identification of Delinquent Loads, Proceedings of the international symposium on Code generation and optimization: feedback-directed and runtime optimization, p.303, March 20-24, 2004, Palo Alto, California Resit Sendag , Ying Chen , David J. Lilja, The Impact of Incorrectly Speculated Memory Operations in a Multithreaded Architecture, IEEE Transactions on Parallel and Distributed Systems, v.16 n.3, p.271-285, March 2005 Franoise Fabret , H. Arno Jacobsen , Franois Llirbat , Joo Pereira , Kenneth A. Ross , Dennis Shasha, Filtering algorithms and implementation for very fast publish/subscribe systems, ACM SIGMOD Record, v.30 n.2, p.115-126, June 2001 Jike Cui , Mansur. H. Samadzadeh, A new hybrid approach to exploit localities: LRFU with adaptive prefetching, ACM SIGMETRICS Performance Evaluation Review, v.31 n.3, p.37-43, December Jean Christophe Beyler , Philippe Clauss, Performance driven data cache prefetching in a dynamic software optimization system, Proceedings of the 21st annual international conference on Supercomputing, June 17-21, 2007, Seattle, Washington Wang , Kuan-Ching Li , Kuo-Jen Wang , Ssu-Hsuan Lu, On the Design and Implementation of an Effective Prefetch Strategy for DSM Systems, The Journal of Supercomputing, v.37 n.1, p.91-112, July 2006 Kenneth A. Ross, Conjunctive selection conditions in main memory, Proceedings of the twenty-first ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems, June 03-05, 2002, Madison, Wisconsin Weidong Shi , Hsien-Hsin S. Lee , Mrinmoy Ghosh , Chenghuai Lu , Alexandra Boldyreva, High Efficiency Counter Mode Security Architecture via Prediction and Precomputation, ACM SIGARCH Computer Architecture News, v.33 n.2, p.14-24, May 2005 Brendon Cahoon , Kathryn S. McKinley, Simple and effective array prefetching in Java, Proceedings of the 2002 joint ACM-ISCOPE conference on Java Grande, p.86-95, November 03-05, 2002, Seattle, Washington, USA Kenneth A. Ross, Selection conditions in main memory, ACM Transactions on Database Systems (TODS), v.29 n.1, p.132-161, March 2004 Luis M. Ramos , Jos Luis Briz , Pablo E. Ibez , Victor Vials, Data prefetching in a cache hierarchy with high bandwidth and capacity, Proceedings of the 2006 workshop on MEmory performance: DEaling with Applications, systems and architectures, p.37-44, September 16-20, 2006, Seattle, Washington Sangyeun Cho , Lei Jin, Managing Distributed, Shared L2 Caches through OS-Level Page Allocation, Proceedings of the 39th Annual IEEE/ACM International Symposium on Microarchitecture, p.455-468, December 09-13, 2006 Zhen Yang , Xudong Shi , Feiqi Su , Jih-Kwon Peir, Overlapping dependent loads with addressless preload, Proceedings of the 15th international conference on Parallel architectures and compilation techniques, September 16-20, 2006, Seattle, Washington, USA Gokul B. Kandiraju , Anand Sivasubramaniam, Going the distance for TLB prefetching: an application-driven study, ACM SIGARCH Computer Architecture News, v.30 n.2, May 2002 David Siegwart , Martin Hirzel, Improving locality with parallel hierarchical copying GC, Proceedings of the 2006 international symposium on Memory management, June 10-11, 2006, Ottawa, Ontario, Canada Ali-Reza Adl-Tabatabai , Richard L. Hudson , Mauricio J. Serrano , Sreenivas Subramoney, Prefetch injection based on hardware monitoring and object metadata, ACM SIGPLAN Notices, v.39 n.6, May 2004 Jun Yan , Wei Zhang, Hybrid multi-core architecture for boosting single-threaded performance, ACM SIGARCH Computer Architecture News, v.35 n.1, p.141-148, March 2007 Weidong Shi , Hsien-Hsin S. Lee, Accelerating memory decryption and authentication with frequent value prediction, Proceedings of the 4th international conference on Computing frontiers, May 07-09, 2007, Ischia, Italy Trishul M. Chilimbi , Martin Hirzel, Dynamic hot data stream prefetching for general-purpose programs, ACM SIGPLAN Notices, v.37 n.5, May 2002 Jacob Leverich , Hideho Arakida , Alex Solomatnikov , Amin Firoozshahian , Mark Horowitz , Christos Kozyrakis, Comparing memory systems for chip multiprocessors, ACM SIGARCH Computer Architecture News, v.35 n.2, May 2007 A. Mahjur , A. H. Jahangir , A. H. Gholamipour, On the performance of trace locality of reference, Performance Evaluation, v.60 n.1-4, p.51-72, May 2005 Kristof Beyls , Erik H. D'Hollander, Generating cache hints for improved program efficiency, Journal of Systems Architecture: the EUROMICRO Journal, v.51 n.4, p.223-250, April 2005

prefetching;memory latency

359194

Dynamically Relaxed Block Incomplete Factorizations for Solving Two- and Three-Dimensional Problems.

To efficiently solve second-order discrete elliptic PDEs by Krylov subspace-like methods, one needs to use some robust preconditioning techniques. Relaxed incomplete factorizations (RILU) are powerful candidates.\ Unfortunately, their efficiency critically depends on the choice of the relaxation parameter $\omega$ whose "optimal" value is not only hard to estimate but also strongly varies from one problem to another. These methods interpolate between the popular ILU and its modified variant (MILU). Concerning the pointwise schemes, a new variant of RILU that dynamically computes variable $\omega=\omega_i$ has been introduced recently. Like its ancestor RILU and unlike standard methods, it is robust with respect to both existence and performance. On top of that, it breaks the problem-dependence of ``$\omega_{opt}$."\ A block version of this dynamically relaxed method is proposed and compared with classical pointwise and blockwise methods as well as with some existing "dynamic" variants, showing that with the new blockwise preconditioning technique, anisotropies are handled more effectively.

Introduction . As model problem, we take the following self-adjoint second order elliptic PDE in\Omega with suitable boundary conditions on where\Omega denotes the unit square (2D case) or cube (3D case); the coefficients p, q and r are positive and bounded while t is nonnegative and bounded. The differential operator is discretized by using the five-point (2D case) or seven-point (3D case) finite difference approximation (point scheme box integration [30, 41]). The mesh points are ordered according to the lexicographic ordering. One then results in a large system of linear equations of the form where A is a block tridiagonal or pentadiagonal diagonally dominant Stieltjes matrix, b is a vector that depends on both the rhs f of (1.1) and the boundary conditions, while u is the vector of unknowns. Combined with some appropriate preconditioning matrix, the conjugate gradient method is (one of) the most widely used method(s) (see e.g. [1, 2, 6, 11, 19, 20, 35]) for solving system (1.2). Relaxed incomplete factorizations (RILU) are powerful preconditioning techniques that interpolate, through a relaxation parameter, between the popular incomplete LU factorization ILU and its modified variant MILU that preserves rowsums of A [4, 13]. As opposed to ILU and MILU, the two main advantages of RILU are following y Research supported by the Commission of the European Communities HCM Contract No. ERB- CHBG-CT93-0420, at Utrecht University, Mathematical Institute, The Netherlands. Current address de Bruxelles, Service des Milieux Continus (CP 194/5), 50, avenue F.D. Roosevelt, B-1050 Brussels, Belgium. magolu@ulb.ac.be z Universit'e Libre de Bruxelles, Service de M'etrologie Nucl'eaire (CP 165), 50 av. F.D. Roosevelt, B-1050 Brussels, Belgium. ynotay@ulb.ac.be. Research supported by the " Fonds National de la Recherche Scientifique", Belgium. M.M. MAGOLU AND Y. NOTAY 1. it does not suffer a lot from existence problem [13, 17, 18]; 2. it is robust with respect to discontinuities and anisotropy [4, 34]. Two major inconveniences are that the "optimal" value of the relaxation parameter strongly varies from a problem to another and the behavior could be very sensitive to variations of ! around the observed "! opt " [12, 39]. In [34], a new variant of RILU has been proposed. There, the relaxation parameter is variable and dynamically computed during the incomplete factorization phase. Like its precursor RILU, it is robust with respect to both existence and performance. In addition, its performance does not critically depends on involved parameter. We intend to propose a block version of this dynamically relaxed method (DRBILU) and to compare it with classical pointwise and blockwise methods as well as with some existing "dynamic" variants. We stress that, even for three-dimensional problems, we consider a linewise partitioning of the unknowns : each block corresponds to a set of gridpoints along a line parallel to the x-axis in the physical domain [28]. Our study is outlined as follows. Needed general terminology and notation are gathered in Section 2. In Section 3, we first review some variants of block incomplete factorizations in Subsection 3.1 and 3.2. We next establish, in Subsection 3.3, some theoretical results that motivate the introduction of dynamically relaxed block preconditioners. Comparative numerical experiments are reported and discussed in Section 4. 2. General terminology and notation. 2.1. Order relation. The order relation between real matrices and vectors is the usual componentwise order : if A is called nonnegative (positive) if A - 0 2.2. Stieltjes matrices. A real square matrix A is called a Stieltjes matrix (or equivalently, a symmetric M-matrix) if it is symmetric positive definite and none of its offdiagonal entries is positive [10]. 2.3. Normalized point LU-factorization. Given a Stieltjes matrix S, by its normalized point LU factorization we understand the factorization s where P s is pointwise diagonal and L s is pointwise lower triangular such that diag(L s 2.4. Miscellaneous symbols. We describe below some symbols that are used in our study. A denotes a given square matrix of order n. the transpose of A the ith smallest eigenvalue of A (A), the smallest eigenvalue of A (A), the largest eigenvalue of A the pointwise diagonal matrix whose diagonal entries coincide with those of A the pointwise tridiagonal matrix whose main three diagonals coincide with those of A DYNAMICALLY RELAXED BLOCK PRECONDITIONERS 3 diag the block diagonal matrix whose block diagonal entries coincide with those of A offdiag e : the vector whose all components are equal to 1 Throughout this work, the term block will refer to the linewise partitioning mentioned in the introduction. 3. Blockwise incomplete factorizations. For simplicity, throughout this work, we consider only blockwise incomplete factorizations with no fill-in allowed outside the main block diagonal part of A. Let n x , n y and n z denote the number of unknowns in respectively the x-, the y- and the z-direction (if any), then the order of the matrix A is According to our assumptions, A is either block tridiagonal (2D case), i.e. or block pentadiagonal (3D case), i.e., A may be split as into its block diagonal part D and strictly block lower (upper) triangular part \GammaL (\GammaL t ). The matrix where P is the block diagonal matrix computed according to Algorithm 3.1, is referred to as the block incomplete LU-factorization (BILU) of A [16]. P stands for the normalized point LU factorization of P i;i . Other choices for g are discussed in [8, 16]. Algorithms that handle more general matrices may be found in [8, 25]. Given that all the blocks A i;i\Gamma1 and A i;i\Gamman y are diagonal, each P i;i is a tridiagonal Stieltjes matrix. It is well known that, if T is a pointwise tridiagonal Stieltjes may be cheaply computed from the normalized point LU factorization of T . This follows from the fact that T \Gamma1 may be rewritten as The lower part of e may then be computed by simple identification of the relevant entries in the above identity. This is performed in Algorithm 3.2 where, for simplicity, we set To improve the performance of Algorithm 3.1, various variants have been proposed in the literature. Most of them differ from the basic method only in the way of computing diag(P i;i ) which is modified so as to satisfy rowsum relations and/or to keep control of the extreme eigenvalues of the preconditioned matrix B \Gamma1 A [4, 21, 22, 26]. Reviewing all existing block variants is beyond the scope of our study. We shall confine ourselves to relaxed and dynamically modified methods. 4 M.M. MAGOLU AND Y. NOTAY Algorithm 3.1 (BILU) Compute, for if (3D) and ny i\Gamman y ;i\Gamman y A i\Gamman y ;i (2) LU-factorization Algorithm 3.2 ( e qn;n normalized point LU-factorization of T 3.1. Relaxed block incomplete factorizations. Now, fill-ins that are neglected in Algorithm 3.1 are accumulated and added to the current main diagonal after multiplication by a relaxation parameter ! 3.3. It amounts to impose the following rowsum relation [22] where M stands for the pointwise diagonal matrix defined as in Algorithm 3.3 with In other words, M satisfies diag block In the 2D case, (3.4) simplifies to DYNAMICALLY RELAXED BLOCK PRECONDITIONERS 5 Algorithm 3.3 (RBILU) Compute, for with if (3D) and ny i\Gamman y ;i\Gamman y A i\Gamman y with A i;i\Gamman y s i\Gamman i\Gamman y ;i\Gamman y A i\Gamman y ;i e i (2) if (3D) and A i;i+1 e i+1 +A i;i+ny e i+ny relaxed block incomplete LU-factorization diagonal (modification) matrix As is well known, multiplication by P \Gamma1 (or by P in Algorithm 3.3) has to be implemented as solution of a (small) linear system. With recovers BILU while with gets the standard modified variant (MBILU) whose conditioning properties are theoretically investigated in, a.o., [9, 27, 32, 23, 5, 25, 28]. In the case under consideration, A is a (nonsingular) irreducible Stieltjes matrix, Algorithm 3.3 cannot breakdown; it gives rise to a (nonsingular) diagonally dominant Stieltjes matrix later purposes, note that the existence analysis discussed in [26, 32] covers any incomplete factorization such that Be Like their pointwise counterparts, both standard methods suffer mainly from robustness problems [17, 22, 25, 26, 34, 40]. Optimal performances are achieved with 0- 1. The trouble is that ! opt strongly depends on the problem Most severe is the fact that performances could be highly sensitive to the variation of ! around ! opt which is very hard to estimate [4, 12, 39]. In the case of 6 M.M. MAGOLU AND Y. NOTAY uniform grid of mesh size h in all directions, it is advocated in [4] to use problems [4]), in which case one has (see Subsection 3.3) that In the light of the theory extensively developed in [9, 22, 23, 28, 32, 34], one should take with ae n y in 2D case case in order to handle any grid. In [39], it has been suggested to try In [34] a pointwise dynamic version of RILU, termed DRILU, that improves the performance stability with respect to the parameter involved, has been introduced. Before discussing the blockwise version of DRILU, we would like to say a few words about dynamically modified block methods. 3.2. Dynamically modified block incomplete factorizations. The standard modified method efficient only in "nice" situations; e.g., in the case of fixed mesh size, Dirichlet boundary conditions and monotonous variation for the PDE coefficients. It is now well-established that the performance strongly depends on the ordering strategy, the variations of both the PDE coefficients and the mesh size, and the boundary conditions [25, 28]. With dynamically modified methods, the goal is to be in more general situations as efficient as, or even more efficient than, classical modified method in nice circumstances, without changing the numbering of the unknowns. To this end, small perturbations are dynamically added to the diagonal entries of P i;i (initially computed with imposed constraints are stand for a O(1) positive parameter independent of n x , n y and n z . If one applies Algorithm 3.4 then the following rowsum relation holds where, at each grid point j, the perturbation - j is defined as in Algorithm 3.4(3). There holds Moreover, one has that [22, 26] In [22, 26], where the 2D case is discussed, it is proposed to take we extend to the 3D case as in agreement with the theoretical argument in [9, 22, 23, 28, 32]. The analysis of the 2D case shows that the performances do not strongly depend on the variation of and that "i opt - 1 range of PDEs [22, 26]. Algorithm 3.4 outperforms by far both basic block methods (Algorithm 3.3 with long as there is isotropy (including strong discontinuities) or moderate anisotropy in the PDE coefficients (see [22]). In the case of strong anisotropy, it gives rise to several degenerate (say, of O(fflh 2 ) with isolated smallest eigenvalues, which slows down the convergence of PCG process [33, 37, 38, 26]. This occurs e.g. in the case of PDEs that involve both isotropy and strong anisotropy (see e.g. [24, 26, 34]). In such a situation, in the 2D case, it is better to use Algorithm 3.5 DYNAMICALLY RELAXED BLOCK PRECONDITIONERS 7 Algorithm 3.4 (DMBILU) Compute, for with if (3D) and ny i\Gamman y ;i\Gamman y A i\Gamman y with +A i;i\Gamman y s i\Gamman i\Gamman y ;i\Gamman y A i\Gamman y ;i e i e if (3D) and e A e dynamically modified block incomplete LU-factorization diagonal (modification) matrix that cancels the perturbation - j at unsafe nodes [24, 26]. Observe that Algorithm 3.5 reduces to Algorithm 3.4 in the absence of "strong" anisotropy. Now there holds where, at each grid point j, the perturbation - j is equal to - j whenever - j is defined, and 0 otherwise [24, 26]. It is worth noting that unsafe nodes are nodes where the coefficients p and q are strongly anisotropic; the factor 10 in Algorithm 3.5 gives a "measure" of the amount of anisotropy beyond which the perturbations should be discarded [26, Subsection 4.4]. Not any generalization of Algorithm 3.5 has been proposed so far to handle 3D problems. This is not a trivial task; one has to take into account the PDE coefficient r too. Contrary to the 2D case [24, 26], it is not easy to 8 M.M. MAGOLU AND Y. NOTAY establish in which of the many different possibilities, with respect to anisotropy, one should drop the perturbations without dramatically increasing the largest eigenvalues of the preconditioned matrix. An alternative solution is investigated in the next section, which is the main contribution of this work. Algorithm 3.5 (DMBILU ) (2D case) Compute, for as in DMBILU then endif as in DMBILU dynamically modified block incomplete LU-factorization with dropping test 3.3. Dynamically relaxed block incomplete factorizations. Relaxed methods have been observed to successfully handle strongly anisotropic problems, whenever the relaxation parameter is properly chosen (see, e.g., Section 4). Unfortunately, no general theory has been provided to date to estimate ! opt . The pointwise variant of standard RILU, introduced in [34] and termed DRILU, dynamically computes variable relaxation parameters; it combines the robustness, with respect to the variation of the PDE coefficients, of optimized RILU with the relative insensitivity, to the variation of the parameters involved, of dynamically perturbed methods. The blockwise version of DRILU that we are going to present in this section is based on a generalization of [34, Theorem 5.1] which motivated the introduction of DRILU. We first give an auxiliary result. Lemma 3.1. [34, Lemma 5.1] Let F - 0 be a pointwise strictly upper triangular matrix and, Q 0 and Q 1 stand for nonnegative pointwise diagonal matrices. If C is a matrix and x a positive vector such that then C is nonnegative definite. DYNAMICALLY RELAXED BLOCK PRECONDITIONERS 9 Theorem 3.2. Let stand for a diagonally dominant Stieltjes matrix, such that while L is strictly block lower triangular. Let P denote a block diagonal diagonally dominant Stieltjes matrix. p be the normalized point LU factorization of P . Assume further that diag block tridiag M is a pointwise diagonal matrix such that f diag block e with for some pointwise diagonal matrix such that W - I. If, for all 1 where then Proof. First, given that to check that, (3.12) and (3.13) imply that Be diag block e offdiag block e diag block offtridiag Next, let be the pointwise diagonal matrix whose diagonal entries are given by One has 0 - \Theta - \Gamma1 I. Set diag block tridiag denotes the pointwise diagonal matrix such that B 1 Taking (3.12) into account, it is an easy matter to show that offdiag(B 1 Hence it follows that is a Stieltjes matrix and therefore nonnegative definite [10]. Now, p is a Stieltjes matrix such that P e - 0, whence (see [10]) L \Gamma1 be the pointwise diagonal matrix defined by ae one has by (3.16) that XP2 denote the pointwise diagonal matrices such that, respectively, By application of Lemma 3.1, successively to B 2 and B 3 , one easily shows that both matrices are nonnegative definite. On the other hand, since readily checks that diag block offtridiag offdiag block where \Delta 4 is a pointwise diagonal matrix whose explicit form does not matter. By (3.18) one has that diag block offtridiag e offdiag block Now, by definition of - (see (3.17)), one easily deduces that 1+! i i, so that either - This implies that I \Gamma W - 2\Theta. Therefore the right hand side of (3.19) is nonnegative definite. The conclusion readily follows. Corollary 3.3. If B corresponds to some relaxed block incomplete LU factorization of A with with Proof. Apply Theorem 3.2 with M defined as in Eq. (3.4) and all Note that, since L \Gamma1 imply for MBILU factorizations with (3. DYNAMICALLY RELAXED BLOCK PRECONDITIONERS 11 (3.22) is nothing but the upper bound on the basis of which both DMBILU and DMBILU have been elaborated, by imposing ~ - i' [22, 24, 26]. An alternative way to achieve the latter imposed upper bound consists in computing P so as to satisfy the assumptions of Theorem 3.2 with for all In view of (3.12) and (3.13), this could be achieved by 1. computing the entries of P as in Algorithm 3.1 (block by block); 2. subtracting, from the pointwise diagonal entries of P , the quantity ( f which is defined by (3.13) with equality symbol, where ! i is defined by (3.24). The corresponding block preconditioner, which we call dynamically relaxed block incomplete LU-factorization (DRBILU), is described in Algorithm 3.6. Note that, if p is a pointwise tridiagonal Stieltjes matrix, then so is . Therefore, e may be computed by means of Algorithm 3.2 with Unlike RBILU, the relaxation parameter is now dynamically modulated in function of - i 's, in a similar way as perturbations are added in DMBILU. This leads us to expect a similar stability with respect to the choice of the parameter i. For all the block preconditioners discussed so far, the parameters involved may be chosen so as to achieve the same upper bound, say i', for the the largest eigenvalues of the preconditioned matrix B \Gamma1 A. In the context of the PCG method, the convergence behavior also depends on the distribution of the smallest eigenvalues. The following estimate, that relates - has been obtained and successfully tested in [7, 26, 31] where X denotes the pointwise diagonal matrix such that Be which satisfies \Gamma1 - ffl h;i - 1, depends weakly upon both the mesh size parameter h and i. ffl rise in general to very accurate estimates in the case of i - n, for both pointwise [7, 31] and blockwise preconditioners [26]. One has (e; that the order of magnitude of the smallest eigenvalues - mostly depends on the sum (e; X e) of all perturbations. As far as DMBILU and DMBILU are concerned one has (e; X O(1) [22, 26], whence it follows that the smallest eigenvalues of B \Gamma1 A do not depend on h, i.e., are O(1). As regards RBILU, with i' , one has from (3.3) that i' M , which shows that all perturbations but the first block nodes ones are O(h). The smallest eigenvalues are then clearly O(h). Therefore, -(B \Gamma1 asymptotically O(h \Gamma2 ). Nevertheless, with a "properly chosen" value for !, RBILU performs very well in practice because of the nice distribution of (interior) eigenvalues [4]. Now, for DRBILU where perturbations (i.e. occur only at selected nodes according to (3.24). Moreover, for all i' . It is then an easy matter to show that, for the same target upper bound, the perturbations introduced in DRBILU are not larger than the ones added in RBILU. Therefore, on the basis of (3.25) one may expect DRBILU to be at least as robust as RBILU. It is of worth noting that for both RBILU and DRBILU, Algorithm 3.6 (DRBILU) Compute, for with e if (3D) and ny i\Gamman y ;i\Gamman y A i\Gamman y with e +A i;i\Gamman y s i\Gamman i\Gamman y ;i\Gamman y A i\Gamman y ;i e i\Gamman y (2) e if (3D) and e dynamically relaxed block incomplete LU-factorization diagonal (relaxation) matrix the perturbations are in direct proportion to the neglected fill-ins which are (very) small; this occurs in particular in 2D problems with strong anisotropy (p - q or Subsection 4.4] (see also (3.28)). For comparison purposes, let us mention that as far as DMBILU is concerned, one has by [22, Lemma 4.4], with O(h), the following sharp upper bound on the perturbations - DYNAMICALLY RELAXED BLOCK PRECONDITIONERS 13 This bound could be large in the case of strong anisotropy. For instance, in 2D case, when p - q (in (1.1)) one could have that - In such a situation, the smallest degenerate eigenvalues of D \Gamma1 A are reproduced, up to some multiplicative constants, by B \Gamma1 A (see (3.25)). If the number - of such degenerate eigenvalues is not very small (i.e., if n), this results in a very slow convergence for the PCG process [33]. In the case of DRBILU, since (3.16) is equivalent to - i , it is straightforward to establish that with Therefore, 1-j-n diag block tridiag e whence it follows that the perturbations involved in DRBILU can never be much larger than the ones involved in DMBILU. It is obvious from all the considerations above that, with DRBILU, one tries to combine the advantages of both RBILU and DMBILU. Observe finally that, in the case of 2D problems, even if the perturbations introduced by DRBILU are very small, their sum (e; X Me) could be larger than the corresponding sum for DMBILU , in particular when the number j of nodes where the perturbations are dropped is large enough, for instance O(n). In order to perform a fair comparison, we give in Algorithm 3.7 a version of DRBILU, termed DRBILU , which uses the same dropping test as in DMBILU . 4. Numerical experiments. The PCG method is run with the zero vector as starting approximate solution and the residual error reduction as convergence criterion. The computations are performed in double precision FORTRAN on a Sun 514MP sparc workstation. For comparison purposes, the precondi- tionings include : 1. RBILU (Algorithm 3.3). Four values of the parameter ! have been tested : respectively, the unmodified and (unper- turbed) modified standard block methods; defined as in Eq. (3.8). For small and moderate problem sizes, this includes ! - 0:95 that has been suggested in [39], while for large problems, this includes ! - 0:99 which is the optimal observed to the nearest 0:01 for minimizing the number of iterations. 14 M.M. MAGOLU AND Y. NOTAY Algorithm 3.7 (DRBILU ) (2D case) Compute, for as in DRBILU then else endif as in DRBILU dynamically relaxed block incomplete LU-factorization with dropping test 2. DMBILU and DMBILU (Algorithms 3.4 and 3.5). We have used 1according to the recommendations made in [26]. 3. DRBILU and DRBILU (Algorithms 3.6 and 3.7). We report the results 4 which we anticipate to be near optimal as in DMBILU and DMBILU . 4. ILU and MILU. The popular pointwise unmodified and modified incomplete LU-factorizations (see, e.g., [6]). 5. DRILU ([34]). The pointwise dynamically relaxed incomplete LU-factorization method. As recommended in [34], the parameter involved is chosen so as to target the upper bound n 1 d where denotes the space dimension. In the first two problems, DMBILU and DRBILU are not considered because they coincide with, respectively, DMBILU and DRBILU. Problem 3 is essentially Stone's problem [36, 26]. The next three problems are some 3D extensions of the first three ones. The last example is intended for comparing the behavior of block and pointwise methods in the case of elongated grids and non uniform mesh size, which arise in 3D simulation problems. DYNAMICALLY RELAXED BLOCK PRECONDITIONERS 15 Problem 1. (2D) ffl The rhs of the linear system to solve is chosen such that the function u 0 (x; generates the solution on the grid. Problem 2. (2D, from [40]) ae 100 in (1=4; 3=4) \Theta (1=4; 3=4) elsewhere ae 100 in (1=4; 3=4) \Theta (1=4; 3=4) elsewhere Problem 3. (2D, essentially from [36]) ffl The coefficients p, q and t are specified in Fig. 1 ffl The rhs of the linear system to solve is chosen such that the function u 0 (x; generates the solution on the grid. -x y region p q d d t0Fig. 1. Problem 3. Configuration and specification of the PDE coefficients; d stands for a positive parameter. Problem 4. (3D) @\Omega M.M. MAGOLU AND Y. NOTAY Problem 5. (3D) ae 100 in (1=4; 3=4) \Theta (1=4; 3=4) \Theta (0; 1) elsewhere ae 100 in (1=4; 3=4) \Theta (1=4; 3=4) \Theta (0; 1) elsewhere Problem 6. (3D) ffl The coefficients p, q, r, t and f depend only on (x,y) as specified in Fig. 2 @\Omega -x y region p q d d r1d Fig. 2. Problem 6. Configuration and specification of the PDE coefficients; all the regions extend from stands for a positive parameter. Problem 7. (3D) ae 1000 in (0; 1=8) \Theta (0; 1=8) \Theta (0; 1=8) elsewhere ae 1 in (0; 1=8) \Theta (0; 1=8) \Theta (0; 1=8) elsewhere For all problems but the last one, we have used a uniform mesh size h in each direction. In the case of Problem 7, we have used non uniform rectangular grids obtained by setting h ny and h z = 4h0 (z) nz , where the function h 0 is defined by 0:25 if 0:25 - t - 0:5 DYNAMICALLY RELAXED BLOCK PRECONDITIONERS 17 We give in Tables 1-14 the extremal eigenvalues and/or the spectral condition numbers as well as the exponent - from the assumed asymptotic relationship -(B \Gamma1 constant. - is estimated from the largest two problems data (say, h \Gamma1 =96 and h \Gamma1 =192, or 40 and 80 in 3D cases). Whenever - min (B \Gamma1 is strongly isolated from the rest of the spectrum, we also include both the second smallest eigenvalue and the effective spectral condition number which accounts for the superlinear convergence of the PCG method (see e.g. [33, 37, 38, 40]). As far as pointwise preconditioners (ILU, MILU and DRBILU) are concerned, whose conditioning analysis is not investigated here, we have computed only the number of iterations to achieve the prescribed accuracy, in order to save space. Problems 3 and 6, with d large enough, are examples of PDE with degenerate smallest eigenvalues. We report in Tables 5 and 11, for case) and h case), the numerically computed smallest and largest four eigenvalues associated to each block preconditioner involved. The pointwise Jacobi (or diagonal) preconditioner whose smallest eigenvalues are connected to those of the other preconditioners through Eq. (3.25), is also considered. In Tables 6, 13 and 14, the number of PCG iterations to reach the target accuracy are collected for each problem and for h of the mesh sizes along the three directions in the case of Problem 7. From all the tables, the following observations can be made. 1. As expected, RBILU (with smallest eigenvalues of O(h) and largest eigenvalues of approximately O(h \Gamma1 ). Nevertheless, the O(h \Gamma2 ) behavior of -(B \Gamma1 compensated by the nice distribution of interior eigen-values [4], which explains the relative good performance of the preconditioner (see Tables 6, 13 and 14, RBILU with 2. For both DMBILU and DRBILU as well as their " versions", the smallest eigenvalues are in general O(1) while the largest ones are O(h \Gamma1 ), so that the (effective) spectral condition numbers are O(h \Gamma1 ). Observe however that, in the case of strong anisotropy (Problems 3 and 6 with smallest eigenvalues associated to DRBILU are O(h), whence it follows that the (effective) spectral condition numbers are O(h \Gamma2 ); as in RBILU, the good behavior of DRBILU is due to the nice distribution of interior eigenvalues. 3. In the case of isotropic problems, DMBILU and DRBILU give better results than classical methods (RBILU with does "optimized" RBILU. All winning methods perform quite similarly, even if there is strong jump in the PDE coefficients (Problems 2 and 5), in which case the small first eigenvalue is the only one that is strongly isolated from the others. 4. In the presence of strong anisotropy (Problems 3 and 6 with d=10 3 ), RBILU essentially reproduce the disastrous distribution of the smallest eigenvalues of D \Gamma1 A, which slows down the convergence of PCG process [33]. It should be noted that for RBILU with eigenvalues are more clustered than for DMBILU as one would expect by comparing the largest eigenvalues. DRBILU, DMBILU and DRBILU successfully break with the dependence upon D \Gamma1 A, which reflects in the number of PCG iterations. As predicted, DRBILU is a little bit more efficient than DRBILU. For RBILU with the rate of convergence mostly depends on M.M. MAGOLU AND Y. NOTAY the distribution of the largest eigenvalues, the smallest ones being known to cluster around 1. 5. In accordance with previous works, [16] (2D), [3, 28] (3D), blockwise (linewise) methods turn out to be more efficient than pointwise counterparts. In the case of 2D problems, the reduction in the number of iterations, from point methods to block methods, is at least about 50%, while it is around 30% in the 3D cases. The gain is even more spectacular in the case of strongly anisotropic problems (see Table 6, Problem 3b and Table 13, Problem 6b). As regards their computational complexity, let us mention that each PCG iteration with blockwise preconditioners needs two more flops per point than for pointwise preconditioners, which is rather small as compared to the total number of flops per PCG iteration [3]. 6. The performances of blockwise methods in general, and in particular, DM- BILU and DRBILU, are (almost) insensitive to the variation of the number of gridpoints along the x-direction, say, the direction which determines the blocks (see Tables 12 and 14). This is in quite agreement which the analysis performed in [28]. 7. DRBILU is the only preconditioner which is always among the best three ones, whatever the problem tested (see Tables 6, 13 and 14). Whenever DRBILU is not absolutely the winner, it is not far from the latter (we have also observed that the variation of the parameter i around 1 4 does not have a significant effect on the behavior of the preconditioners). From our discussion together with the analysis made in Section 3, we conclude that the most promising method is DRBILU (with It perfoms quite well in a wide range of situations. Its main merit is that, contrary to DMBILU and DMBILU , it does not (strongly) depend on a dropping test that has been set up on the basis of experiments performed only on five-diagonal 2D problems [24, 26]. The dropping test involved has not yet been extended to 3D PDEs, while the performances of DRBILU are quite satisfactory for both two- and three-dimensional problems. Even though all our computations were performed on well structured grids for which natural blockwise partitionings are available, the block methods that we have investigated could be applied to finite element unstructured grids. The problem of defining blocks in such irregular grids has been solved recently in [14, 15]. Combining the techniques developed in the latter papers with the preconditoners discussed here would result in robust preconditioners to tackle real life engineering problems. This awaits further investigation. DYNAMICALLY RELAXED BLOCK PRECONDITIONERS 19 Table Problem 1. Extremal eigenvalues (-min and -max) and/or spectral condition number (-) of B exponent - corresponding to the (estimated) asymptotic relationship -=Ch \Gamma- , C denoting a constant. RBILU 48 16.2 4.02 0.58 2.39 4.09 0.39 1.96 5.05 DMBILU DRBILU 48 0.65 2.52 3.88 0.77 2.80 3.64 0.98 3.78 3.84 Table Problem 2. Extremal eigenvalues (-min , -2 and -max) and/or (effective) spectral condition number corresponding to the (estimated) asymptotic relationship denoting a constant. RBILU 48 2300 13.7 51.3 70E-4 0.61 2.97 422 4.86 48 38E-4 0.44 2.28 608 5.21 54E-4 0.56 2.94 549 5.27 DRBILU 48 76E-4 0.66 3.29 435 4.98 20E-3 0.85 5.42 271 6.39 M.M. MAGOLU AND Y. NOTAY Table Problem 3 with Extremal eigenvalues (-min , -2 and -max) and/or (effective) spectral condition number (- (2) ) - of B \Gamma1 A; exponent - corresponding to the (estimated) asymptotic relationship -=Ch \Gamma- , C denoting a constant. RBILU 48 7298 38.4 65.8 25E-4 0.42 3.54 8.52 192 116422 614. 309. 57E-5 0.11 6.85 64.0 48 13E-4 0.24 2.76 11.5 50E-5 0.13 3.22 24.8 DRBILU 48 19E-4 0.34 3.63 10.6 44E-4 0.67 5.98 8.95 DYNAMICALLY RELAXED BLOCK PRECONDITIONERS 21 Table Problem 3 with Extremal eigenvalues (-min , -2 and -max) and/or (effective) spectral condition number (- (2) ) - of B \Gamma1 A; exponent - corresponding to the (estimated) asymptotic relationship -=Ch \Gamma- , C denoting a constant. RBILU 48 7302 36.2 75.0 26E-4 0.43 3.14 7.22 48 14E-4 0.28 2.45 8.88 11E-6 37E-4 3.84 1029 DRBILU 48 19E-4 0.34 4.20 12.4 43E-4 0.64 6.49 10.2 22 M.M. MAGOLU AND Y. NOTAY Table Problem 3 with Distribution of extremal eigenvalues of B \Gamma1 A for different preconditioners. Preconditioning smallest eigenvalues largest eigenvalues 1. 1. 1. 1. 159. 204. 248. 368. Point Jacobi 3E-9 59E-8 17E-7 46E-7 2. 2. 2. 2. Table Number of PCG iterations to achieve kr (i) Problem 1 Problem 2 Problem 3a Problem 3b ILU DYNAMICALLY RELAXED BLOCK PRECONDITIONERS 23 Table Problem 4. Extremal eigenvalues (-min and -max) and/or spectral condition number (-) of B exponent - corresponding to the (estimated) asymptotic relationship -=Ch \Gamma- , C denoting a constant. RBILU DMBILU DRBILU Table Problem 5. Extremal eigenvalues (-min , -2 and -max) and/or (effective) spectral condition number corresponding to the (estimated) asymptotic relationship denoting a constant. RBILU DRBILU M.M. MAGOLU AND Y. NOTAY Table Problem 6 with Extremal eigenvalues (-min , -2 and -max) and/or (effective) spectral condition number (- (2) ) - of B \Gamma1 A; exponent - corresponding to the (estimated) asymptotic relationship -=Ch \Gamma- , C denoting a constant. RBILU DRBILU Table Problem 6 with Extremal eigenvalues (-min , -2 and -max) and/or (effective) spectral condition number (- (2) ) - of B \Gamma1 A; exponent - corresponding to the (estimated) asymptotic relationship -=Ch \Gamma- , C denoting a constant. RBILU DRBILU Table Problem 6 with Distribution of extremal eigenvalues of B \Gamma1 A for different preconditioners. Preconditioning smallest eigenvalues largest eigenvalues 1. 1. 1. 1. 444. 511. 807. 3129 Point Jacobi 17E-9 39E-7 57 E-7 15E-6 2. 2. 2. 2. 26 M.M. MAGOLU AND Y. NOTAY Table Problem 7. Extremal eigenvalues (-min and -max) and/or spectral condition number (-) of B \Gamma1 A. RBILU grid -min -max -min -max - 160 \Theta 80 \Theta 40 374. 30.2 0.12 10.6 91.7 62E-3 7.5 121. DMBILU DRBILU grid -min -max -min -max -min -max - 160 \Theta 80 \Theta 40 0.29 10.6 36.7 0.33 11.0 32.4 0.63 15.3 24.1 Table Number of PCG iterations to achieve kr (i) Problems 4, 5, and 6 (6a: Problem 4 Problem 5 Problem 6a Problem 6b ILU DRILU Table Number of PCG iterations to achieve kr (i) k2 =kr (0) k2 -10 \Gamma7 for Problem 7. Grids: (a)=40 \Theta 40 \Theta (g)=160 \Theta 80 \Theta 40 , (h)=80 \Theta 160 \Theta 40 . grid (a) (b) (c) (d) (e) (f) (g) (h) 43 43 61 28 42 28 28 28 ILU 76 158 114 128 175 138 205 205 Acknowledgments . Part of this work was done while the first author was holding a postdoctoral position at the Mathematical Institute of Utrecht University. He thanks Henk van der Vorst for his warm hospitality, and for suggesting to include the three-dimensional case in this study. Thanks are also due to the referees for their constructive comments. --R Cambridge University Press Finite Element Solution of Boundary Value Problems. Vectorizable preconditioners for elliptic difference equations in three space dimensions On the eigenvalue distribution of class of preconditioning methods On eigenvalue estimates for block incomplete factorization methods Templates for the Solution of Linear Systems Modified incomplete factorization strategies On sparse block factorization iterative methods Existence and conditioning properties of sparse approximate block factorizations Nonnegative Matrices in the Mathematical Sciences A survey of preconditioned iterative methods Fourier analysis of relaxed incomplete factorization preconditioners Approximate and incomplete factorizations An object-oriented framework for block preconditioning BPKIT block preconditioning tool kit Block preconditioning for the conjugate gradient method Beware of unperturbed modified incomplete factorizations Relaxed and stabilized incomplete factorizations for nonself-adjoint linear sys- tems Matrix Computations Closer to the solution: iterative linear solvers Compensative block incomplete factorizations Modified block-approximate factorization strategies Analytical bounds for block approximate factorization methods Empirically modified block incomplete factorizations Ordering strategies for modified block incomplete factorizations Taking advantage of the potentialities of dynamically modified block incomplete factorizations On the conditioning analysis of block approximate factorization methods Theoretical comparison of pointwise Efficient planewise like preconditioners to cope with 3D prob- lems Computational Methods in Engineering and Science Conditioning analysis of modified block incomplete factorizations On the convergence rate of the conjugate gradients in the presence of rounding errors Iterative Methods for Sparse Linear Systems Iterative solution of implicit approximation of multidimensional partial differential equations The convergence behaviour of conjugate gradients and ritz values in various circumstances The rate of convergence of conjugate gradients ICCG and related methods for 3D problems on vector computers The convergence behaviour of preconditioned CG and CG-S Iterative Solution of Elliptic Systems and Application to the Neutron Diffusion Equations of Reactor Physics --TR

large sparse linear systems;preconditioned conjugate gradient;incomplete factorizations;diagonal relaxation;discretized partial differential equations

359199

The Finite Mass Method.

The finite mass method, a new Lagrangian method for the numerical simulation of gas flows, is presented and analyzed. In contrast to the finite volume and the finite element method, the finite mass method is founded on a discretization of mass, not of space. Mass is subdivided into small mass packets of finite extension, each of which is equipped with finitely many internal degrees of freedom. These mass packets move under the influence of internal and external forces and the laws of thermodynamics and can undergo arbitrary linear deformations. The method is based on an approach recently developed by Yserentant and can attain a very high accuracy.

Introduction . Fluid mechanics is usually stated in terms of conservation laws that link the change of a quantity like mass or momentum inside a given volume to a flux of this quantity across the boundary of the volume. The finite volume method is directly based on this formulation. Space is subdivided into little cells, and the balance laws for mass, momentum and energy are set up for each of these cells separately. Similarly, also the finite element method is based on a discretization of space and a choice for the trial functions on the resulting cells. In contrast, the finite mass method is founded on a discretization of mass, an idea which is at least as obvious and that can be traced back to the work of von Neumann [10] or of Pasta and Ulam [9] in the late 1940's and the 1950's. Instead of dividing space into elementary cells, we divide mass into a finite number of mass packets of finite extension, each of which equipped with a given number of internal degrees of freedom. These mass packets move under the influence of internal and external forces and the laws of thermodynamics and can intersect and penetrate each other. They can contract, expand, rotate, and even change their shape. Their internal mass distribution is described by a fixed shape function, similarly as with finite elements. Although the finite mass method is a purely Lagrangian approach, it has not much to do with particle methods as used for Boltzmann-like transport equations; in some way, it is much closer to finite element and finite volume schemes. The approximations it produces are differentiable functions and not discrete measures. The method is basically of second order, and in some experiments we observed even fourth order convergence! The Lagrangian form of description of fluid flows can have many advantages. For example, there are no problems with free surfaces, and no convection terms arise. Such features make numerical methods based on the Lagrangian view especially attractive unbounded space. In fact, one of the most popular methods of this type, Monaghan's smoothed particle hydrodynamics [8], has its origins in astrophysics. Like the smoothed particle hydrodynamics, the finite mass method is a completely grid-free approach, but it is not a method de facto imitating statistical mechanics and possesses a sounder mathematical and physical foundation. The finite mass method is a generalization and extension of the particle model of compressible fluids that had been proposed by the last author in [11], [12] and [13] and is based on the principles developed there. The compactness and convergence results Mathematisches Institut der Universitat Tubingen, 72076 Tubingen, Germany obtained in [11] and [13] concerning the transition to the continuum limit transfer to the present situation. One of the essential differences to the approach in the articles mentioned above is that the single mass packets can now undergo arbitrary linear deformations and not only rotations and changes of size. Although the dimension of the configuration manifold of the single mass packet increases because of that, this strongly simplifies the equations of motion the mass packets are subject to because the configuration manifold is now a linear space. The equations of motion take the same form for all space dimensions. The main advantage, however, are the superior approximation properties, due to the fact that the mass packets can now be deformed by the flow. The main issue with a method like ours is how the equations of motion for the mass packets or particles, as we often prefer to say, are set up. We start from the basic physical principles that finally lead to the Euler and Navier-Stokes equations, not from these equations themselves and a least squares or Galerkin approach. In consequence, the equations of motion also do not break down when particles completely cover each other. In the most simple case of an adiabatic, inviscid flow, the equations of motion for the particles are derived from a Lagrange-function with the internal energy as potential energy. To damp the fluctuation part of the local kinetic energy that necessarily arises with every such model, frictional forces vanishing in the limit of particle sizes tending to zero are added to these potential forces. They break the invariance to time reversal and make the method consistent with the second law of thermodynamics. The paper is organized as follows. In x2, the finite mass method is explained and derived in detail, both for inviscid and for viscous fluids. It is shown how the mass density, the velocity field and, as second independent thermodynamic quantity, the entropy are discretized, and it is discussed what sort of approximation properties can be expected. The equations of motion describing the local interaction of the mass packets and the time evolution of the system are set up. A main feature of the approach is that mass, momentum, angular momentum and energy are exactly conserved. The conservation of mass follows immediately from the construction and is discussed in x2. The conservation of energy, momentum and angular momentum is studied in x3. The conservation of angular momentum is a remarkable fact as in continuum mechanics the conservation of angular momentum is hidden in the symmetry of the stress tensor and does not appear explicitly as a conservation law. Consequently, most discretizations violate this principle. The finite mass method as described in x2 is invariant to arbitrary translations and rotations and, as it concerns the shape the mass packets can attain, even to every linear transformation of space. These properties are reflected by the quadrature formulas that are developed in x4 to evaluate the integrals which define the forces acting upon the particles. Conservation of momentum, angular momentumand energy are maintained. In x4, we also discuss how the forces and moments acting upon the particles can be calculated on the computer. In x5, a suitable time discretization is presented. The proposed scheme is an exponential integrator in the spirit of the recent paper [4] by Hochbruck and Lubich. In case of pure pressure forces, it transfers to the well-known Stormer-Verlet method for second order equations and conserves momentum and angular momentum exactly. Finally, in x6, some typical test calculations for flows in two space dimensions are documented. These examples clearly demonstrate the potential and the high accuracy of the method. We restrict our attention in this article to free flows in vacuum. For the inviscid case, the reflection laws needed for particles touching the walls of bounded volumes have already been given in [11], [12] and [13]. Their numerical realization will be discussed elsewhere. For a certain background in mechanics and fluid dynamics, we refer to the text-books [5], [6] by Landau and Lifschitz on one hand and [1] by Chorin and Marsden on the other hand, and to the classical monograph [2] by Courant and Friedrichs. 2. The particle model of compressible fluids. The basic ingredient of the finite mass method is a continuously differentiable shape function / : R d ! R, d the space dimension, with compact support that attains only values 0. This function describes the internal mass distribution inside the mass packets into which the fluid is subdivided. We assume that Z Z The second property states that the origin of the body coordinate system attached to a particle is its center of mass. Further we suppose that Z with the y k the components of y. For example, the function / may be built up from a piecewise polynomial function e in one space variable. If we let d Y e the conditions (2.1) are equivalent to Z e Z e The second condition in (2.4) also implies that the integrals (2.2) vanish for k 6= l. The constant J is given by Z e and does not depend on the space dimension. A suitable choice for e /, that we have used in the numerical computations documented in x6, is the normalized third order B-spline given by e for jj 1 and by e 1. It can be composed of smaller copies and be used to build up a basis for the cubic spline functions on a uniform grid. This e fulfills the conditions (2.4), and the constant (2.5) takes the value An advantage of such tensor product like shape functions are their good approximation properties. The points y of the particle i move along the trajectories The vector q i (t) determines the position of the particle and the matrix H i (t) its size, shape and orientation in space. Correspondingly, are the body coordinates of the point at position x in space at time t. Let denote the mass of the particle i. The total mass density then results from the superposition of the mass densities of the single particles. The points y of the particle i have the velocity Inserting the expression (2.9) for y, one gets the velocity field of the particle i related to the space coordinates. The total mass flux density again results from the superposition of the mass flux densities of the single particles. With the local mass fractions the velocity field v of the flow, that is defined by the relation is the convex combination of the velocity fields of the single particles. To simplify notation, we introduce the abbreviation where again The gradient of / i , that is the derivative of / i with respect to x, is and can, as the derivatives be expressed in terms of the internal variables y. To derive the expression above for the derivative with respect to the matrix H i , we suppress the index i serving for the distinction of the particles and write / instead of / i . First we observe that, for H fixed and E tending to zero, E) and, because of for A tending to zero, similarly E) where k;l denotes the inner product of two matrices A and B. One findsdet(H +E) / det H (r/)(y) This means that the linear mapping det H /(y) is the total derivative of det H / at given H. Inserting Ej l for the entries of E and using (2.17), (2.18), @/ follows. The continuity equation expressing the conservation of mass is automatically satisfied with the given ansatz. Independently of the size and shape of the particles and their distribution in space, @ae @t This follows from the fact that such an equation holds for every single particle, as one proves using (2.19) and div v 6 CHRISTOPH GAUGER, PETER LEINEN, HARRY YSERENTANT To study the potential accuracy of the approach, we start from a given twice continuously differentiable velocity field u. Then fixing the particle trajectories q i (t) and the matrices H i (t) for t t 0 , with given initial values, as solutions of the differential equations the velocity field (2.12) of the particle i reads and is therefore a second order approximation of u in a neighborhood of As the mass fractions (2.14) form a partition of unity, the resulting overall velocity field remains a second order approximation of u on the region occupied by mass, independent of where the particles are located. The mass density is an exact solution of the transport equation (2.20) with respect to this perturbed velocity field and therefore, with corresponding initial values, a good approximation of the true density in the sense of a backward error analysis. For a more careful and detailed analysis of this type also covering the motion in an external force field, we refer to [14]. For a complete description of the thermodynamic state of a compressible fluid, besides the mass density ae a second thermodynamic quantity like temperature is needed. Most convenient for our purposes is the entropy density s that is given here in the form where the S i (t) denote the specific entropies of the single particles. In particular, the specific entropy is the convex combination of the specific entropies of the single particles. As with the velocity, one recognizes that this representation potentially leads to a first order approximation of the exact specific entropy independent of the distribution of the particles, where the actual accuracy can again be much higher. The pressure, the absolute temperature and the internal energy per unit volume are functions of the mass density and the entropy density. These functions are not independent of each other but are connected by the Gibbs fundamental relation of thermodynamics taking the form @ae @s @s in the present variables. We assume that the internal energy "(ae; s) is defined for all ae ? 0 and all real s and is twice continuously differentiable on this set. We suppose that @" @s for all these ae and s and require that "=ae and the first order partial derivatives of " can be extended by the value 0 at (ae; to functions that are continuous on the sectors jsj S ae, S ? 0. Barytropic ideal gases with ae 0 s ae represent a simple example. The constant 0 ? 0 is a characteristic pressure, the constant mass density, and the constant c v ? 0 the characteristic heat for constant volume. The constant fl ? 1 is the ratio of the specific heats for constant pressure and constant volume. A typical value is The particle i has the kinetic energy that attains the closed representation denotes the Euclidean norm of a vector and the Frobenius norm of a matrix, respectively. The total kinetic energy of the system is From the kinetic energy (2.33) and the internal energy Z the Lagrange-function of the system is formed. For the completely adiabatic case that is described by the Euler-equations of gas dynamics, the equations of motion d dt d dt are derived from this Lagrange-function. With the normalized forces explicitly given by @ae @" @s @ae @" @s these equations read Note that the masses m i of the particles cancel out and appear only implicitly in the mass density ae and the entropy density s. The derivatives of / i occurring in the expressions above can be written as functions of the internal particle coordinates (2.9) and are given by (2.19). The pressure forces (2.37) acting upon a group of particles can be interpreted as a kind of surface force [11]. As long as the specific entropies S i of the single particles are assumed to be constant, the time evolution of the system is completely determined by the equations of motion (2.36). This system is time reversible, a fact that contradicts the behavior of actual fluids where heat is generated in shock fronts. Therefore it has been proposed in [13] to add the (normalized) frictional force F (r) Z to the right-hand side of the first equation in (2.40). This force damps local velocity fluctuations and couples the particles softly together. The difference between the velocity of the given particle and the velocity field of the surrounding flow determines the direction of the force and the scalar function R 0 its size. The quantity in the equation for the H i corresponding to the force (2.41) is In smooth flows, the forces (2.41) and (2.42) will be comparatively small and will vanish with the second power of the local particle size, but they can dominate in shocks. For a better quantitative understanding of the frictional forces (2.41), (2.42), we pause for a moment and consider a little model problem. We neglect the pressure forces, keep R constant, and fix the velocity field The equations of motion for a single particle then read that is, within the time the velocity of the particle is reduced by the factor 1=e. This justifies calling the local relaxation time of the system. With the trajectory of the particle is Thus the particle is stopped in distance T jv 0 j from its position at the given time t 0 . The friction among the particles generates heat. Therefore the specific entropies are no longer constant and increase in time. They obey the differential equations The quantity ' i defined by is the mean value Z of the absolute temperature around the particle and Z the specific heat supply. The quantity is the density of the fluctuation energy. It can be rewritten as which is more convenient for computational purposes. The equation (2.43) is the second law of thermodynamics. It ensures conservation of energy as will be shown in x3. Note that the specific entropy of a particle never decreases. If the q i and H i and the function R are kept fixed, is a linear mapping. To study this mapping, we utilize the energy norm given by and the corresponding energy inner product, respectively. Note that the square of the energy norm of the vector consisting of the velocity components q 0 i is the total kinetic energy (2.33) of the system. Denoting by b v i the velocities (2.16) formed with b i instead of q 0 , one has Z with the symmetric bilinear form in the velocities v i and b v i . This shows that the linear mapping (2.49) is negative semidefinite and selfadjoint with respect to the energy inner product. For R constant, its eigenvalues range in the interval [\Gamma1=T; 0]. The proof of (2.51) is based on the same arguments as the proof of Theorem 1 below. In viscous fluids, an additional force Z acts upon the mass contained in a volume W t transported with the flow, that is the mass initially contained in a given volume inside the region occupied by mass. The symmetric tensor T is the viscous part of the stress tensor and is a function of the density, the temperature and of the symmetric part of the gradient of the velocity field. In Newtonian fluids, it depends linearly on D and v, respectively, and is given by d in d space dimensions. The first term on the right-hand side has trace zero and describes the shear forces in the fluid. The second part comes from the bulk viscos- ity. The viscosity coefficients j and i are nonnegative functions of ae and ', which guarantees a fundamental property needed to satisfy the second law of thermodynamics. For most gases, can be assumed. To model the viscous forces, one starts from the observation that Z Z where the ffi are continuously differentiable functions with values between 0 and 1 vanishing outside W t and tending pointwise to the characteristic function of W t for ffi tending to 0. If I denotes the set of indices of the particles initially contained in the region W , the mass fraction i2I serves as our discrete counterpart of the functions ffi . This leads to the viscous force Z acting upon this group of particles, or to the normalized force F (v) Z acting upon the particle i, where the integrals extend over the region occupied by mass at the given time t. Note that the expression (2.59) correctly takes into account only that part of the boundary of the moving volume that is touched by mass from its exterior. Therefore the construction also covers the case not directly included above when a part of the boundary of the moving volume belongs to the boundary of the region occupied by mass. The quantities M i corresponding to the forces (2.60) are Z Z and the additional specific heat supply to the particle i is Z Provided the given particle is located in the interior of the region occupied by mass, the forces (2.60) and (2.61) formally vanish for stress tensors T of divergence zero, as continuum mechanics requires. The proof is by partial integration, where the second integral on the right-hand side of (2.61) cancels. The problem with this approach is that, in more than one space dimension, the derivatives of the mass fractions i can have singularities and are not always square integrable. Fortunately, this effect is often compensated by the behavior of T and the properties of the shape function /. Assume that T is of the form (2.55) and that the viscosity coefficients behave like with ff ? 0. Let the gradient of / satisfy the pointwise inverse estimate holds for the cubic splines (2.3), (2.6) with 3. Then there exist constants C i depending on the number of the locally interacting particles and their mass, size, and shape with ae ff jr Provided that ff \Gamma 2=k ? 0, this demonstrates that the integrands above are continuous and tend to zero at the points at which ae tends to zero. Therefore, under the given circumstances, one can refrain from replacing the i by regularized mass fractions such as proposed in [11] and [12]. However, our considerations would immediately transfer to this case. To incorporate heat conduction, the right-hand side of the entropy equation (2.43) has to be supplemented to Z The heat flux vector k is mostly given by Fourier's law where the coefficient function 0 is the heat conductivity. For the heat flux to be well defined, k has to satisfy an estimate similar to the estimate above for the viscosity coefficients. For the rest of this section, we restrict ourselves to Newtonian fluids (2.55). For given q i and H i , the mapping is then linear. Denoting by b D the tensor (2.54) formed with the velocities b q 0 instead of q 0 Z Ddx holds, with the integrals discretized as described above. Because D \Gammad (tr D)(tr b this proves that also the linear mapping (2.68) is symmetric and negative semidefinite with respect to the energy inner product given by (2.50). For the proof of (2.69), we refer again to the proof of Theorem 1 below. 3. The conservation of energy, momentum, and angular momentum. The conservation of energy, momentum, and angular momentum are basic physical properties of any closed system and must therefore be reproduced by the finite mass method. Moreover, energy estimates are basic for the mathematical examination of the model and, in particular, are needed to transfer the compactness and convergence results from [11] and [13] to the present situation of particles underlying arbitrary linear deformations. The conservation of energy also prevents the determinants of the H i from becoming arbitrarily small since, with equations of state like (2.30), this would require too much energy. Our first result is: Theorem 1. The total energy composed of the kinetic energy (2.33) of the particles and the internal energy (2.34) is a constant of motion. Proof. Utilizing the representation (2.32) of the kinetic energy of a single particle, one first obtains d dt Therefore the equations of motion yield d where the terms coming from the pressure forces (2.37) have canceled with the corresponding derivatives of V , comes from the frictional forces (2.41), (2.42), and from the viscous forces (2.60), (2.61). Before we start calculating 3 , we state two simple algebraic relations that are repeatedly used in this section, namely that A for all square matrices A and B and all vectors a and b and that A for all square matrices A, B and C. For example, with I the first of these two relations yields for arbitrary vectors f . Choosing Z follows. Because, by the definition (2.16) of v,2 with (2.48) this leads to the representation Z Rq dx of the first of the two terms above. Inserting Tr i for f in (3.3), one finds Z for the second part coming from the viscous forces (2.60) and (2.61). As by the symmetry of T and by the definition of v i and (3.2), the second integral in the formula above cancels and the sum attains the value Z 14 CHRISTOPH GAUGER, PETER LEINEN, HARRY YSERENTANT As by (2.43), (2.44), (2.46), (2.62) and (2.66) Z Z this proves the proposition and demonstrates that exactly the right amount of kinetic energy is converted to heat. Surprisingly, for vanishing frictional and viscous forces, one has both conservation of energy and entropy, which contradicts the usual conception of gas flows and would not be possible in continuum mechanics, but which is explained by the fact that the kinetic energy (2.33) considered here is composed of the kinetic energies of the single particles and is not identical with the mean kinetic energy E(t) =2 Z known from continuum mechanics; it consists additionally of the local fluctuation energy Z ae In smooth flows, this fluctuation energy is a negligibly small part of the total kinetic energy and will vanish with the fourth power of the particle size, but it can dominate where the particles clash. The role of the frictional forces (2.41) and (2.42) is to convert this kind of fluctuation energy into true internal energy. The total momentum of the system is defined as Z Because of (2.1), it has the closed representation Theorem 2. The total momentum (3.6) is a constant of motion. Proof. The representation (3.7) and the equations of motion yield d dt 0with the three parts coming from the pressure forces (2.38), the frictional forces (2.41), and the viscous forces (2.60). With (2.19), the first part reads @ae @" @s By the definitions (2.10) of the mass density and (2.24) of the entropy density, this means Z As ", under the given assumptions, is a continuously differentiable function with compact support, this yields P 0 2 , with (2.41) one obtains Z As, by the definition (2.16) of the velocity field v, As the i form a partition of unity, finally also the part Z resulting from the viscous forces vanishes. Last, we consider the scalar quantities Z ae(x; with fixed skew-symmetric matrices W, that have, by the definition of v and because of (2.1) and (2.2), the closed representation In three space dimensions, the components of the total angular momentum Z are quantities of this form, and vice versa all quantities of the form (3.8) can be composed of the components of the angular momentum (3.10). Therefore, for the three-dimensional case, the following theorem states that the angular momentum of the system is a constant of motion. In other space dimensions, one gets less or more first integrals, corresponding to the dimension of the space of the skew-symmetric matrices. Theorem 3. For arbitrarily given skew-symmetric matrices W, the scalar quantity L defined by (3.8) is a constant of motion. Proof. As a \Delta vectors a and A \Delta one gets dL dt from the representation (3.9) of L. The equations of motion therefore yield dL 0where the first part comes from the pressure forces (2.38) and (2.39), the second part from the frictional forces (2.41) and (2.42), and the third part from the viscous forces (2.60) and (2.61). We show that each of these three parts vanishes separately. From the representation of the partial derivatives of / i , (3.1) and (3.2), and the skew-symmetry of W, first follows. Thus (2.38) and (2.39) yield @ae @" @s or, again taking into account the definitions of ae and s, summed up Z As " is a continuously differentiable function with compact support, partial integration gives and therefore L 0 because W is skew-symmetric. In the same way, (3.1) yields and therefore, with (2.41) and (2.42), Z As, by the definition (2.16) of the velocity field v, one obtains L 0 (3.2), the symmetry of the viscous stress tensor T and the skew-symmetry of W give and therefore as above Z As the i form a partition of unity, finally also L 0 4. The discretization of the integrals. Our particle model of compressible fluids is purely Lagrangian. It is invariant to arbitrary translations and rotations and, as it concerns the shape the particles can attain, even to every linear transformation of space. These properties must be reflected by the quadrature formula needed to evaluate the integrals that define the forces acting upon the particles. We start from the observation that the integral of a scalar or vector function f weighted by the mass density (2.10) can be written as a sum Z Z of contributions from the single mass packets. The integrals on the right-hand side of (4.1) live on the reference domain on which the shape function / is strictly positive. They are replaced by a fixed quadrature formula Z ff g(a ) with weights ff ? 0 and nodes a inside the support of the shape function /, which is considered as the weight function here. This results in the composite rule Z Z f d with weights m j ff and nodes q j +H j a that is based on the given discretization of mass and not on a subdivision of space into elementary cells. Therefore it has the desired invariance properties. Utilizing the quadrature rule (4.3), one replaces the potential energy (2.34) by the fully discrete expression Z denotes the specific internal energy. With this discrete potential, the normalized forces (4. and the temperatures are formed. As both the specific internal energy itself and the quadrature points depend on the q i and H i , the forces F i and M i correspondingly split into two parts that are of different structure. Into F (1) @ae @~" @s @ae @~" @s d all quadrature points q j +H j a contained in the support of the given particle i enter whereas F (2) and the corresponding term M (2) are completely determined by the function values and the first order derivatives of the mass density and the entropy density at the quadrature points q i +H i a assigned to the particle i itself. The local temperature (4.6) around the particle i transfers to Z @~" @s The forces (4.7) replace the forces (2.38) and (2.39) in the Lagrangian equations of motion (2.36) and (2.40), respectively, and the local temperature (4.6) the local temperature (2.44) in the entropy equation (2.43). Note that, through the assumptions we made on "(ae; s) and because ff the quantities above behave numerically well. The frictional forces (2.41) and (2.42) and the heat supply (2.46) are directly discretized using the quadrature rule (4.3). For the viscous forces (2.60) and (2.61), the heat supply (2.62) and the heat flux in (2.66) one can proceed correspondingly. The method reacts sensitively on the choice of the weighted quadrature rule (4.2) for the reference domain. Experience has shown that quite a few quadrature points are needed to exploit the full accuracy of the approach. A too small number of quadrature points leads to instabilities, in particular when the quadrature points are not properly spaced; a high polynomial accuracy alone does not suffice. For the Table The basic one-dimensional quadrature rule for cubic B-splines nodes 4131656631641tensor-product third order B-splines described at the beginning of x2, we had good experience with the tensor-product counterpart of the one-dimensional quadrature rule given by Table 1. This quadrature formula is exact for fifth order polynomials and assigns 5 2 or quadrature points to each particle in two space dimensions. The total number of quadrature points q j +H j a entering into the computation for a given particle i depends on the overlap of the B-splines and will generally be much higher, not much less than in two space dimensions. The conservation of energy, momentum, and angular momentum transfers to the present case of discretized integrals. The proofs from the last section can be taken over with minor changes concerning only the pressure terms. For the proof that the total momentum (3.6) remains a constant of motion, one has only to utilize that Z Z r~" d such that the quantity vanishes. In the proof that the quantities (3.8) associated with the angular momentum are constants of motion, the relations Z Z enter, giving L 0 The essential idea to compute the forces acting upon the particles is to separate the quadrature points from the particles. This reflects the fact that particles do not interact directly with each other but only with global fields like the mass density or the velocity. Two different data structures are used. The first data structure is associated with the particles themselves. It contains the fixed particles masses m i , the values q i , H i , i and S i finally to be determined, storage for the local temperatures ' i , and for the forces acting upon the particles and the heat supply to the particles, of course. The second data structure is associated with the quadrature points. First, it contains their positions and the weights and then all necessary global information at xZ like the values ae Z of the mass density, s Z of the entropy density, j Z of the mass flux density or vZ of the velocity, quantities associated with the viscous stress tensor T, should the occasion arise, and the other field information needed in the computation. The procedure to compute the integrals involved in the differential equations is then quite simple and consists of three phases. We will describe this procedure only for the inviscid case; the case of additional viscous forces is similar. In the first phase, the quadrature points are generated and the values ae Z of the mass density (2.10), s Z of the entropy density (2.24), j Z of the mass flux density of the intermediate quantity as well as the gradients of ae and s at the quadrature points are assembled. This is a loop on all particles. In this phase, information is transferred from the particles to the quadrature points. In the second phase, the values of the derivatives @ae @s and, to incorporate the frictional forces (2.41) and (2.42), of ae Z are computed. These operations work only on the single quadrature points. In the final third phase, the discrete integrals determining the forces, the heat supplies and the local temperatures are computed using the results of phase 1 and phase 2. In this phase, information is transferred from the quadrature points back to the particles. Phase 1 and phase 3 require an efficient access to the quadrature points contained in the support of a given particle. Search trees can be used for this purpose, which have to be set up after the quadrature points have been generated. The information needed to access the quadrature points assigned to a given particle i has to be stored separately. 5. A time-stepping procedure. The remaining big system of differential equations for the particle positions q i , the deformation matrices H i and the entropies S i has to be solved numerically. In this section, we present a simple, robust second order method for that which is adapted to the structure of this system and the properties of its solutions. The method has been proposed to us by Christian Lubich [7] and is inspired by the recent work of Hochbruck and Lubich [4] on exponential integrators. Its stability range is de facto independent of the strength of the frictional and viscous forces but shrinks, as with all schemes for this type of equation, when approaching incompressibility. First, we combine the vectors q i and the matrices H i to a big vector of dimension (d and the S i to a vector z of dimension N . The system of differential equations (2.40) together with the additional frictional forces (2.41) and (2.42) and the viscous forces (2.60) and (2.61) can then be written in the form and the entropy equation reads The first term on the right-hand side of (5.2) corresponds to with the F i and M i the discrete counterparts (4.7) of the pressure forces (2.38) and (2.39). The second term corresponds to the discretized versions of the frictional forces (2.41) and (2.42) and of the viscous forces (2.60) and (2.61) in Newtonian fluids (2.55). The notation (5.5) reflects that these forces depend only linearly on q 0 . As the considerations in x2 have shown, the matrices are symmetric negative semidefinite with respect to the energy inner product determined by (2.50). Finally, the right-hand side of (5.3) corresponds to with the ffi Q i the discretized versions of (2.46) and (2.62), respectively, plus eventually a term coming from heat conduction. This function depends nonlinearly on all its arguments. The proposed method for the numerical solution of the system (5.2), (5.3) is a leap-frog scheme. To come from the approximations for y 0 at time t k \Gamma =2 and for y and z at time t k to the new approximations at times t k +=2 and t k respectively, one first computes The new value for the derivative of y is then and the new value for y itself is where OE 0 is given by \Theta OE 2 and OE 1 and OE 2 are the entire functions The new value z k+1 finally is implicitly determined by the equation z 22 CHRISTOPH GAUGER, PETER LEINEN, HARRY YSERENTANT Using the new b k+1 from the next time step, the approximation is obtained at little additional cost. This approximation does not enter into the further computations. The computation starts with the approximation for y 0 at t 0 +=2 and the approximation for y itself at the quantities z 1 and y 0 are then computed as above. The method is constructed such that it would yield the exact values of y and y 0 for f and A constant. For vanishing frictional and viscous forces, i.e. for the scheme transfers to the time-reversible integrator known as Verlet method in molecular dynamics, where (5.13) reduces to With the starting step above, the Verlet method is equivalent to the one-step method k+1=2 is now considered as the intermediate value The Verlet method is symplectic and has therefore very favorable properties for Hamiltonian systems as arising in the present case of pure pressure forces [3]. In particular, it preserves momentum and angular momentum: Let denote the total momentum (3.7) and the scalar quantity (3.9). As\Omega is a skew-symmetric matrix and by Theorem 2 and Theorem 3 and the considerations in x4, respectively, both quantities retain their value in the transition from one time level to the next. The matrix-vector products OE( A)b can be computed by a Krylov-space method based on the symmetric Lanczos process using the energy inner product. Started with the vector b, after m steps the Lanczos algorithm delivers a matrix V consisting of columns orthonormal with respect to this inner product, a symmetric tridiagonal matrix H of dimension m \Theta m, and a rank one matrix R such that The matrix-vector products are then approximated by With a spectral decomposition of the small symmetric tridiagonal matrix H, OE( H) is given by Note that the computation of Ax corresponds to an evaluation of the discretized frictional forces (2.41) and (2.42) and, if necessary, of the discretized viscous forces (2.60) and (2.61) and can be performed without any explicit knowledge of A. The approximation (5.23) converges very fast for entire functions OE, much faster than the conjugate gradient method for the solution of a linear system with the coefficient Usually, very few Krylov steps suffice. In the examples presented in the next section, we solved the equation (5.12) approximately by two steps of a simple fixed point iteration. More sophisticated methods will be needed when heat conduction is present. 6. Examples. In this section, we have compiled some two-dimensional examples that exhibit a typical behavior and compare the numerical with the known exact solutions. As shape function / of the particles we have used the bicubic B-spline (2.3), (2.6) together with the 25-point quadrature rule described in x4. To find a good approximation of the given initial data, we assume that the region occupied by mass at time contained inside an axiparallel rectangle and cover this rectangle by a regular grid of gridsize h. The initial matrices are then H i and the initial positions are selected from the gridpoints, that we denote by x k here. To determine the particle masses, we first compute the coefficients a k 0 minimizing an appropriately chosen distance between the linear combination a k and the given initial mass density ae, for example by the projected Gau-Seidel method. At the points x k for which a k ? 0, particles with are located. The remaining gridpoints are ignored in the sequel. The initial velocities can be determined by solving the interpolation problem at the particle positions q Setting the discrete mass flux will virtually be a fourth order approximation to the continuous mass flux aev at time sufficiently smooth. The specific entropies S i (0) are computed in the same way interpolating the entropy density s. The alternative is to choose the initial velocities correspondingly to (2.21) as which works better with dominating external forces and has the advantage that velocity fields linearly depending on x like rigid body motions are exactly reproduced. For reference, we recall the Navier-Stokes equations @ae @t ae @t @s @t that govern the time evolution of a smooth compressible flow in the absence of heat conduction and that transfer to the Euler equations for the inviscid case These equations have to be completed by material laws like (2.30) and (2.55) and the basic thermodynamic relation (2.28). Often, the equations are stated in a different, but mathematically equivalent form, with (6.6) also written as conservation law and replaced by the conservation law for the energy. For the special case of inviscid ideal gases (2.30), the entropy equation (6.7) can be replaced by the equation @ @t coupling pressure and velocity. 6.1. Gas clouds. The first example serves, more or less, as consistency check and shows how accurate the method can be. We reproduce the self-similar solutions of finite mass of the Euler equations. The functions (6.9), (6.10) satisfy the continuity equation (6.5) regardless of the choice of the density profile \Psi and the matrix function H(t), and of course also the entropy equation (6.7). Starting from the internal energy of a barytropic ideal gas (2.30), the momentum equation (6.6) reads ae 0 has been set and H is an abbreviation for the determinant of H. Up to normalization, ae is the only function with compact support that satisfies the relation (6.11) for an appropriately chosen function H(t). For 1! 2, the function (6.12) is continuously differentiable at 1. The corresponding matrix functions H(t) are the solutions of the differential equation ae 0 det H(t) with initial values det H(0) ? 0. This differential equation coincides with the differential equation from (2.40) one would obtain for the motion of a single with the shape function (6.12). If one lets the particles move according to the differential equations (2.21) in the velocity field (6.10), the whole configuration would simply undergo a series of linear transformations. So it is no wonder that a high accuracy can be reached in this example. In the computation presented here, we started from the material constants and the initial values yielding a rotating, first contracting and then again expanding gas ball. The initial positions q i (0), the masses m i and the matrices H i (0) have been determined as described in the introduction to this section. To reproduce the initial velocity field (6.10) exactly, the initial velocities q 0 fixed by (6.4), and not via the linear system (6.2) as in the other examples. The initial entropies are S i We set R = 250 in this example. We solved the problem over the period 0 t 100 with the stepsize in the time-stepping procedure from x4, starting from a grid of gridsize 21 \Theta 21 B-splines finally yielding 325 particles. Fig. 1 shows the initial configuration of the particles at time and the final configuration at time 100, the latter rescaled from the radius determined by (6.13) and (6.15) to the initial radius. The Fig. 1. The particles in the gas cloud example at times initial configuration of the particles is almost retained over this long period, although the gas ball first shrinks to the radius 0:515 and then again expands to the final radius 516:9, that is by more than the factor thousand. The exact and the approximate solution cannot be distinguished with the naked eye; the maximum norm of the error is about one per thousand of the maximum norm of the solution. Fig. 2 shows a cross section of the mass density together with the differences to the approximate mass 26 CHRISTOPH GAUGER, PETER LEINEN, HARRY YSERENTANT densities along the line x are again rescaled to the same radius and the errors have been multiplied by the factor thousand in order to compare them with the exact solution. Astonishingly, the Fig. 2. Mass density and absolute error in the gas cloud example at times relative size of the error does not increase. Probably one could run this example to infinity. There is a viscous counterpart of the solutions above, with the same density profile and the viscosity coefficients j and i constant multiples of 'ae. Also these solutions are well reproduced, even if with a slightly reduced accuracy. In examples like these, the shear viscosity must counterbalance the bulk viscosity; otherwise physical instabilities occur. 6.2. Shock fronts. The second example shows how the particle method behaves when shocks develop in an inviscid fluid and demonstrates how the frictional forces (2.41), (2.42) work. In this example, the transformation of kinetic energy into internal energy plays a dominant role. We consider a spherically symmetric shock front ct in an ideal gas (2.30) and make for the velocity, the density, and the pressure the ansatz r x r r r for ct and with constant values ae 1 and 1 for jxj ! ct. To determine the values c, ae 1 and 1 and the functions V (), M () and P () for 1=c from the initial state and we first observe that the differential equations (6.5), (6.6) and (6.8) transfer to where d is again the space dimension. For d ? 1, these differential equations can be used to compute V , M and P numerically. For The next step is to express c, ae 1 and 1 as functions of the values fixed for the time being. Let u c be the radial components of the velocity on both sides of the shock front relative to the velocity of the front itself. As continuum mechanics teaches (see [1], for example), the mass flux, the momentum flux and the energy flux across the shock front are continuous. This is equivalent to the relations With help of (6.24) and (6.25), the equation (6.26) can be replaced by the Hugoniot equation that equivalently expresses the conservation of energy. For an ideal gas (2.30), the pressure and the internal energy are coupled by the equation of state The conservation of mass (6.24) leads to c and with (6.28), the Hugoniot equation (6.27) takes the form Inserting (6.29) and (6.30) into (6.25), one finally obtains a quadratic equation for the velocity of the shock front. Only the positive solution of this equation is of interest; the negative solution leads to a rarefraction shock that is not compatible with the second law of thermodynamics. Taking into account the functional dependence (6.23), (6.31) represents an equation for the unknown velocity c and the corresponding value This equation can be solved numerically. With (6.29) and (6.30), one then yields also the values ae 1 and 1 for the density and the pressure inside the shock. As an example, with 28 CHRISTOPH GAUGER, PETER LEINEN, HARRY YSERENTANT one obtains in two space dimensions. The values (6.32) mean that, at time the fluid collides in the origin with nearly the double speed of sound, which takes the value p here. We set R = 250ae in this example. The idea behind this form of R is that friction should grow when the density of matter increases, as physical intuition suggests. In our actual computation we placed 251 \Theta 251 particles on a regular grid covering the square [\Gamma5; 5] 2 and followed their motion in the time interval 0 t 1. Approximately 10000 of these particles are contained in the region [\Gamma2; 2] 2 of interest here at time slightly more than 20000 at time 1. With the time stepsize the exponential integrator needed 955 Krylov steps, at most 5 for each of the 200 time steps, a typical behavior also observed in other examples. Fig. 3 shows a cross section of the exact and the approximate solution along the on the interval \Gamma2 x 1 2 and Fig. 4 the corresponding contour lines of the approximate density at time 1. Both the position and the height of the shock and the solution profile outside the shock are very well reproduced and the approximate solution retains its spherical symmetry. The shock is practically as well resolved as possible with particles of the given size, and as a closer look at the results of the single time steps shows, the shock does not smear during the computation. Except for the shock region itself and a small neighborhood of the origin, the error of density and pressure is less than one per thousand. The same holds for the velocity field outside the shock. When passing the shock, the particles almost loose all their kinetic energy such that the velocity is practically zero inside the shock. The error in the mass density around the origin is probably due to the discontinuous initial data; the region over which this error extends seems to shrink proportionally to the particle size. An interesting observation is that behind the shock the particles again rearrange to a rectangular grid, as can be seen in Fig. 5. Fig. 3. Density, pressure, and the velocity in the shock front example at time Fig. 4. Contour lines of the density in the shock front example at time Fig. 5. Particle contours in a neighborhood of the shock. 6.3. Shear flows. Our last example serves as a test for the modeling of the shear viscosity. We keep the mass density and the pressure constant, neglect the heat production, that is set the right-hand side of (6.7) to zero, and consider velocity fields of the form where n is a given unit vector fixing the direction of the flow and component of x in direction of a unit vector e orthogonal to n. Again, the continuity equation is fulfilled independent of the choice of the functions OE and c. For a Newtonian fluid (2.55) with constant shear viscosity j, the momentum equation (6.6) reads This means that either OE() is linear and c(t) constant, that is up to a translation in direction of e, or correspondingly with The solution corresponding to the velocity field (6.37) is remarkable in so far as it is reproduced exactly provided that the initial data are exact and the integrals defining the forces are evaluated exactly. The reason is that all forces acting upon the particles then vanish such that all particles are distorted in the same way and mass density and pressure remain constant. The solution with the velocity field (6.38) represents a more severe test. If one lets the particles move according to the differential equations (2.21) in the velocity field (6.38), the particle positions q i and the matrices H i at time t would be with the function ff(t) given by Thus, in a little while, the particles have practically lost all their kinetic energy and get stuck. In our experiments, we used the data fixing the physical properties of the fluid, set v as in the first example. To exclude influences coming from an alignment of the particles with the flow and to avoid problems with the infinite extension of the flow region, we set p' 3' p: The problem then becomes periodic with period interval [0; 3] \Theta [0; 2]. All computations were for the time interval 0 t 1 with time stepsize The particles in Fig. 6 stem from such a computation. They have initially been located on an axiparallel grid of sidelength Fig. 7 shows how the velocity Fig. 6. The deformation of particles in the shear flow example. field and the errors corresponding to the initial gridsizes evolve in time, where the function values have been sampled on a 500 \Theta 500 grid and the associated l 2 -norm has been taken as distance measure. For better comparison, the errors have been multiplied by the factors 25, respectively. The picture demonstrates that the viscous time scale is Fig. 7. The time evolution of the velocity and the rescaled errors in the shear flow example. perfectly reproduced. In the transition from 30 \Theta 20 to 60 \Theta 40 and from 60 \Theta 40 to 120 \Theta 80 particles, the error decreases approximately by the factor 16, a clear fourth order convergence, and a fine confirmation of the finite mass method. --R A mathematical introduction to fluid mechanics New York Solving ordinary differential equations On Krylov subspace approximations to the matrix exponential operator Lehrbuch der Theoretischen Physik Lehrbuch der Theoretischen Physik Heuristic numerical work in some problems of hydrodynamics Proposal and analysis of a new numerical method in the treatment of hydrodynamical shock problems A particle model of compressible fluids Particles of variable size Entropy generation and shock resolution in the particle model of compressible fluids A convergence analysis for the finite mass method for flows in external force and velocity fields --TR

finite mass method;gridless discretizations;compressible fluids

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Discrete Kinetic Schemes for Multidimensional Systems of Conservation Laws.

We present here some numerical schemes for general multidimensional systems of conservation laws based on a class of discrete kinetic approximations, which includes the relaxation schemes by S. Jin and Z. Xin. These schemes have a simple formulation even in the multidimensional case and do not need the solution of the local Riemann problems. For these approximations we give a suitable multidimensional generalization of the Whitham's stability subcharacteristic condition. In the scalar multidimensional case we establish the rigorous convergence of the approximated solutions to the unique entropy solution of the equilibrium Cauchy problem.

Introduction . In this paper we present a new class of numerical schemes based on a discrete kinetic approximation for multidimensional hyperbolic systems of conservation laws. Consider a weak solution u K to the Cauchy problem where the system is hyperbolic (symmetrizable) and the flux functions A d are locally Lipschitz continuous on R K with values in R K . We approximate problem (1.1), (1.2) by a sequence of semilinear systems with Cauchy data Here # is a positive number, # d are real diagonal L - L matrices, P is a real constant coe#cients K - L matrix, and M is a Lipschitz continuous function defined on R K # Received by the editors August 5, 1998; accepted for publication (in revised form) November 15, 1999; published electronically May 23, 2000. This work was partially supported by TMR project http://www.siam.org/journals/sinum/37-6/34307.html Math-ematiques Appliqu-ees de Bordeaux, Universit-e de Bordeaux 1, 351 cours de la Lib-eration, Talence, France (aregba@math.u-bordeaux.fr). # Istituto per le Applicazioni del Calcolo "M. Picone," Consiglio Nazionale delle Ricerche, Viale del Policlinico 137, I-00161 Rome, Italy (natalini@iac.rm.cnr.it). 1974 DENISE AREGBA-DRIOLLET AND ROBERTO NATALINI with values in R L . Moreover we suppose the following relations are satisfied for all fixed rectangle# in R K : It is easy to see that, if f # converges in some strong topology to a limit f and if 0 converges to u 0 , then Pf is a solution of problem (1.1), (1.2). In fact system (1.3) is just a BGK approximation for (1.1); see [5, 12] and references therein. The interaction term on the right-hand side is given by the di#erence between a nonlinear function, which describes the equilibria of the system, in our case M(Pf ), and the unknown f . Our purpose here is to construct numerical schemes for system (1.3) in order to obtain a numerical approximation of (1.1) in the relaxed limit As well known for general relaxation problems, see [71, 44, 55], approximation needs a suitable stability condition to produce the correct limits. In the frame-work of general 2 - 2 quasi-linear hyperbolic relaxation problems, this condition is known as the subcharacteristic condition. In section 2, we shall argue in the spirit of the Chapman-Enskog analysis [71, 44, 14], to find the following stability condition for for all # 1 , . , # D ) # (R K ) D and every u belonging to some fixed rectangle# R K . Actually, in [54] and for the scalar case convergence of Pf # towards the Kruzkov entropy solution of (1.1), (1.2) has been obtained under a slightly stronger version of condition (1.6): every component of the Maxwellian function M is monotone nondecreasing on the interval I. The main tool in that case is the fact that under this condition the right-hand side in system (1.3) is quasi-monotone in the sense of [25] and this implies special comparison and stability properties on the corresponding system. Unfortunately, similar properties are not verified for nontrivial examples in the general case K > 1. Therefore, for the systems, we shall use just condition (1.6). Note that for certain families of kinetic approximations, the results of [6] show that (1.6) is also a necessary condition for (1.3) to be compatible with the entropies of (see section 3). The continuous kinetic approximation of systems of conservation laws in gas dynamics is classical. In particular, Euler equations can be formally obtained as the fluid dynamical limit of the Boltzmann equation; see [12, 13]. The rigorous theory of kinetic approximations for solutions with shocks is recent and the main results were obtained only in the scalar case. The first result of convergence of a fractional step BGK approximation with continuous velocities, with an entropy condition for the limit (weak) solution, was proven in [7] (see also [22]). Another convergence result was given later in [60], using a continuous velocities BGK model. An important related kinetic formulation can be found in [41]. Other results have been established for special systems or partially kinetic approximations [42, 29, 9, 40]. Related numerical schemes can be found in [19, 59]; for a general overview and many other references see [24]. Discrete velocities models and their fluid dynamical limits have also been considered by many people; see the review paper [61]. In particular we mention the studies on the Broadwell model [11, 72]. Convergence for various relaxation models was also investigated in [15, 14, 17, 47, 68, 74, 32, 33, 67]. The analysis of the stability of various nonlinear waves for relaxation models, and in particular for the Jin-Xin relaxation approximation, can be found in [44, 16, 50, 43, 46, 49]. A general survey of recent results on relaxation hyperbolic problems is given in [55]. Let us also point out some numerical references related to our approach. A lot of computational work has been done in the last ten years in the very closed framework of lattice Boltzmann and BGK models; see [21, 62] and references therein. Let us also mention the monotone schemes of [8], which are an example of numerical (relaxed, i.e., discretization of our construction in the scalar case. Other numerical investigations for hyperbolic problems with relaxation can be found in [37, 57, 58, 73, 4, 3, 18, 70, 31]. Our numerical schemes are constructed by splitting (1.3) into a homogeneous linear part and an ordinary di#erential system, which is exactly solved thanks to the particular structure of the source term. In the scalar case this construction allows us to preserve the monotonicity properties of (1.3) and to prove convergence results. Our approximation framework generalizes to systems the construction presented in [54] for the scalar case, and shares most of the advantages of the relaxation approximation as proposed in [30] (see also [53, 2, 70]): simple formulation even for general multidimensional systems of conservation laws and easy numerical implementation, hyperbolicity, regular approximating solutions. Actually the main advantage, especially in the multidimensional case, of both the approximations, seems to be the possibility of avoiding the resolution of local Riemann problems in the design of numerical schemes. Moreover our framework presents some special properties: - the scalar and the system cases are treated in the same way at the numerical - all the approximating problems are in diagonal form, which is very likely for numerical and theoretical purposes; - we can easily change the number and the geometry of the velocities involved in our construction to improve the accuracy of the method. In this sense our work shares most of the spirit of [56, 34, 45], where very flexible and simple schemes, which do not need Riemann solvers in their construction, were proposed to approximate general multidimensional systems of conservation laws. Let us also observe that the presented algorithms are surely not optimal, but they just illustrate how to construct an e#cient and simple approximation even for very complicated systems. This could be useful, for example, in the numerical investigation of large systems like those arising in the extended thermodynamics and other generalized moment closures hierarchies for kinetic theories [52, 38, 1]. Further investigations will be addressed to the construction of high order schemes. The plan of the article is as follows: In section 2 we establish the stability condition and define the monotone Maxwellian functions for the scalar case. In section 3 we propose some examples of stable approximations in the class (1.3). The issue of entropy is discussed. In section 4 we set the numerical schemes and section 5 is devoted to the convergence results in the scalar multidimensional case. Some numerical experiments are given in section 6. After the completion of this work we received a preprint from Serre [64], where he proves, by using the methods of compensated compactness, the convergence for the Jin-Xin relaxation approximation and some of the discrete kinetic approximations contained in the present paper to (one-dimensional) genuinely nonlinear hyperbolic systems of conservation laws having a positively invariant domain. The convergence of related first-order numerical schemes has been proved by Lattanzio and Serre in [35]. The stability conditions are in both cases strongly related to our condition (1.6). 1976 DENISE AREGBA-DRIOLLET AND ROBERTO NATALINI 2. Chapman-Enskog analysis and monotone Maxwellian functions. In this section we discuss the stability conditions for the discrete kinetic approximation (1.3). Since the local equilibrium for that system is given by the hyperbolic system (1.1), it is natural to seek for a dissipative first-order approximation to (1.3), which is the analogue of the compressible Navier-Stokes equations in the classical kinetic theory. In principle we could try to use the theory developed in a more general context in [14]. Unfortunately it is easy to realize that their main assumption, namely the existence of a strictly convex dissipative entropy for the relaxing system (1.3), which verifies in particular the requirement (iii) of Definition 2.1 of [14], is not satisfied in the present case and we need it for a di#erent construction. Let f # be a sequence of solutions to (1.3)-(1.4) parametrized by #, for a fixed initial data f 0 , which for simplicity we can choose as a local equilibrium, i.e., f 0 for some u 0 # L # (R D , R K ). Set Then, from (1.3) and the compatibility assumptions (1.5), we have Consider a formal expansion of f # in the form Then Reporting in (2.1) yields Now we have Then, up to the higher order terms in (2.4), we obtain where is a K-K matrix. We can state now our stability condition. Proposition 2.1. The first-order approximation to system (1.3) takes the form and it is dissipative provided that the following condition is verified: for all # 1 # R K , . , # D # R K and every u belonging to some fixed rectangle# R K . As we shall see in the next section parabolicity of (2.6) is, at least in some cases, necessary for the compatibility of (1.3) with the entropies of (1.1). But let us recall that, even in the scalar case, the expansion (2.6) cannot be considered in any way as a rigorous asymptotic description of system (1.3). Actually, to prove our rigorous convergence results, we need a slightly stronger version of condition (2.8). Definition 2.2. Take I # R be a fixed interval. A Lipschitz continuous function R L is a monotone Maxwellian function (MMF) for (1.1) and with respect to the interval I if conditions (1.5) are verified and if, moreover, M i is a monotone (nondecreasing) function on I, for every i # {1, . , L}. This condition was used in [54] to show the convergence of approximation (1.3), (1.4) at the continuous level and in the multidimensional scalar case. In the following section we present some examples of di#erent approximations according to the choices of the matrices of velocities # j and the local Maxwellian function M . We discuss the issue of entropy and we investigate both stability and (only for K=1) monotonicity conditions. 3. Examples of discrete kinetic approximations. In order to construct the system (1.3) one must find P , M and # such that the consistency relations (1.5) are satisfied. The first three examples presented here own a block structure. Keeping notations of the introduction we take blocks I K , the identity matrix in R K . Each matrix # d is constituted of N diagonal blocks of size K- K: I K , # (d) With this formalism, (1.3) can be written as and f # Considering that A # setting 1978 DENISE AREGBA-DRIOLLET AND ROBERTO NATALINI we have the following equivalent expression for (2.8): In the case when the M # are symmetric we obtain System (3.1) enters the framework proposed by Bouchut [6]. Suppose that there exists at least one smooth strictly convex entropy for (1.1), and that the Maxwellian functions are of form nd A d (u), where a n , b nd are scalar. Denote #(M # n (u)) the set of eigenvalues of M # under some technical assumptions, it is shown in [6] that the M # n (u) are diagonalizable. Moreover if then (3.1) is compatible with any convex entropy # of (1.1): there exists a kinetic entropy for (3.1) associated with # and Lax entropy inequalities are satisfied in the hydrodynamic limit # 0. As well known [36, 10] these inequalities characterize the admissible weak solutions of (1.1). Moreover, in this case, (2.6) is parabolic. More general results for Maxwellian functions not in the form (3.4) can be found in [51]. Remark that for a general hyperbolic system of conservation laws neither (3.5) nor imply convergence of the kinetic approximation. In the scalar case it is always possible to write (1.3) under the form (3.1), and (3.5) is the monotonicity condition (2.9). Under this condition convergence holds [54]. Moreover one can use the fact that # N and the discrete Jensen inequality to prove directly the following proposition [54]. Proposition 3.1. Let suppose that M is an MMF. Then (2.8) is In the general case, denote B the K - D-K - D matrix defined by the blocks B dj , D. The stability condition means positively defined for all u # where# is some fixed rectangle of R K . Example 1. The diagonal relaxation method (DRM). We first consider the minimal case 1. The system (1.5) is then a squared linear one. We take # > 0 and Let us denote by e j the canonical unit vector of R D . The characteristic speeds of system (1.3) are -#e 1 , . , -#eD , # D . The Maxwellian function is given by means of the following A d (u) # /(D Therefore the result of [6] applies to this model as described above. For a one-dimensional system of conservation laws this formulation coincides with the relaxation approximation of [30]. In fact we can set Then we have We recall that in this case and for convergence to the unique entropy solution was proved in [53], and convergence of the associated numerical relaxed schemes was done in [2]. For other convergence and error estimate results on this model, see also [33, 67]. However, in several space dimensions, there is no diagonal form for the formulation of [30], as already pointed out in [54]. For a one-dimensional system the stability condition (2.8) is here: for all # R K , which, if A # (u) is symmetric with spectral ray #(u), gives #(u). This coincides with Bouchut's condition (3.5) so that here both (3.5) and (2.8) are equivalent. In two space dimensions, (2.8) becomes # . In fact in the scalar case we are able to prove that (2.8) and (2.9) coincide. Proposition 3.2. Let us suppose that and that the choice of P , and M is as above. Then the stability condition (2.8) and the monotonicity condition coincide and can be written as 1980 DENISE AREGBA-DRIOLLET AND ROBERTO NATALINI Proof. Here 3B is symmetric and its characteristic polynomial is of form where A and C are defined by the following relations: Both eigenvalues of B are positive if and only if A and C are positive. A is positive if and only if # 8 . Now we remark that Six cases have to be studied for the values of A # 1 and A # 2 . As A # 1 and A # 2 play symmetric roles in the formulas, we can suppose that A # 1 # A # 2 and study the three following cases: 1. 2. 3. Let us study the first case. C is positive if and only if It remains to determine the position of 0: two cases are under consideration. and one can see that Consequently the stability condition is satisfied if and only if which is easily seen to be the monotonicity condition (2.9). so that we again recover the same condition. Cases 2 and 3 are similar and we omit the proof. Example 2. Flux decomposition method (FDM). In this example, in view of a more accurate approximation, we take a greater number of equations, and, following an idea due to Brenier [8], we decompose the Jacobians of the fluxes in positive and negative part. Denoting by B d the diagonal matrix of eigenvalues of A # d and by Q d the associated matrix of the right eigenvectors we set - . Then we can define with A d for some # d > 0 and In the scalar convex case and for an appropriate (first-order) discretization, this choice corresponds, in the relaxation limit # 0, to the Engquist-Osher numerical scheme [8]. For a one-dimensional system the stability condition (2.8) is here: for all # R K , (|A #I K - |A # 0 so that in the one-dimensional scalar case it coincides with the monotonicity condition. The two-dimensional case condition (2.8) reads (R K and # . For K=1 this matrix is positive if and only if and this is exactly the monotonicity condition. Proposition 3.3. Let us suppose that K=1 and that the choice of P , and M is as above. Then the stability condition (2.8) and the monotonicity condition coincide and can be written as (3.14). Concerning the entropy properties of this model we refer to [51]. Example 3. Orthogonal velocities method (OVM). This example works with any number of blocks N # D+ 1. We take the velocities such that and 1982 DENISE AREGBA-DRIOLLET AND ROBERTO NATALINI This means that we choose an orthogonal family of D vectors (# 1d , . , #Nd 1, . , D) in the orthogonal space of the vector . The corresponding Maxwellian function is now given by A d (u) a 2 d where a 2 id . Here again the Maxwellian functions are of type (3.4) and the result of [6] applies as described in the preceding section. The one-dimensional case. Let us examine the stability condition in one space dimension for this approximation. We take # > 0 and # I K # 1#n#N diag Then We obtain the following expression for B: I K - For instance if is symmetric with spectral ray # 2 (u), the stability condition reads However Bouchut's condition (3.5) now reads This approximation therefore gives an example where, even in the scalar case, the monotonicity condition is strictly stronger than the stability one. The two-dimensional case. In two space dimensions we take length and direction varying velocities. Fix J, N # 1, # > 0. Set diag # n diag # cos( i# diag # n diag # sin( i# Here we have J . It is easy to see that (3.15) and (3.16) are satisfied and that a 2 For the Maxwellian functions are Proposition 3.4. Let us suppose that and that the choice of P , # 1 , # 2 and M is as above. We denote # the argument of sin #. If then the monotonicity condition (2.9) is satisfied. The stability condition (2.8) is satisfied as soon as Proof. The first part of the proposition is immediate. Let us examine the stability condition (2.8). We remark first that in addition to (3.15), (3.16) we also have sin 2 i# cos 3 i# sin 3 i# Consequently for all j, d # {1, 2}, P# d jd a 2 d N . Denoting -A -A #2 2 # . This matrix is positive if and only if which ends the proof. Example 4. Suliciu's method. Let us conclude this presentation by giving an example which is a bit di#erent from those presented above. For K=2, D=1, let us consider a system of the form Take and # . 1984 DENISE AREGBA-DRIOLLET AND ROBERTO NATALINI For A 2 we have a p-system, and in this case the present approximation was first introduced by Suliciu [65, 66] to study instability problems in phase transitions described by elastic or viscoelastic constitutive equations. In this case the Chapman-Enskog analysis gives the stability condition Some preliminary investigations on this model can be found in [26, 48]. More recent results have been found in [28, 27, 39, 69]. In particular see [69] for an entropy condition. In fact, this model enters the preceding block structure (3.1) by equivalence with the following one: # . The Maxwellian functions M are of type (3.4) so that the result of [6] applies. For an easy calculation shows that both conditions (2.8) and (3.5) coincide with (3.26). 4. Discrete kinetic schemes. In this section we construct numerical schemes for the relaxing semilinear problem (1.3), (1.4) associated with (1.1), (1.2). Of course a lot of numerical schemes are available for this problem including those presented in [30]. We present here a finite volume scheme on structured mesh based on a splitting method. The space time domain R D discretized by a rectangular grid: I # , [0, T d be the canonical d th vector in R D . As usual we denote by x# the center of I #x #,d the length of I # in the direction d, #t #x #,d , #x #,d #t n . Finally we set f #,n f #,n f #,n If then f # 0 is approximated by f #,0 We use for u the same notation as for f here above. Let us recall that and System (1.3) is split into a linear diagonal hyperbolic part and an ordinary di#erential system. For a given f #,n # , the function f #,n+1/2 # is an exact or approximate solution at time t n+1 of the problem # . As the system is diagonal, we may consider each equation separately. We suppose that the scheme can be put in conservation form: f #,n+1/2 #x #,d #,n ed - #,n ed # , where #,n ed #-k1+ed ,1 , . , f #,n #-kL+ed ,L , . , f #,n #,n ed ,l # 1#l#L Here k l # Z D and # l,d (g, . , In the following, the scheme on the linear part will be referred to as homogeneous scheme (HS) and the associated evolution operator will be denoted by H# : f #,n+1/2 # . To take into account the contribution of the singular perturbation term on the right-hand side, we solve on [t n , t n+1 ] the ordinary di#erential system with initial data 1986 DENISE AREGBA-DRIOLLET AND ROBERTO NATALINI for all # Z D . Using (1.5) we obtain so that the solution of (4.9) with data F (t n at t n can be explicitly obtained as Hence f #,n+1 where u is defined by Note that Therefore we have constructed a wide family of numerical schemes for the semilinear system (1.3), which di#er by the choice of the HS. In the scalar case (K=1), thanks to the monotonicity properties of the interaction, we show in the following sections that in fact the properties of each scheme are roughly speaking the same as those of HS and the estimates are uniform in #. We shall often refer to these properties, which are now classical; see [23]: preservation of extrema, monotonicity, total variation diminishing (TVD), and L 1 -contraction. In particular recall that monotonicity implies all the other properties and the TVD property implies the preservation of extrema for initial data in L 1 (R D , R) # L # (R D , R) #BV(R D , R). In the following, the numerical scheme given by (4.1), (4.2), (4.7), (4.13) will be referred to as discrete kinetic scheme (DKS). When # 0 in DKS, we obtain the relaxed limit of the scheme: # . This last scheme can be written in conservation form: #x #,d ed -A n ed # . The numerical fluxes are defined by A n ed ed with ed #-k l +ed ), . , M l (u n #+k l In the following section, for the scalar case K=1, we specify in what sense this relaxed scheme is the limit of DKS. Moreover all the estimates for DKS are uniform and pass to the limit, so that we obtain strong convergence for the limit scheme. 5. Convergence of discrete kinetic schemes for multidimensional scalar conservation laws. In this section we use monotonicity to prove rigorous convergence results for DKS and the associated relaxed scheme. We consider a scalar conservation law (K=1) and {u # a family of initial data for (1.1) such that (R D , R) # L # (R D , R) # BV(R D , R) and ld be such that (1.5) is satisfied. Throughout this section we suppose without loss of generality that and for sake of simplicity we consider a uniform mesh for all # Z D #x #,d = #x d . Moreover, as shown in section 3, M and the # d can be found satisfying is an MMF on [-# ]. The two following relations will be useful in that which follows: for all and for all u, Therefore we have for all # > 0 #f #,0 #f #,0 #f #,0 #,l # , TV(f #,0 TV(f #,0 #. 5.1. The supremum norm bound. Proposition 5.1. Suppose (H1), (H2) are satisfied and that HS preserves the extrema. Then the scheme DKS is L # stable and there holds for all # > 0 for all n # 0, #f #,n # . Moreover, if HS is monotone, then the same is true for DKS. Proof. Using (1.5) and (4.12), we have S(t, 1988 DENISE AREGBA-DRIOLLET AND ROBERTO NATALINI By (H2) we have also for all # Z D , for all l # {1, . , L}, M l (-# f #,0 Suppose that for all # Z D , for all l # {1, . , L}, M l (-# f #,n Since HS preserves the extrema, we have l (-# f #,n+1/2 for every # Z D and l # {1, . , L}. Then for all # Z D , -# u #,n+1 # . Using expression (4.12) we obtain l (-# S l # t, t n , f #,n+1/2 for every t # [t n , t n+1 ], # Z D and l # {1, . , L}. Recalling (5.2) we obtain (5.4). To conclude we remark that if M is an MMF on the interval [-# ], then the RHS of (4.9) is a quasi-monotone application on for all F # [M(-# We can then apply the results of [25]: the flow given by (4.9) preserves the order. 5.2. BV and L 1 estimates. The following lemma follows easily from the monotonicity properties of the interaction. Lemma 5.2. Let F 0 and G 0 be two initial conditions for system (4.9) with corresponding solutions F (t), G(t). If F 0 , G for all t # 0 |F l (t) -G l (t)| # |F 0,l -G 0,l | . As a consequence we obtain another important result. Proposition 5.3. Suppose that (H1), (H2) are satisfied for {u # (1) If HS is TVD then DKS is TVD. (2) If HS is L 1 contracting then DKS is L 1 contracting and there holds #f #,n+1 In order to prove the equicontinuity property and the boundedness of the inter-action we need to estimate the RHS of (1.3). Lemma 5.4. Suppose that (H1), (H2) are satisfied. Suppose HS is TVD and can be put in conservation form (4.7) with a Lipschitz continuous flux #. Then there exists a constant C not depending on # such that exp #t n Proof. From (4.13) and the fact that u #,n+ # we have exp #t n Moreover, by (4.7), ld #x d #,n ed ,l - #,n ed ,l # . Hence for 1 # l # L |M l (u #,n+ 1# ) -f #,n+ 1#,l | # |M l (u #,n #,l | +C #x d sup l |# ld ||#,n ed ,l -#,n ed ,l | . Using the fact that the flux is Lipschitz continuous and that for we have # |w #+ed - w# | , we obtain the desired inequality. Proposition 5.5. Suppose that (H1), (H2) are satisfied. Suppose HS is TVD. Then there exists a constant C not depending on # such that for all t # [0, T for all t, t # [0, T Proof. Inequality (5.8) is immediate by (5.7), thanks to a recursive argument. Let us now prove (5.9). As the numerical flux of HS is Lipschitz continuous, as in the proof of Lemma 5.4, we obtain #f #,n+ 1# - f #,n and #f #,n+1 As M(u #,0 # we have, by a recursive argument, #f #,n+1 The above estimates allow us to prove the convergence of our numerical schemes. First we have convergence towards the unique solution of (1.3), (1.4) when # is fixed. Theorem 5.6. Let T > 0, # > 0, suppose that (H1), (H2) are satisfied for Suppose that HS 1990 DENISE AREGBA-DRIOLLET AND ROBERTO NATALINI is TVD. For any T > 0, let #t #x be constant. As #t # 0 the sequence f # # converges in to the unique solution f # of (1.3), (1.4), f # sup sup As a corollary we obtain global existence of the solution of (1.3), (1.4) (see [54] for a direct proof). Proof of Theorem 5.6. We follow exactly the method of [20] and just give a sketch of the proof: for all t # 0, {f # is bounded in L 1 # L # . Moreover, by Proposition 5.3, we can apply Fr-echet-Kolmogorov theorem and obtain a relatively compact set of L 1 loc . Using the equicontinuity property (5.9) as in the proof of Ascoli- Arzela's theorem we obtain convergence in L # (0, almost everywhere, and the limit is a solution of the problem by Lax-Wendro#'s theorem. Estimates (5.10), (5.11), (5.12) follow from (5.6), (5.8), (5.9). 5.3. The relaxed limit of the scheme. In this section we are interested in the behavior of our numerical schemes as the parameter # tends to zero. The above estimates imply boundedness and TVD properties for the relaxed discrete kinetic schemes. We need one further assumption on the convergence of the initial data: (H3) the sequence {u # converges towards a function u 0 in L 1 . As a consequence and f #,0 converges towards f As above we set and by (1.5) this notation is compatible with what happens at Theorem 5.7. Let T > 0 and suppose that (H1), (H2), and (H3) are satisfied and HS is TVD. As # 0, f # # converges in L # ((0, T ); L 1 loc (R) L ) to a limit f # which satisfies #x #,d ed -A n ed # , where the numerical fluxes are defined by A n ed ld # n ed ,l (5. and ed #-k l +ed ), . , M l (u n #+k l The resulting numerical scheme is TVD and converges to a weak solution of (1.1), (1.2). Moreover, if HS is monotone, the limit scheme is also monotone and converging to the unique entropy solution of (1.1), (1.2). Remark 5.8. (5.17)-(5.19) are the formulas (4.16)-(4.18) obtained formally at the end of section 4. Proof. L 1 stability and the TVD property imply that for n # {0, . , N}, {f #,n 0} is bounded in L 1 # BV, so there exists a sequence # k # 0 such that in L 1 loc and f # k ,n # for every #. Consequently in L # (0, loc ). Estimates (5.14), (5.15), (5.16), monotonicity, and TVD properties are immediate consequences of Propositions 5.1-5.5. Moreover the resulting scheme can be written as (5.16)-(5.19). Thus f # is unique and the whole sequence converges. The consistency with the conservation law (1.1) is the consequence of the consistency of HS with the linear part of (1.3). Therefore we can use the same arguments as in the proof of Theorem 5.6 to obtain convergence and, when HS is monotone, the monotonicity property ensures that the limit is the unique entropy solution of (1.1), (1.2). 6. Numerical experiments. 6.1. The numerical schemes. System (1.3) is split into a linear diagonal hyperbolic part and an ordinary di#erential system. For given f #,n # is an approximate solution at time t n+1 of the problem ld # xd f # . As the system is diagonal, we may consider each equation separately so that we have to approximate the scalar problem where the # d are real and w n # is a piecewise constant function given in L 1 (R D (R D ) #BV (R D ). We present here two methods. Both are constructed on a cartesian grid; see notations in section 4. The first is a straightforward generalization of the upwind scheme: one solves exactly (6.3) on [t n , t n+1 ], obtaining a piecewise constant 1992 DENISE AREGBA-DRIOLLET AND ROBERTO NATALINI function - w# . w n+1 # is then calculated by taking the average of - w# on each cell. We obtain the following explicit formulations: For D=1, where For D=2, we set . For D=3, we set i-u,j-v,k-w i-u,j-v,k i-u,j,k . A second order MUSCL type method can also be applied, generalizing the one-dimensional schemes, by the following steps: (1) given the piecewise constant function w n # we construct a piecewise linear function - #,d for x #]x #- 1, x #+ 1[ ; (2) we solve on [t n , t n+1 ] the linear system (6.2) exactly with initial data: (3) we compute cell average of the resulting solution to obtain w n+1 . The method depends on the choice of # n # . For example we can choose #x# #+ed - w n #-ed d (w n #-ed , w n #+ed where the minmod function is defined by and e This choice corresponds to a linear interpolation of the piecewise constant function # on the neighboring cells of I # with slope limiter. The resulting formulas for are, respectively, (6.4), (6.5) to which one adds the following correction terms: For D=1, For D=2 and A similar formula holds for D=3. Let us write down the one-dimensional numerical flux for the relaxed scheme A n l,# l <0 l,# l >0 Note that each Maxwellian function appears individually in the computation of the slopes #. 6.2. Application to some models. Let us apply the above formulas to some of the approximations given in section 3. One has to write down (6.4), (6.8), (6.5), for each of the N blocks of K equations and to apply (5.17)-(5.19). For the approximation DRM in one space dimension, actually the Jin-Xin ap- proximation, (6.4) gives the following scheme: #x #t #x As already pointed out in [8] and [2], in the scalar case, if #t Lax-Friedrichs scheme. Actually, at first order the approximation OVM of Example 3 in section 3 may be thought of as a generalization of the relaxation approximation. A straightforward calculation gives the following scheme associated with (6.4): #x #t #x i-1, 1994 DENISE AREGBA-DRIOLLET AND ROBERTO NATALINI where if N is even, and if N is odd. The viscosity coe#cient is #N #t #x and it is easy to verify that for a scalar linear conservation law the monotonicity condition, as well as the stability condition, ensure L 2 stability. The model OVM is an interesting case where the monotonicity condition (3.19) is strictly stronger than the stability condition (3.18) issued from the Chapman-Enskog analysis. Actually if we just impose (3.18) the TVD property is lost. Nevertheless the numerical experiments are satisfying (see also Remark 6.1 below). Another point is that (3.19) is not the minimal condition for (6.12) to be monotone. It is su#cient to impose the intermediate condition if N # 3 is odd. With this condition (6.12) can be also viewed as scheme (6.11) where # is replaced by #N # |A # (u)|. Refer now to the second example of section 3, the FDM. The numerical scheme corresponding to the discretization (6.4) is given by Remark that # does not appear explicitly but is needed to ensure that the CFL condition is satisfied. As already pointed out in [8], in the convex scalar (one-dimensional) case we recover the Engquist-Osher's scheme. Our construction improves this scheme (in the general case) by the correction terms (6.8) of the MUSCL discretization. As far as one is concerned with first-order approximations our method retrieves some known schemes. This is not true for the second-order MUSCL discretizations: formula (6.10) shows that the computation of the slopes # involves the Maxwellian functions. Therefore we do not recover in a direct way any known scheme. However our work shares the spirit of the central, Riemann solver free schemes of [56, 34] and a closer comparison should be useful. We write the two-dimensional schemes only for the FDM, which has been found to be the most e#cient. Set We have #t second order correction terms. Here again # does not appear explicitly. 6.3. The numerical tests. In this section we perform some numerical tests with our schemes. The systems considered are a simple p-system, the one-dimensional Euler equations, a two-dimensional scalar conservation law, and the two-dimensional Euler equations. As we have observed that the relaxed numerical solution is in all cases better than the numerical solution of (1.3), we present here only computations performed with The HS is always chosen to be the MUSCL type one exposed above. (1) The p-system. We consider the system In all our computations we take Exact solutions are known for this system; see, for example, [63]. We compute the 1-shock, 2-rarefaction solution of the Riemann problem with data: 0.4 for x > 0, Here we discretize the di#erent approximations: DRM, 4 velocities, and FDM, 6 velocities, and we compute the L 1 error between exact and calculated solutions. We have also computed the solution for Example 4 in section 3, Suliciu's method, but there is almost no di#erence with that of DRM. For the same system we also test the approximation OVM, with 16 and 26 velocities (N=8 and N=13, respectively). Let us recall that this system has the eigenvalues - # (u). At each time step we take the minimal value provided by the stability condition (2.8). Here we have so that the stability condition (3.18) for the OVM reads The computation has been performed on a space interval [-2, 2] and with a maximal 1. The space step has been kept constant and equal to 0.01 and the ratio #t/#x varies between 1 and 0.1. The results are given in Table 6.1. Remark 6.1. In view of (6.12) one may think that for #t small it is possible to take # smaller than # (u) # 3(N-1) N+1 . We tried to perform the computation with # (u). For ratios # > 0.5 we observed oscillations around discontinuities and rarefactions, but then when # 0.5 the results are correct and even better than those obtained with the right condition. 1996 DENISE AREGBA-DRIOLLET AND ROBERTO NATALINI Table exact and numerical solution, 1.0 8.35E-03 2.984E-02 6.302E-03 6.122E-03 0.7 7.34E-03 7.262E-03 6.045E-03 6.035E-03 0.4 6.85E-03 5.281E-03 6.019E-03 5.925E-03 Then we observe the evolution of the L 1 error when the space length step varies and the ratio #t/#x is kept constant and equal to 0.5. Here the convergence speed is defined by #k where e k is the L 1 error. The results are the following. Table exact and numerical solution. 2.00E-02 1.226E-02 .87 9.854E-03 .94 1.058E-02 1.031E-02 5.00E-03 4.035E-03 .84 3.177E-03 .84 3.314E-03 3.287E-03 2.50E-03 2.237E-03 .85 1.769E-03 .84 1.777E-03 1.760E-03 Tables 6.1 and 6.2 show that the relaxation model is the worst. The other three seem to be comparable but the computation is faster for the FDM, since there are fewer equations in the model and also because the value of # is smaller, allowing a greater time step to reach the same value of t. Actually for the same value of #t the results are better for the FDM. We complete this results by the graphic representation of the computed and exact solutions (Figures 6.1-6.2). we represent the v component of the solution, the u component being similar. It appears that the OVM with 26 velocities is nearly the same as the relaxation approximation. The scheme FDM improves the approximation of the one-shock but not that of the rarefaction, while the OVM improves both slightly. (2) One-dimensional Euler system. We now consider the one-dimensional Euler system where #, v, p and E are, respectively, the density, velocity, pressure, and total energy of a perfect gas: (DRM) exact solution Fig. 6.1. p-system: diagonal relaxation method (DRM) and orthogonal velocities method (OVM).0.450.550.65-1 -0.5 0 0.5 1 Fig. 6.2. p-system: flux decomposition method (FDM) and orthogonal velocities method (OVM). We have tested our schemes with a Sod shock tube. Here the solution is constituted by three constant states connected by a one-rarefaction, a two-contact discontinuity, and a three-shock. We have analyzed the L 1 error for the same three models as for the p-system. In Table 6.3 #t is defined as above. Table exact and numerical solution. 2.50E-03 exact solution0.20.61-0.4 -0.3 -0.2 -0.1 0 exact solution Fig. 6.3. One-dimensional Euler system: density and velocity with flux decomposition method exact solution Fig. 6.4. One-dimensional Euler system: pressure with flux decomposition method (FDM). In view of this table, it appears that the approximation FDM and the 39 velocities model of the OVM give similar results. Actually the graphic representation shows an oscillation around the three-shock for the 39 velocities model while the first is stable (see Figures 6.3, 6.4, 6.5(a)). On Figure 6.5(b) we have suppressed these oscillations by taking #t All the computations show that the contact discontinuity is not well approximated. This is a general feature of kinetic schemes (see, for example, [24]). A smaller time step does not improve the contact discontinuity very much but a smaller space step does, as shown in Figure 6.6. (3) A two-dimensional scalar conservation law. In order to compare exact solutions with numerical computations we also consider a one-dimensional conservation law with one-dimensional initial data v 0 . This problem is solved on a two-dimensional cartesian grid not parallel to the axis. We solve u(x, y, where # [0, #/2[. Putting u(t, x, R is the rotation of angle # we have v(X, Y, exact solution0.20.61-0.4 -0.3 -0.2 -0.1 0 (a) (b) Fig. 6.5. One-dimensional Euler system: density with orthogonal velocities method (OVM), -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 exact solution Fig. 6.6. One-dimensional Euler system: density with flux decomposition method (FDM), two di#erent space steps, Here we take #/5 and a rectangular mesh such that #y = 0.8#x. The CFL number is 0.8 for x and y. We consider Burgers' equation: a shock wave: v As the relaxation model has been proved to be worse, we concentrate our attention on the second-order model FDM, here with five velocities, and on the model OVM. Recall that for these last models one-dimensional tests show that it is not very e#cient to take a large number J of modulus for the velocities. The two-dimensional computations confirm this analysis. Figure 6.7 represents the solution for at In the first we compare the cases 1. In the second we compare the cases It appears that the best choice for this model is to take the minimal values N Figure 6.8 represents the same solution with, respectively, but here we compare the model FDM to the one with Here again the five velocities model is the best one. We want to point out that the computation durations are nearly the same in both cases. 2000 DENISE AREGBA-DRIOLLET AND ROBERTO NATALINI0.20.61 exact solution0.20.61 Fig. 6.7. 2D scalar conservation law at Fig. 6.8. Two-dimensional scalar conservation law at Two-dimensional Euler system. We end this section with the two-dimensional Euler system where #, v, p and E are, respectively, the density, velocity, pressure, and total energy of a perfect gas: The initial data are chosen in order to represent a "double Sod tube": We have chosen here the approximation FDM, here with 20 velocities. The calculation has been performed with a rectangular mesh where and the CFL Fig. 6.9. Two-dimensional Euler system: density and velocity x-component. Fig. 6.10. Two-dimensional Euler system: velocity y-component and energy. condition has been fixed to 0.4 in x and y. Figures 6.9 and 6.10 represent the isolines of, respectively, the density, the first and second component of the velocity, and the energy at time 0.16. Acknowledgment . The authors would like to thank Vuk Milisic for performing the numerical tests for the two-dimensional Euler system and for many useful discussions. --R An extended thermodynamic framework for the hydrodynamic modeling of semicon- ductors Convergence of relaxation schemes for conservation laws A model for collision processes in gases. 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Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics Numerical passage from kinetic to fluid equations Monotone di Special issue on lattice gas methods for PDEs: Theory A kinetic construction of global solutions of first order quasilinear equations Hyperbolic Systems of Conservation Laws Numerical Approximation of Hyperbolic Systems of Conservation Laws Weakly coupled systems of quasilinear hyperbolic equations Stability of traveling wave solutions for a rate-type viscoelastic system Zero relaxation limit to centered rarefaction waves for a rate-type viscoelastic system Nonlinear stability of rarefaction waves for a rate-type viscoelastic system Kinetic formulation of the chromatography and some other hyperbolic systems The relaxation schemes for systems of conservation laws in arbitrary space dimensions Convergence and error estimates of relaxation schemes for multidimensional conservation laws Contractive relaxation systems and the scalar multidimensional conservation law New high-resolution central schemes for nonlinear conservation laws and convection-di#usion equations Convergence of a relaxation scheme for n-n hyperbolic systems of conservation laws Shock waves and entropy A linear hyperbolic system with a sti Moment closure hierarchies for kinetic theories Zero Relaxation Limit to Piecewise Smooth Solutions for a Rate-Type Viscoelastic System in the Presence of Shocks Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates A kinetic formulation of multidimensional scalar conservation laws and related equations Kinetic formulation of the isentropic gas dynamics and p-systems Stability of a relaxation model with a nonconvex flux Hyperbolic conservation laws with relaxation Positive schemes for solving multi-dimensional hyperbolic systems of conservation laws Asymptotic stability of planar rarefaction waves for the relaxation approximation of conservation laws in several dimensions BV solutions and relaxation limit for a model in viscoelasticity Linear stability of shock profiles for a rate-type viscoelastic system with relaxation Nonlinear stability of shock fronts for a relaxation system in several space dimensions stability of travelling waves for a hyperbolic system with relaxation Springer Tracts Nat. Convergence to equilibrium for the relaxation approximations of conservation laws A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws Recent results on hyperbolic relaxation problems Nonoscillatory central di Numerical methods for hyperbolic conservation laws with sti Numerical methods for hyperbolic conservation laws with sti A kinetic equation with kinetic entropy functions for scalar conservation laws Discrete velocity models of the Boltzmann equation: A survey on the mathematical aspects of the theory Recent advances in lattice Boltzmann computing Relaxation semi-lin-eaire et cin-etique des syst-emes de lois de conservation On modelling phase transitions by means of rate-type constitutive equations Some stability-instability problems in phase transitions modelled by piecewise linear elastic or viscoelastic constitutive equations Pointwise error estimates for relaxation approximations to conservation laws On the rate of convergence to equilibrium for a system of conservation laws with a relaxation term Viscosity and relaxation approximations for hyperbolic systems of conservation laws Convergence of relaxing schemes for conservation laws Linear and Nonlinear Waves The fluid dynamical limit of the Broadwell model of the nonlinear Boltzmann equation in the presence of shocks Numerical Analysis of Relaxation Schemes for Scalar Conservation Laws --TR --CTR Mapundi K. Banda, Variants of relaxed schemes and two-dimensional gas dynamics, Proceedings of the international conference on Computational methods in sciences and engineering, p.56-59, September 12-16, 2003, Kastoria, Greece Mapundi Kondwani Banda, Variants of relaxed schemes and two-dimensional gas dynamics, Journal of Computational and Applied Mathematics, v.175 n.1, p.41-62, 1 March 2005 Ansgar Jngel , Shaoqiang Tang, Numerical approximation of the viscous quantum hydrodynamic model for semiconductors, Applied Numerical Mathematics, v.56 n.7, p.899-915, July 2006 D. Aregba-Driollet , R. Natalini , S. Tang, Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems, Mathematics of Computation, v.73 n.245, p.63-94, January 2004 Ansgar Jngel , Shaoqiang Tang, A relaxation scheme for the hydrodynamic equations for semiconductors, Applied Numerical Mathematics, v.43 n.3, p.229-252, November 2002

conservation laws;hyperbolic systems;kinetic schemes;BGK models;numerical convergence

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An architecture for more realistic conversational systems.

In this paper, we describe an architecture for conversational systems that enables human-like performance along several important dimensions. First, interpretation is incremental, multi-level, and involves both general and task- and domain-specific knowledge. Second, generation is also incremental, proceeds in parallel with interpretation, and accounts for phenomena such as turn-taking, grounding and interruptions. Finally, the overall behavior of the system in the task at hand is determined by the (incremental) results of interpretation, the persistent goals and obligations of the system, and exogenous events of which it becomes aware. As a practical matter, the architecture supports a separation of responsibilities that enhances portability to new tasks and domains.

INTRODUCTION Our goal is to design and build systems that approach human performance in conversational interaction. We limit our study to practical dialogues: dialogues in which the conversants are cooperatively pursuing specific goals or tasks. Applications involving practical dialogues include planning (e.g. designing a kitchen), information retrieval (e.g. finding out the weather), customer service (e.g. booking an airline advice-giving (e.g. helping assemble modular furni- ture) or crisis management (e.g. a 911 center). In fact, the class of practical dialogues includes almost anything about which people might want to interact with a computer. TRIPS, The Rochester Interactive Planning System [6], is an end-to-end system that can interact robustly and in near real-time using spoken language and other modalities. It has Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for pro-t or commercial advantage and that copies bear this notice and the full citation on the -rst page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior speci-c permission and/or a fee. IUI'00, January 14-17, 2001, Sante Fe, New Mexico. participated successfully in dialogues with untrained users in several di#erent simple problem solving domains. Our experience building this system, however, revealed several problems that motivated the current work. 1.1 Incrementality Like most other dialogue systems that have been built, TRIPS enforces strict turn taking between the user and system, and processes each utterance sequentially through three stages: interpretation-dialogue management-generation. Unfortunately, these restrictions make the interaction un-natural and stilted and will ultimately interfere with the human's ability to focus on the problem itself rather than on making the interaction work. We want an architecture that allows a more natural form of interaction; this requires incremental understanding and generation with flexible turn-taking Here are several examples of human conversation that illustrate some of the problems with processing in stages. All examples are taken from a corpus collected in an emergency management task set in Monroe County, NY [17]. Plus signs (+) denote simultaneous speech, and "#" denotes silence. First, in human-human conversation participants frequently ground (confirm their understanding of) each other's contributions using utterances such as "okay" and "mm-hm". Clearly, incremental understanding and generation are required if we are to capture this behavior. In the following example, A acknowledges each item in B's answers about locations where there are road outages. Excerpt from Dialogue s16 A: can you give me the first uh # outage B: okay B: so Elmwood bridge A: okay um # Thurston road A: mm-hm Three at Brooks A: mm-hm B: and Four Ninety at the inner # loop A: okay Second, in human-human dialogues the responder frequently acknowledges the initiator's utterance immediately after it is completed and before they have performed the tasks they need to do to fully respond. In the next excerpt, A asks for problems other than road outages. B responds with an immediate acknowledgment. Evidence of problem solving activity is revealed by the user smacking their lips ("lips- mack") and silence, and then B starts to respond to the request. Excerpt from Dialogue s16 A: and what are the # other um # did you have just beside road +1 outages +1 B: +1 okay +1 # um Three Eighty Three and Brooks +2 # is +2 a # road out # and an electric line down A: +2 Brooks mm hm +2 A: okay A sequential architecture, requiring interpretation and problem solving to be complete before generation begins, cannot produce this behavior in any principled way. A third example involves interruptions, where the initiator starts to speak again after the responder has formulated a response and possibly started to produce it. In the example starts to respond to A's initial statement but then A continues speaking. Excerpt from Dialogue s6 A: and he's going to pull the tree # B: mm hm done at # um # he's going to be done at # in forty minutes # We believe e#ective conversational systems are going to have to be able to interact in these ways, which are perfectly natural (in fact are the usual mode of operation) for humans. It may be that machines will not duplicate human behavior exactly, but they will realize the same conversational goals using the communication modalities they have. Rather than saying "uh-huh," for instance, the system might ground a referring expression by highlighting it on a display. Note also that the interruption example requires much more than a "barge-in" capability. B needs to interpret A's second utterance as a continuation of the first, and does not simply abandon its goal of responding to the first. When B gets the turn, B may decide to still respond in the same way, or to modify its response to account for new information. 1.2 Initiative Another reason TRIPS does not currently support completely natural dialogue is that, like most other dialogue systems, it is quite limited in the form of mixed-initiative interaction it supports. It supports taking discourse-level initiative (cf. [3]) for clarifications and corrections, but does not allow shifting of task-level initiative during the inter- action. The reason is that system behavior is driven by the dialogue manager, which focuses on interpreting user input. This means that the system's own independent goals are deemphasized. The behavior of a conversational agent should ideally be determined by three factors, not just one: the interpretation of the last user utterance (if any), the sys- tem's own persistent goals and obligations, and exogenous events of which the agent becomes aware. For instance, in the Monroe domain, one person often chooses to ignore the other person's last utterance and leads the conversation to discuss some other issue. Sometimes they explicitly acknowledge the other's request and promise to address it later, as in the following: Excerpt from Dialogue s16 A: can you # +1 can +1 you go over the # the thing +2 for me again +2 B: +2 yeah in one +2 minute have to # clarify at the end here . In other cases, they simply address some issue they apparently think is more important than following the other's lead, as in the following example where B does not address A's suggestion about using helicopters in any explicit way. Excerpt from Dialogue s12 A: we can we # we can either uh guess we have to decide how to break up # this A: we can # make three trips with a helicopter B: so i guess we should send one ambulance straight o# to # marketplace right # now # right 1.3 Portability Finally, on a practical note, while TRIPS was designed to separate discourse interpretation from task and domain reasoning, in practice domain- and task-specific knowledge ended up being used directly in the dialogue manager. This made it more di#cult to port the system to di#erent domains and also hid the di#erence between general domain-independent discourse behavior and task-specific behavior in a particular domain. To address these problems, we have developed a new architecture for the "core" of our conversational system that involves asynchronous interpretation, generation, and system planning/acting processes. This design simplifies the incremental development of new conversational behaviors. In addition, our architecture has a clean separation between discourse/dialogue modeling and task/domain levels of rea- soning, which (a) enhances our ability to handle more complex domains, (b) improves portability between domains; and (c) allows for richer forms of task-level initiative. The remainder of this paper describes our new architecture in detail. The next section presents an overview of the design and detailed descriptions of the major components. A brief but detailed example illustrates the architecture in action. We conclude with a discussion of related work on conversational systems and the current status of our implementation 2. ARCHITECTURE DESCRIPTION As mentioned previously, we have been developing conversational agents for some years as part of the TRAINS [7] and TRIPS [6] projects. TRIPS is designed as a loosely-coupled collection of components that exchange information by passing messages. There are components for speech processing (both recognition and synthesis), language understanding, dialogue management, problem solving, and so on. In previous versions of the TRIPS system, the Dialogue Manager component (DM) performed several functions: . Interpretation of user input in context . Maintenance of discourse context . Planning the content (but not the form) of system response . Managing problem solving and planning Having all these functions performed by one component led to several disadvantages. The distinction between domain planning and discourse planning was obscured. It became di#cult to improve interpretation and response planning, because the two were so closely knit. Incremental processing was di#cult to achieve, because all input had to pass through the DM (even if no domain reasoning was going to occur, but only discourse planning). Finally, porting the system to new tasks and domains was hampered by the inter-connections between the various types of knowledge within the DM. The new core architecture of TRIPS is shown in Figure 1. There are three main processing components. The Interpretation Manager (IM) interprets user input as it arises. It broadcasts the recognized speech acts and their interpretation as problem solving actions, and incrementally updates the Discourse Context. The Behavioral Agent (BA) is most closely related to the autonomous "heart" of the agent. It plans system behavior based on its goals and obligations, the user's utterances and actions, and changes in the world state. Actions that involve communication and collaboration with the user are sent to the Generation Manager (GM). The GM plans the specific content of utterances and display updates. Its behavior is driven by discourse obligations (from the Discourse Context), and directives it receives from the BA. The glue between the layers is an abstract model of problem solving in which both user and system contributions to the collaborative task can be expressed. All three components operate asynchronously. For in- stance, the GM might be generating an acknowledgment while the BA is still deciding what to do. And if the user starts speaking again, the IM will start interpreting these new actions. The Discourse Context maintains the shared state needed to coordinate interpretation and generation. In the remainder of this section, we describe the major components in more detail, including descriptions of the Discourse Context, Problem Solving Model, and Task Manager. 2.1 Discourse Context The TRIPS Discourse Context provides the information to coordinate the system's conversational behavior. First, it supplies su#cient information to generate and interpret anaphoric expressions and to interpret forms of ellipsis. Given the real-time nature of the interactions, and the fact that the system may have its own goals and receive reports about external events, the discourse context must also provide information about the status of the turn (i.e. can I speak now or should I wait?), and what discourse obligations are currently outstanding (cf. [19]). The latter is especially important when the system chooses to pursue some other goal (e.g. notifying the user of an accident) rather than perform the expected dialogue act (e.g. answering a question); to be coherent and cooperative, the system should usually still outstanding discourse obligations, even if this is done Behavioral Agent Interpretation Manager Generation Manager Parser Speech Planner Scheduler Monitors Events Task- and Domain-specific Knowledge Sources Exogenous Event Sources Response Planner Graphics Speech Task Manager Reference Discourse Context Interpretation Generation Behavior Task Interpretation Requests Problem-Solving Acts recognized from user Problem-Solving Acts to perform Task Execution Requests Figure 1: New Core Architecture simply by means of an apology. Finally, as we move towards open-mike interactive systems, we must also identify and generate appropriate grounding behaviors. To support these needs, the TRIPS discourse context contains the following information: 1. A model of the current salient entities in the discourse, to support interpretation and generation of anaphoric 2. The structure and interpretation of the immediately preceding utterance, to support ellipsis resolution and clarification questions; 3. The current status of the turn-whether it is assigned to one conversant or currently open. 4. A discourse history consisting of the speech-act interpretations of the utterances in the conversation so far, together with an indication of what utterances have been grounded; 5. The current discourse obligations, typically to respond to the other conversant's last utterance. Obligations may act as a stack during clarification subdialogues, or short-term interruptions, but this stack never becomes very large. This is a richer discourse model than found in most systems (although see [12] for a model of similar richness). 2.2 Abstract Problem Solving Model The core modules of the conversational agent, the IM, BA and GM, use general models of collaborative problem solv- ing, but these models remain at an abstract level, common to all practical dialogues. This model is formalized as a set of actions that can be performed on problem solving objects. The problem solving objects include objectives (goals being pursued), solutions (proposed courses of action or structures that may achieve an objective), resources (objects used in solutions, such as trucks for transportation, space in kitchen design), and situations (settings in which solutions are used to attain objectives). In general, there are a number of di#erent actions agents can perform as they collaboratively solve problems. Many of these can apply to any problem solving object. For exam- ple, agents may create new objectives, new solutions, new situations (for hypothetical reasoning) and new resources (for resource planning). Other actions in our abstract problem solving model include select (e.g.focus on a particular objective), evaluate (e.g. determine how long a solution might take), compare (e.g. compare two solutions to the same objective), modify (e.g. change some aspect of a so- lution, change what resources are available), repair (e.g. fix an old solution so that it works) and abandon (e.g. give up on an objective, throw out a possible solution) 1 . Note that because we are dealing with collaborative problem solving, not all of these actions can be accomplished by one agent alone. Rather, one agent needs to propose an action (the agent is said to initiate the collaborative act), and the other accept it (the other agent is said to complete the collaborative act). There are also explicit communication acts involved in collaborative problem solving. Like all communicative acts, these acts are performed by a single agent, but are only successful if the other agent understands the communication. The main communication acts for problem solving include describe (e.g. elaborate on an objective, describe a particular solution), explain (e.g. provide a rationale for a solution or decision), and identify (e.g. communicate the existence of a resource, select a goal to work on). These communication acts, of course, may be used to accomplish other problem solving goals as well. For instance, one might initiate the creation of an objective by describing it. 2.3 Task Manager The behaviors of the IM, BA and GM are defined in terms of the abstract problem solving model. The details of what these objects are in a particular domain, and how operations are performed, are specified in the Task Manager (TM). The TM supports operations intended to assist in both the recognition of what the user is doing with respect to the task at hand and the execution of problem solving steps intended to further progress on the task at hand. Specifically, the Task Manager must be able to: 1. Answer queries about objects and their role in the task/domain (e.g. is an ambulance a resource? Is loading a truck an in-domain plannable/executable action? Is "evacuating a city" a possible in-domain goal?) 2. Provide the interface between the generic problem solving acts used by the BA (e.g. create a solution) and the actual task-specific agents that perform the tasks (e.g. build a course of action to evacuate the city using two trucks) 3. Provide intention recognition services to the IM (e.g. can "going to Avon" plausibly be an extension of the current course of action?) This list is not meant to be exhaustive, although it has been developed based on our experiences building systems in several problem-solving domains. In our architecture, the Task Manager maps abstract problem solving acts onto the capabilities of the knowledge-based agents at its disposal. For example, in one of our planning domains, the Task Manager uses a planner, router, sched- uler, and temporal knowledge base to answer queries and create or modify plans. 2.4 Interpretation Manager The Interpretation Manager (IM) interprets incoming parsed utterances and generates updates to the Discourse Context. First, it produces turn-taking information. With a push- to-talk interface this is simple. When the user presses the button they have taken the turn; when they release it they have released the turn. As we move to open-mike, identifying turn-taking behavior will require more sophisticated in- terpretation. TRIPS uses an incremental chart parser that will assist in this process by broadcasting constituents as they are recognized. The principal task of the IM, however, is to identify the intended speech act, the collaborative problem solving act that it furthers, and the system's obligations arising from the interaction. For instance, the utterance "The bridge over the Genesee is blocked" would be interpreted in some circumstances as a problem statement, the intention being to initiate replanning. The IM would broadcast a discourse-level obligation to respond to a statement, and announce that the user has initiated the collaborative problem solving act of identifying a problem as a means of initiating replanning (say, to change the route currently planned). In other circumstances, the same utterance might be recognized as the introduction of a new goal (i.e. to reopen the bridge). The rules to construct these interpretations are based on the abstract problem solving model and specific decisions are made by querying the Task Manager. For instance, in the above example, key questions might be "is there an existing plan using the bridge?" (an a#rmative answer indicates the replanning interpretation) and "is making the bridge available a reasonable high-level goal in this domain?" (an a#rmative answer indicates the introduce-goal interpreta- tion). 2.5 Generation Manager The Generation Manager (GM), which performs content planning, receives problem solving goals requiring generation from the Behavioral Agent (BA) and discourse obligations from the Discourse Context. The GM's task is to synthesize these input sources and produce plans (sequences of discourse acts) for the system's discourse contributions. Because the GM operates asynchronously from the IM, it can be continuously planning. For instance, it is informed when the user's turn ends and can plan simple take-turn and keep-turn acts even in the absence of further information from the IM or the BA, using timing information. In the case of grounding behaviors and some conventional interactions (e.g. greetings), the GM uses simple rules based on adjacency pairs; no reference to the problem solving state is necessary. In other cases, it may need information from the BA in order to satisfy a discourse obligation. It may also receive goals from the Behavioral Agent that it can plan to satisfy even in the absence of a discourse obligation, for instance when something important changes in the world and the BA wants to notify the user. The GM can also plan more extensive discourse contributions using rhetorical relations expressed as schemas, for instance to explain a fact or proposal or to motivate a proposed action. It has access to the discourse context as well as to sources for task and domain-level knowledge. When the GM has constructed a discourse act or set of acts for production, it sends the act(s) and associated content to the Response Planner, which performs surface gen- eration. The RP comprises several subcomponents; some are template-based, some use a TAG-based grammar, and one performs output selection and coordination. It can realize turn-taking, grounding and speech acts in parallel and in real-time, employing di#erent modalities where useful. It can produce incremental output at two levels: it can produce the output for one speech act before others in a plan are realized; and where there is propositional content, it can produce incremental output within the sentence [10]. If a discourse act is realized and produced successfully, the GM is informed and sends an update to the Discourse Context. 2.6 Behavioral Agent As described above, the Behavioral Agent (BA) is responsible for the overall problem solving behavior of the system. This behavior is a function of three aspects of the BA's environment: (1) the interpretation of user utterances and actions in terms of problem solving acts, as produced by the Interpretation Manager; (2) the persistent goals and obligations of the system, in terms of furthering the problem solving task; (3) Exogenous events of which the BA becomes aware, perhaps by means of other agents monitoring the state of the world or performing actions on the BA's behalf. As we noted previously, most dialogue systems (including previous versions of TRIPS) respond primarily to the first of these sources of input, namely the user's utterances. In some systems (including previous versions of TRIPS) there is some notion of the persistent goals and/or obligations of the system. Often this is implicit and "hard-coded" into the rules governing the behavior of the system. In realistic conversational systems, however, these would take on a much more central role. Just as people do, the system must juggle its various needs and obligations and be able to talk about them explicitly. Finally, we think it is crucial that conversational systems get out into the world. Rather than simply looking up answers in a database or even conducting web queries, a conversational system helping a user with a real-world task is truly an agent embedded in the world. Events occur that are both exogenous (beyond its control) and asynchronous (occurring at unpredictable times). The system must take account of these events and integrate them into the conver- sation. Indeed in many real-world tasks, this "monitoring" function constitutes a significant part of the system's role. The Behavioral Agent operates by reacting to incoming events and managing its persistent goals and obligations. In the case of user-initiated problem solving acts, the BA determines whether to be cooperative and how much initiative to take in solving the joint problem. For example, if the user initiates creating a new objective, the system can complete the act by adopting a new problem solving obligation to find a solution. It could, however, take more initiative, get the Task Manager to compute a solution (perhaps a partial or tentative one), and further the problem solving by proposing the solution to the user. The BA also receives notification about events in the world and chooses whether to communicate them to the user and/or adopt problem solving obligations about them. For exam- ple, if the system receives a report of a heart attack victim needing attention, it can choose to simply inform the user of this fact (and let them decide what to do about it). More likely, it can decide that something should be done about the situation, and so adopt the intention to solve the problem (i.e. get the victim to a hospital). Thus the system's task-level initiative-taking behavior is determined by the BA, based on the relative priorities of its goals and obligations. These problem-solving obligations determine how the system will respond to new events, including interpretations of user input. 2.7 Infrastructure The architecture described in this paper is built on an extensive infrastructure that we have developed to support e#ective communication between the various components making up the conversational system. Space precludes an extended discussion of these facilities, but see [1] for further details. System components communicate using the Knowledge Query and Manipulation Language (KQML [11]), which provides a syntax and high-level semantics for messages exchanged between agents. KQML message tra#c is mediated by a Facilitator that sits at the hub of a star topology network of components. While a hub may seem to be a bottleneck, in practice this has not been a problem. On the contrary, the Facilitator provides a variety of services that have proven indispensable to the design and development of the overall system. These include: robust initial- ization, KQML message validation, naming and lookup ser- vices, broadcast facilities, subscription (clients can subscribe in order to receive messages sent by other clients), and advertisement (clients may advertise their capabilities). The bottom line is that an architecture for conversational systems such as the one we are proposing in this paper would be impractical, if not impossible, without extensive infrastructure support. While these may seem like "just implementation details," in fact the power and flexibility of the TRIPS infrastructure enables us to design the architecture to meet the needs of realistic conversation and to make it work. 3. EXAMPLE An example will help clarify the relationships between the various components of our architecture and the information that flows between them, as well as the necessity for each. Consider the situation in which the user asks "Where are the ambulances?" First, the speech recognition components notice that the user has started speaking. This is interpreted by the Interpretation Manager as taking the turn, so it indicates that a TAKE-TURN event has occurred: (tell (done (take-turn :who user))) The Generation Manager might use this information to cancel or delay a planned response to a previous utterance. It can also be used to generate various grounding behaviors (e.g. changing a facial expression, if such a capability is sup- ported). When the utterance is completed, the IM interprets the user's having stopped speaking as releasing the turn: (tell (done (release-turn :who user))) At this point, the GM may start planning (or executing) an appropriate response. The Interpretation Manager also receives a logical form describing the surface structure of this request for infor- mation. It performs interpretation in context, interacting with the Task Manager. In this case, it asks the Task Manager if ambulances are considered resources in this domain. With an a#rmative response, it interprets this question as initiating the problem solving act of identifying relevant re- sources. Note that contextual interpretation is critical-the user wants to know where the usable ambulances are, not where all known ambulances might be. The IM then generates 1. A message to the Discourse Context recording the user's utterance in the discourse history together with its structural analysis from the parser. 2. A message to the Discourse Context that the system now has an obligation to respond to the question: (tell (introduce-obligation :id OBLIG1 :who system :what (respond-to (wh-question :id :who user :what (at-loc (the-set ?x (type ?x ambulance)) (wh-term ?l (type ?l location))) :why (initiate PS1))))) This message includes the system's obligation, a representation of the content of the question, and a connection to the recognized problem solving act (defined in the message described next). The IM does not specify how the obligation to respond to the question should be discharged. 3. A message to the Behavioral Agent that the user has initiated a collaborative problem solving act, namely attempting to identify a resource: (tell (done (initiate :who user :what (identify-resource :id PS1 :what (set-of ?x (type ?x ambulance)))))) This message includes the problem solving act recognized by the IM as the user's intention, and a representation of the content of the question. When the Discourse Context receives notification of the new discourse obligation, this fact is broadcast to any subscribed components, including the Generation Manager. The GM cannot answer the question without getting a response from the Behavioral Agent. So it adopts the goal of answer- ing, and waits for information from the BA. While waiting, it may plan and produce an acknowledgment of the question. When the Behavioral Agent receives notification that the user has initiated a problem solving act, one of four things can happen depending on the situation. We will consider each one in sequence. Do the Right Thing It may decide to "do its part" and try to complete (or at least further) the problem solv- ing. In this case, it would communicate with other components to answer the query about the location of the ambulances, and then send the GM a message like: (request (identify-resource :who system :what (and (at-loc amb-1 rochester) .) :why (complete :who system :what PS1))) The BA expects that this will satisfy its problem solving goal of completing the identify-resources act initiated by the user, although it can't be sure until it hears back from the IM that the user understood the response. Clarification The BA may try to identify the resource but fail to do so. If a specific problem can be identified as having caused the failure, then it could decide to initiate a clarification to obtain the information needed. For instance, say the dialogue has so far concerned a particular subtask involving a particular type of am- bulances. It might be that the BA cannot decide if it should identify just the ambulances of the type for this subtask, or whether the user wants to know where all usable ambulances are. So it might choose to tell the GM to request a clarification. In this case, the BA retains its obligation to perform the identify-resources act. Failure On the other hand, the BA may simply fail to identify the resources that the user needs. For instance, the agents that it uses to answer may not be responding, or it may be that the question cannot be answered. In this case, it requests the GM to notify the user of fail- ure, and abandons (at least temporarily) its problem solving obligation. Ignoring the Question Finally, the BA might decide that some other information is more important, and send that information to the GM (e.g. if a report from the world indicates a new and more urgent task for the user and system to respond to). In this case, the BA retains the obligation to work on the pending problem solving action, and will return to it when circumstances permit. Whatever the situation, the Generation Manager receives some abstract problem solving act to perform. It then needs to reconcile this act with its discourse obligation OBLIG1. Of course, it can satisfy OBLIG1 by answering the ques- tion. It can also satisfy OBLIG1 by generating a clarification request, since the clarification request is a satisfactory response to the question. (Note that the obligation to answer the original question is maintained as a problem solving goal, not a discourse obligation). In the case of a failure, OBLIG1 could be satisfied by generating an apology and a description of the reason the request could not be satisfied. If the BA ignores the question, the GM might apologize and add a promise to address the issue later, before producing the unrelated information. The apology would satisfy OBLIG1. For a very urgent message (e.g. a time critical warning), it might generate the warning immediately, leaving the discourse obligation OBLIG1 unsatisfied, at least temporarily. The GM sends discourse acts with associated content to the Response Planner, which produces prosodically-annotated text for speech synthesis together with multimodal display commands. When these have been successfully (or partially in the case of a user interruption) produced, the GM is informed and notifies the Discourse Context as to which discourse obligations should have been met. It also gives the Discourse Context any expected user obligations that result from the system's utterances. The Interpretation Manager uses knowledge of these expectations to aid subsequent interpretation. For example, if an answer to the user's question is successfully produced, then the user has an obligation to acknowledge the answer. Upon receiving an acknowledgment (or inferring an implicit acknowledge), the IM notifies the Discourse Context that the obligation to respond to the question has truly been discharged, and might notify the BA that the collaborative "Identify-Resource" act PS1 has been completed. 4. IMPLEMENTATION The architecture described in this paper arose from a long-term e#ort in building spoken dialogue systems. Because we have been able to easily port most components from our previous system into the new one, the system itself has a wide range of capabilities that were already present in earlier versions. Specifically, it handles robust, near real-time spontaneous dialogue with untrained users as they solve simple tasks such as trying to find routes on a train map and planning evacuation of personnel from an island (see [1] for an overview of the di#erent domains we have implemented). The system supports cooperative, incremental development of plans with clarifications, corrections, modifications and comparison of di#erent options, using unrestricted, natural language (as long as the user stays focussed on the task at hand). The new architecture extends our capabilities to better handle the incremental nature of interpretation, the fact that interpretation and generation must be interleaved, and the fact that realistic dialogue systems must also be part of a broader outside world that is not static. A clean separation between linguistic and discourse knowledge on one hand, and task and domain knowledge on the other, both clarifies the role of the individual components and improves portability to new tasks. We produced an initial demonstration of our new architecture in August 2000, in which we provided the dialogue capabilities for an emergency relief planning domain which used simulation, scheduling, and planning components built by research groups at other institutions. Current work involves extending the capabilities of individual components (the BA and GM in particular) and porting the system to the TRIPS-911 domain. 5. RELATED WORK Dialogue systems are now in use in many applications. Due to space constraints, we have selected only some of these for comparison to our work. They cover a range of domains, modalities and dialogue management types: . Information-seeking systems [2, 5, 8, 9, 13, 15, 16] and planning systems [4, 14, 18]; . Speech systems [13, 14, 15, 16], multi-modal systems [5, 8, 18] and embodied conversational agents [2, 9]; . Systems that use schemas or frames to manage the dialogue [9, 13, 14, 16], ones that use planning [4], ones that use models of rational interaction [15], and ones that use dialogue grammars or finite state models [5, 8, 18]. Most of the systems we looked at use a standard interpretation- dialogue management-generation core, with the architecture being either a pipeline or organized around a message-passing hub with a pipeline-like information flow. Our architecture uses a more fluid processing model, which enables the di#erences we outline below. 5.1 Separation of domain/task reasoning from discourse reasoning Since many dialogue systems are information-retrieval sys- tems, there may be fairly little task reasoning to perform. For that reason, although many of these systems have domain models or databases separate from the dialogue manager [5, 8, 9, 13, 15, 16], they do not have separate task models. By contrast, our system is designed to be used in domains such as planning, monitoring, and design, where task-level reasoning is crucial not just for performing the task but also for interpreting the user's actions and utter- ances. Separation of domain knowledge and task reasoning from discourse reasoning - through the use of our Task Manager, various world models, the abstract problem solving model and the Behavioral Agent - allows us access to this information without compromising portability and flexibility CommandTalk [18], because it is a thin layer over a stand-alone planner-simulator, has little direct involvement in task reasoning. However, the dialogue manager incorporates some domain-dependent task reasoning, e.g. in the discourse states for certain structured form-filling dialogues. In the work of Cassell et al [2], the response planner performs deliberative task and discourse reasoning to achieve communicative and task-related goals. In our architecture, there is a separation between task- and discourse-level plan- ning, with the Behavioral Agent handling the first type of goal and the Generation Manager the other. Chu-Carroll and Carberry's CORE [4] is not a complete system, but does have a specification for input to the response planner that presumably would come from a dialogue manager. The input specification allows for domain, problem solving, belief and discourse-level intentions. Our generation manager reasons over discourse-level intentions; it obtains information about domain, problem solving and belief intentions from other modules. The CMU Communicator systems have a dialogue man- ager, but use a set of domain agents to "handle all domain-specific information access and interpretation, with the goal of excluding such computation from the dialogue management component" [14]. However, the dialogue manager uses task or domain-dependent schemas to determine its behavior 5.2 Separation of interpretation from response- planning Almost all the systems we examined combine interpretation with response planning in the dialogue manager. The architecture outlined by Cassell et al [2], however, separates the two. It includes an understanding module (performing the same kinds of processing performed by our Interpretation Manager); a response planner (performing deliberative reasoning); and a reaction module (which performs action coordination and handles reactive behaviors such as turn- taking). We do not have a separate component to process reactive behaviors; we get reactive behaviors because different types of goals take di#erent paths through our sys- tem. Cassell et al's ``interactional'' goals (e.g. turn-taking, grounding) are handled completely by the discourse components of our system (the discourse interpretation and response planner); the handling of their "propositional" goals may involve domain or task reasoning and therefore will involve our Behavioral Agent and problem-solving modules. Fujisaki et al [8] divide discourse processing into a user model and a system model. As in other work [4, 15], this is an attempt to model the beliefs and knowledge of the agents participating in the discourse, rather than the discourse it- self. However, interpretation must still be completed before response planning begins. Furthermore, the models of user and system are finite-state models; for general conversational agents more flexible models may be necessary. 6. CONCLUSIONS We have described an architecture for the design and implementation of conversational systems that participate effectively in realistic practical dialogues. We have emphasized the fact that interpretation and generation must be interleaved and the fact that dialogue systems in realistic settings must be part of and respond to a broader "world outside." These considerations have led us to an architecture in which interpretation, generation, and system behavior are functions of autonomous components that exchange information about both the discourse and the task at hand. A clean separation between linguistic and discourse knowledge on the one hand, and task- and domain-specific information on the other hand, both clarifies the roles of the individual components and improves portability to new tasks and domains. 7. ACKNOWLEDGMENTS This work is supported by ONR grant no. N00014-95-1- 1088, DARPA grant no. F30602-98-2-0133, and NSF grant no. IRI-9711009. 8. --R An architecture for a generic dialogue shell. Requirements for an architecture for embodied conversational characters. Initiative in collaborative interactions - its cues and e#ects Collaborative response generation in planning dialogues. An architecture for multi-modal natural dialogue systems TRIPS: An integrated intelligent problem-solving assistant The design and implementation of the TRAINS-96 system: A prototype mixed-initiative planning assistant Principles and design of an intelligent system for information retrieval over the internet with a multimodal dialogue interface. The August spoken dialogue system. Incremental generation for real-time applications A proposal for a new KQML specification. Modelling grounding and discourse obligations using update rules. Design strategies for spoken dialog systems. Creating natural dialogs in the Carnegie Mellon Communicator system. ARTIMIS: Natural dialogue meets rational agency. The Monroe corpus. The CommandTalk spoken dialogue system. Discourse obligations in dialogue processing. --TR TRIPs The Design and Implementation of the TRAINS-96 System: A Prototype Mixed-Initiative Planning Assistant The Monroe Corpus --CTR Javier Calle-Gmez , Ana Garca-Serrano , Paloma Martnez, Intentional processing as a key for rational behaviour through Natural Interaction, Interacting with Computers, v.18 n.6, p.1419-1446, December, 2006 Emerson Cabrera Paraiso , Jean-Paul A. Barths, Une interface conversationnelle pour les agents assistants appliqus des activits professionnelles, Proceedings of the 16th conference on Association Francophone d'Interaction Homme-Machine, p.243-246, August 30-September 03, 2004, Namur, Belgium Robert Porzel , Iryna Gurevych, Towards context-adaptive utterance interpretation, Proceedings of the 3rd SIGdial workshop on Discourse and dialogue, p.154-161, July 11-12, 2002, Philadelphia, Pennsylvania James Allen , Nate Blaylock , George Ferguson, A problem solving model for collaborative agents, Proceedings of the first international joint conference on Autonomous agents and multiagent systems: part 2, July 15-19, 2002, Bologna, Italy Judith Hochberg , Nanda Kambhatla , Salim Roukos, A flexible framework for developing mixed-initiative dialog systems, Proceedings of the 3rd SIGdial workshop on Discourse and dialogue, p.60-63, July 11-12, 2002, Philadelphia, Pennsylvania Kenneth D. Forbus , Thomas R. Hinrichs, Companion cognitive systems: a step toward human-level AI, AI Magazine, v.27 n.2, p.83-95, July 2006 Ana-Maria Popescu , Oren Etzioni , Henry Kautz, Towards a theory of natural language interfaces to databases, Proceedings of the 8th international conference on Intelligent user interfaces, January 12-15, 2003, Miami, Florida, USA Meriam Horchani , Laurence Nigay , Franck Panaget, A platform for output dialogic strategies in natural multimodal dialogue systems, Proceedings of the 12th international conference on Intelligent user interfaces, January 28-31, 2007, Honolulu, Hawaii, USA Sheila Garfield , Stefan Wermter, Call classification using recurrent neural networks, support vector machines and finite state automata, Knowledge and Information Systems, v.9 n.2, p.131-156, February 2006 Michelle X. Zhou , Keith Houck , Shimei Pan , James Shaw , Vikram Aggarwal , Zhen Wen, Enabling context-sensitive information seeking, Proceedings of the 11th international conference on Intelligent user interfaces, January 29-February 01, 2006, Sydney, Australia Michelle X. Zhou , Vikram Aggarwal, An optimization-based approach to dynamic data content selection in intelligent multimedia interfaces, Proceedings of the 17th annual ACM symposium on User interface software and technology, October 24-27, 2004, Santa Fe, NM, USA Iryna Gurevych , Rainer Malaka , Robert Porzel , Hans-Pter Zorn, Semantic coherence scoring using an ontology, Proceedings of the Conference of the North American Chapter of the Association for Computational Linguistics on Human Language Technology, p.9-16, May 27-June 01, 2003, Edmonton, Canada Ryuichiro Higashinaka , Mikio Nakano , Kiyoaki Aikawa, Corpus-based discourse understanding in spoken dialogue systems, Proceedings of the 41st Annual Meeting on Association for Computational Linguistics, p.240-247, July 07-12, 2003, Sapporo, Japan G. Aist , J. Dowding , B. A. Hockey , M. Rayner , J. Hieronymus , D. Bohus , B. Boven , N. Blaylock , E. Campana , S. Early , G. Gorrell , S. Phan, Talking through procedures: an intelligent space station procedure assistant, Proceedings of the tenth conference on European chapter of the Association for Computational Linguistics, April 12-17, 2003, Budapest, Hungary Nate Blaylock , James Allen , George Ferguson, Synchronization in an asynchronous agent-based architecture for dialogue systems, Proceedings of the 3rd SIGdial workshop on Discourse and dialogue, p.1-10, July 11-12, 2002, Philadelphia, Pennsylvania Ryuichiro Higashinaka , Noboru Miyazaki , Mikio Nakano , Kiyoaki Aikawa, Evaluating discourse understanding in spoken dialogue systems, ACM Transactions on Speech and Language Processing (TSLP), 1, p.1-20, 2004 Jeremy Goecks , Dan Cosley, NuggetMine: intelligent groupware for opportunistically sharing information nuggets, Proceedings of the 7th international conference on Intelligent user interfaces, January 13-16, 2002, San Francisco, California, USA Phillipa Oaks , Arthur Hofstede, Guided interaction: A mechanism to enable ad hoc service interaction, Information Systems Frontiers, v.9 n.1, p.29-51, March 2007 Emerson Cabrera Paraiso , Jean-Paul A. Barths, An intelligent speech interface for personal assistants applied to knowledge management, Web Intelligence and Agent System, v.3 n.4, p.217-230, October 2005 Emerson Cabrera Paraiso , Jean-Paul A. Barths, An intelligent speech interface for personal assistants applied to knowledge management, Web Intelligence and Agent System, v.3 n.4, p.217-230, January 2005 Alexander Yates , Oren Etzioni , Daniel Weld, A reliable natural language interface to household appliances, Proceedings of the 8th international conference on Intelligent user interfaces, January 12-15, 2003, Miami, Florida, USA M. Turunen , J. Hakulinen , K.-J. Rih , E.-P. Salonen , A. Kainulainen , P. Prusi, An architecture and applications for speech-based accessibility systems, IBM Systems Journal, v.44 n.3, p.485-504, August 2005 Dawn N. Jutla , Dimitri Kanevsky, Adding User-Level SPACe: Security, Privacy, and Context to Intelligent Multimedia Information Architectures, Proceedings of the 2006 IEEE/WIC/ACM international conference on Web Intelligence and Intelligent Agent Technology, p.77-84, December 18-22, 2006 Jamal Bentahar , Karim Bouzoubaa , Bernard Moulin, A Computational Framework for Human/Agent Communication Using Argumentation, Implicit Information, and Social Influence, Proceedings of the 2006 IEEE/WIC/ACM international conference on Web Intelligence and Intelligent Agent Technology, p.372-377, December 18-22, 2006 James Allen , George Ferguson , Nate Blaylock , Donna Byron , Nathanael Chambers , Myroslava Dzikovska , Lucian Galescu , Mary Swift, Chester: towards a personal medication advisor, Journal of Biomedical Informatics, v.39 n.5, p.500-513, October 2006 Martin Beveridge , John Fox, Automatic generation of spoken dialogue from medical plans and ontologies, Journal of Biomedical Informatics, v.39 n.5, p.482-499, October 2006

architectures for intelligent;cooperative;distributed;conversational systems;multimodal interfaces

360091

Separation of Transparent Layers using Focus.

Consider situations where the depth at each point in the scene is multi-valued, due to the presence of a virtual image semi-reflected by a transparent surface. The semi-reflected image is linearly superimposed on the image of an object that is behind the transparent surface. A novel approach is proposed for the separation of the superimposed layers. Focusing on either of the layers yields initial separation, but crosstalk remains. The separation is enhanced by mutual blurring of the perturbing components in the images. However, this blurring requires the estimation of the defocus blur kernels. We thus propose a method for self calibration of the blur kernels, given the raw images. The kernels are sought to minimize the mutual information of the recovered layers. Autofocusing and depth estimation in the presence of semi-reflections are also considered. Experimental results are presented.

Introduction The situation in which several (typically two) linearly superimposed contributions exist is often encountered in real-world scenes. For example [12, 20], looking out of a car (or room) window, we see both the outside world (termed real object [35, 36, 41, 42, 43, 45]), and a semi-reflection of the objects inside, termed virtual objects. The treatment of such cases is important, since the combination of several unrelated images is likely to degrade the ability to analyze and understand them. The detection of the phenomenon is of importance itself, since it indicates the presence of a clear, transparent surface in front of the camera, at a distance closer than the imaged objects [35, 42, 45]. The term transparent layers has been used to describe situations in which a scene is semi- reflected from a transparent surface [6, 12, 58]. It means that the image is decomposed into depth ordered layers, each with an associated map describing its intensity (and, if applicable, its motion [58]). We adopt this terminology, but stress the fact that this work does not deal with imaging through an object with variable opacity. Approaches to recovering each of the layers by nulling the others relied mainly on triangulation methods like motion [6, 12, 13, 22, 36, 49], and stereo [7, 48]. Algorithms were developed to cope with multiple superimposed motion fields [6, 49] and ambiguities in the solutions were discovered [47, 60]. Another approach to the problem has been based on polarization cues [18, 20, 35, 41, 42, 43, 45]). However, that approach needs a polarizing filter to be operated with the camera, may be unstable when the angle of incidence is very low, and is di#cult to generalize to cases in which more than two layers exist. In recent years, range imaging relying on the limited depth of field (DOF) of lenses has been gaining popularity. An approach for depth estimation using a monocular system based on focus sensing [14, 16, 25, 31, 32, 33, 34, 52, 53, 61] is termed Depth from Focus (DFF) in the computer-vision literature. In that approach, the scene is imaged with di#erent focus settings (e.g., by axially moving the sensor, the object or the lens), thus obtaining image slices of the scene. In each slice, a limited range of depth is in focus. Depth is extracted by a search for the slice that maximizes some focus criterion [21, 25, 31, 32, 34, 52, 55, 62] (usually related to the two dimensional intensity variations in the region), and corresponds to the plane of best focus. DFF and image-based rendering based on focused slices has usually been performed on opaque (and occluding) layers. In particular, just recently a method has been presented for generating arbitrarily focused images and other special e#ects performed separately on each occluding layer [5]. Physical modeling of DOF as applied to processing images of transparent objects has long been considered in the field of microscopy [2, 3, 8, 10, 15, 17, 19, 23, 29, 30, 37, 51], where the defocus e#ect is most pronounced. An algorithm for DFF was demonstrated [23] on a layered microscopic object, but due to the very small depth of field used, the interfering layer was very blurred so no reconstruction process was necessary. Note that microscopic specimens usually contain detail in a continuum of depth, and there is correlation between adjacent layers, so their crosstalk is not as disturbing as in semi-reflections. Fundamental consequences of the imaging operation (e.g. the loss of biconic regions in the three dimensional frequency domain) that pose limits on the reconstruction ability, and the relation to tomography, were discovered [9, 29, 51, 50, 54]. Some of the three dimensional reconstruction methods used in microscopy [2, 3, 10] may be applicable to the case of discrete layers as well. We study the possibility of exploiting the limited depth of field to detect, separate and recover the intensity distribution of transparent, multi-valued layers. Focusing yields an initial separation, but crosstalk remains. The layers are separated based on the focused images, or by changing the lens aperture. The crosstalk is attenuated by mutual blurring of the disturbing components in the images (Section 2). Proper blurring requires the point spread functions (PSF) in the images to be well estimated. A wrong PSF will leave each recovered layer contaminated by its complementary. We therefore study the e#ect of error in the PSFs. Then, we propose a method for estimating the PSFs from the raw images (Section 3). It is based on seeking the minimum of the mutual information between the recovered layers. Recovery experiments are described in Section 4. We also discuss the implication of semi-reflections on the focusing process and the depth extracted from it (Section 5). Preliminary and partial results were presented in [40, 44]. 2 Recovery from focused slices 2.1 Using two focused slices Consider a two-layered scene. Suppose that either manually or by some automatic procedure (see Section 5), we acquire two images, such that in each image one of the layers is in focus. Assume for the moment that we also have an estimate of the blur kernel operating on each layer, when the camera is focused on the other one. This assumption may be satisfied if the imaging system is of our design, or by calibration. Due to the change of focus settings, the images may undergo a scale change. If a telecentric imaging system (Fig. 1) is used, this problem is avoided [33, 59]. Otherwise, we assume that the scale change 1 is corrected during preprocessing [27]. Let layer f 1 be superimposed 2 on layer f 2 . We consider only the slices g a and g b , in which either layer f 1 or layer f 2 , respectively, is in focus. The other layer is blurred. Modeling the blur as convolution with blur kernels, (The assumption of a space-invariant response to constant depth objects is very common in analysis of defocused images, and is approximately true for paraxial systems or in systems corrected for aberrations). If a telecentric system is used, h 1 The depth dependence of the scale change can typically be neglected. 2 The superposition is linear, since the real/virtual layers are the images of the objects multiplied by the transmission/reflection coe#cients of the semi-reflecting surface, and these coe#cients do not depend on the light intensities. The physical processes in transparent/semi-reflected scenes are described in Refs. [41, 42, 43, 45]. Nonlinear transmission and reflection e#ects (as appear in photorefractive crystals) are negligible at intensities and materials typical to imaging applications. d F Figure 1: A telecentric imaging system [59]. An aperture D is situated at distance F (the focal length) in front of the lens. An object point at distance u is at best focus if the sensor is at v. If the sensor is at - v, the image of the point is a blurred spot parameterized by its e#ective diameter d. In the frequency domain Eqs. (1) take the form Assuming that the kernels are symmetric, ImH so the real components of G a and G b are respectively ReG with similar expressions for the imaginary components of the images. These equations can be visualized as two pairs of straight lines (see Fig. 2). The solution, which corresponds to the line intersection, uniquely exists for H 2a H 1b #= 1. Since the imaging system cannot amplify any component (H 1b , H 2a # 1), a unique intersection exists unless H To gain insight, consider a telecentric system (the generalization is straightforward). In this case, H H, and the slopes of the lines in Fig. 2 (representing the constraints) are reciprocal to each other. As H # 1 the slopes of the two lines become similar, hence the solution is more sensitive to noise in G a and G b . As the frequency decreases, H # 1, hence at frequency Transversal domain+H F G G a F FG a G a F Figure 2: Visualization of the constraints on reconstruction from focused slices and the convergence of a suggested algorithm. For each frequency, the relations (3) between the real components of G a , G b , F 1 and F 2 take the form of two straight lines. The visualization of the imaginary parts is similar. low frequencies the recovery is ill conditioned. Due to energy conservation, the average gray level (DC) is not a#ected by defocusing. Thus, at DC, H = 1. In the noiseless case the constraints on the DC component coincide into a single line, implying an infinite number of solutions. In the presence of noise the lines become parallel and there is no solution. The recovery of the DC component is thus ill posed. This phenomenon is also seen in the three dimensional frequency domain. The image space is band limited by a missing cone of frequencies [9, 29], whose axis is in the axial frequency direction # v and its apex is at the origin. Recovery of the average intensity in each individual layer is impossible since the information about inter-layer variations of the average transversal intensity is in the missing cone [46]. A similar conclusion may be derived from observing the three dimensional frequency domain support that relies on di#raction limited optics [54]. In order to obtain another point of view on these di#culties, consider the naive inverse filtering approach to the problem given by Eq. (2). In the transversal spatial frequency domain, the reconstruction is where As hence the solution is instable. Note, however, that the problem is well posed and stable at the high frequencies. Since H is a LPF, then B # 1 at high frequencies. As seen in Eqs. (4), the high frequency contents of the slice in which a layer is in focus are retained, while those of the other slice are diminished. Even if high frequency noise is added during image acquisition, it is amplified only slightly in the reconstruction. This behavior is quite opposite to typical reconstruction problems, in which instability and noise amplification appear in the high frequencies. Iterative solutions have been suggested to similar inversion problems in microscopy [2, 3, 15] and in other fields. A similar approach was used in [5] to generate special e#ects on occluding layers, when the inverse filtering needed special care in the low frequency components. The method that we consider can be visualized as progression along vectors in alternating directions parallel to the axes in Fig. 2. It converges to the solution from any initial hypothesis for |H| < 1. As |H| decreases (roughly speaking, as the frequency increases), the constraint lines approach orthogonality, thus convergence is faster. A single iteration is described in Fig. 3. This is a version of the Van-Cittert restoration algorithm [24]. With slices g a and g b as the initial hypotheses for respectively, at the l'th iteration SchechnerIJCVfig3.ps Figure 3: A step in the iterative process. Initial hypotheses for the layers serve as input images for a processing step, based on Eq. (1). The new estimates are fed back as input for the next iteration. for odd l, where . (7) B(m) has a major e#ect on the amplification of noise added to the raw images g a and g b (with the noise of the unfocused slice attenuated by H). Again, we see that at high frequencies the amplification of additive noise approaches 1. As the frequency decreases, noise amplification increases. The additive DC error increases linearly with m. Let us define the basic solution as that result of using indicates that we can do the recovery directly, without iterations, by calculating the kernel (filter) beforehand. m is a parameter that controls how close the filter B(m) is to the inverse filter, and is analogous to regularization parameters in typical inversion methods. In the spatial domain, Eq. (7) turns into a convolution kernel | {z } once | {z } # h 1b # h 2a | {z } | {z } twice | {z } #h 1b # h 2a # h 1b # h 2a | {z } | {z } times The spatial support of - b m is approximately 2dm pixels wide, where d is the blur diameter (assuming for a moment that both kernels have a similar support). Here, the finite support of the image has to be taken into account. The larger m is, the larger the disturbing e#ect of the image boundaries. The unknown surroundings a#ect larger portions of the image. It is therefore preferable to limit m even in the absence of noise. This di#culty seems to indicate at a basic limit to the ability to recover the layers. If the blur diameter d is very large, only a small m can be used, and the initial layer estimation achieved only by focusing cannot be improved much. In this case the initial slices already show a good separation of the individual layers, since in each of the two slices, one layer is very blurred and thus hardly disturbs the other one. On the other hand, if d is small, then in each slice one layer is focused, while the other is nearly focused - creating confusing images. But then, we are able to enhance the recovery using a larger m with only a small e#ect of the image boundaries. Using a larger m leads, however, to noise amplification and to greater sensitivity to errors in the assumed PSF (see subsection 2.4). Example A simulated scene consists of the image of Lena, as the close object, seen reflected through a window out of which Mt. Shuksan 3 is seen. The original layers appear in the top of Fig. 4. While any of the layers is focused, the other is blurred by a Gaussian kernel with standard deviation (STD) of 2.5 pixels. The slices in which each of the layers is focused are shown in the second row of Fig. 4 (all the images in this work are presented contrast-stretched). During reconstruction, "mirror" [4] extrapolation was used for the surroundings of the image in order to reduce the e#ect of the boundaries. The basic solution removes the crosstalk between the images, but it lacks contrast due to the attenuation of the low frequencies. Using which is equivalent to 13 iterations, improves the balance between the low frequency components to the high ones. With larger m's the results are similar. 3 Courtesy of Bonnie Lorimer SchechnerIJCVfig4.ps solution Basic Originals Far layer Close layer solution Enhanced slices Focused Figure 4: Simulation results. In the focused slices one of the original layers is focused while the other is defocus blurred. The basic solution with the correct kernel removes the crosstalk, but the low frequency content of the images is too low. Approximating the inverse filter with 6 terms amplifies the low frequency components. 2.2 Similarity to motion-based separation In separating transparent layers, the fact that the high frequencies can be easily recovered, while the low ones are noisy or lost, is not unique to this approach. It also appears in results obtained using motion. Note that, like focus changes, motion leaves the DC component unvaried. In [6], the results of motion-based recovery of semi-reflected scenes are clearly highpass filtered versions of the superimposing components. An algorithm presented in [22] was demonstrated in a setup similar to [6]. In [22], one of the objects is "dominant". It can easily be seen there that even as the dominant object is faded out in the recovery, considerable low-frequency contamination remains. Shizawa and Mase [49] have shown that, in regions of translational motion, the spatiotemporal energy of each layer resides in a plane, which passes through the origin in the spatiotemporal frequency domain. This idea was used [12, 13] to generate "nulling" filters to eliminate the contribution of layers, thus isolating a single one. However, any two of these frequency planes have a common frequency "line" passing through the origin (the DC), whose components are thus generally inseparable. These similarities are examples of the unification of triangulation and DOF approaches discussed in [38]. In general, Ref. [38] shows that the depth from focus or defocus approaches are manifestations of the geometric triangulation principle. For example, it was shown that for the same system dimensions, the depth sensitivity of stereo, motion blur and defocus blur systems are basically the same. Along these lines, the similarity of the inherent instabilities of separation based on motion and focus is not surprising. 2.3 Using a focused slice and a pinhole image Another approach to layer separation is based on using as input a pinhole image and a focused slice, rather than two focused slices. Acquiring one image via a very small aperture ("pinhole camera") leads to a simpler algorithm, since just a single slice with one of the layers in focus is needed. The advantage is that the two images are taken without changing the axial positions of the system components, hence no geometric distortions arise. Acquisition of such images is practically impossible in microscopy (due to the significant di#raction e#ects associated with small objects) but is possible in systems inspecting macroscopic objects. The "pinhole" image is described by where 1/a is the attenuation of the intensity due to contraction of the aperture. This image is used in conjunction with one of the focused slices of Eq. (1), for example g a . The inverse filtering solution is where As in subsection 2.1, S can be approximated by 2a . (12) 2.4 E#ect of error in the PSF The algorithm suggested in subsection 2.1 computes B(m)[G a -G b H 2a ]. We normally assume (Eq. (2)) that G a . If the assumption holds, B(m) . (13) Note that, regardless of the precise form of the PSFs, had the imaging PSFs and the PSFs used in the recovery been equal, the reconstruction would have converged to F 1 as m # when |H 1b |, |H 2a | < 1. In practice, the imaging PSFs are slightly di#erent, i.e., G a f f f are some functions of the spatial frequency. This di#erence may be due to inaccurate prior modeling of the imaging PSFs or due to errors in depth estimation. The reconstruction process leads to e similar relation is obtained for the other layer. An error in the PSF leads to contamination of the recovered layer by its complementary. The larger is, the stronger is the amplification of this disturbance. Note that - monotonically increases with m, within the support of the blur transfer function if H 1b H 2a > 0, as is the case when the recovery PSF's are Gaussians. Note that usually in the low frequencies (which is the regime of the crosstalk) H 1b , H 2a > 0. Thus, we may expect that the best sense of separation will be achieved using a small m, actually, one iteration should provide the least contamination. This is so although the uncontaminated solution obeys - increases. In other words, decreasing the reconstruction error does not necessarily lead to less crosstalk. Both H and f H (of any layer) are low-pass filters that conserve the average value of the images. Hence, E # 0 at the very low and at the very high frequencies, i.e., E is a bandpass filter. However, B(m) amplifies the low frequencies. At the low frequencies, their combined e#ect may have a finite or infinite limit as m #, depending on the PSF models used. Continuing with the example shown in Fig. 4, where the imaging PSF had an STD of 2.5 pixels, the e#ects of using a wrong PSF in the reconstruction are demonstrated in Fig. 5. When the PSF used in the reconstruction has STD of 1.25 pixels, negative traces remain (i.e., brighter areas in one image appear as darker areas in the other). When the PSF used in the reconstruction has STD of 5 pixels, positive traces remain (i.e., brighter areas in one image appear brighter in the other). The contamination is slight in the basic solution but is more noticeable with larger m's, that is, when - B. So, the separation seems worse, even though each of the images has a better balance (due to the enhancement of the low frequencies). SchechnerIJCVfig5.ps r =1.25 r =5 r =5 Far layer Close layer r =1.25 Figure 5: Simulated images when using the wrong PSF in the reconstruction. The original blur kernel had a STD of r = 2.5 pixels. Crosstalk between the recovered layers is seen clearly if the STD of the kernels used is 1.5 or 5 pixels. The contamination increases with m. We can perform the same analysis for the method described in subsection 2.3. Now there is only one filter involved, H 2a , since the layer f 1 is focused. Suppose that, in addition to using H 2a in the reconstruction rather than the true imaging transfer function f H 2a , we inaccurately use the scalar a rather than the true value - a used in the imaging process. Let e denote the relative error in this parameter, e = (a - a)/-a. We obtain that e e where here are the results had the imaging defocus kernel been the same as the one used in the reconstruction and had a. Note the importance of the estimation of e defocused layer) is recovered uncontaminated by F 1 . However, even in this case e focused layer) will have a contamination of F 2 , amplified by 3 Seeking the blur kernels The recovery methods outlined in Section 2 are based on the use of known, or estimated blur kernels. If the imaging system is of our design, or if it is calibrated, and in addition we have depth estimates of the layers obtained during the focusing process (e.g., as will be described in Section 5), we may know the kernels a-priori. Generally, however, the kernels are unknown. Even a-priori knowledge is sometimes inaccurate. We thus wish to achieve self-calibration, i.e., to estimate the kernels out of the images themselves. This will enable blind separation and restoration of the layers. To do that, we need a criterion for layer separation. Note that the method for estimating the blur kernels based on minimizing the fitting error in di#erent layers as in [5] may fail in this case as the layers are transparent and there is no unique blur kernel at each point. Moreover, the fitting error is not a criterion for separation. Assume that the statistical dependence of the real and virtual layers is small (even zero). This is reasonable since they usually originate from unrelated scenes. The Kullback-Leibler distance measures how far the images are from statistical independence, indicating their mutual information [11]. Let the probabilities for certain values - In practice these probabilities are estimated by the histograms of the recovered images. The joint probability is is in practice estimated by the joint histogram of the images, that is, the relative number of pixels in which - f 1 has a certain value - f 2 has a certain value - f 2 at corresponding pixels. The mutual information is then . (18) In this approach we assume that if the layers are correctly separated, each of their estimates contains minimum information about the other. Mutual information was suggested and used as a criterion for alignment in [56, 57], where its maximum was sought. We use this measure to look for the highest discrepancy between images, thus minimizing it. The distance (Eq. 18) depends on the quantization of - , and on their dynamic range, which in turn depends on the brightness of the individual layers f 1 and f 2 . To decrease the dependence on these parameters, we performed two normalizations. First, each estimated layer was contrast-stretched to a standard dynamic range. Then, I was normalized by the mean entropy of the estimated layers, when treated as individual images. The self information [11] (entropy) of - f 1 is and the expression for - f 2 is similar. The measure we used is I indicating the ratio of mutual information to the self information of a layer. The recovered layers depend on the kernels used. Therefore, the problem of seeking the kernels can be stated as a minimization problem: According to subsection 2.4, errors in the kernels lead to crosstalk (contamination) of the estimated layers, which is expected to increase their mutual information. There are generally many degrees of freedom in the form of the kernels. On the other hand, the kernels are constrained: they are non-negative, they conserve energy etc. To simplify the problem, the kernels can be assumed to be Gaussians. Then, the kernels are parameterized only by their standard deviations (proportional to the blur radii). This limitation may lead to a solution that is suboptimal but easier to obtain. Another possible criterion for separation is decorrelation. Decorrelation was a necessary condition for the recovery of semi-reflected layers by independent components analysis in [18], and by polarization analysis in [42, 43]. Note that requiring decorrelation between the estimated layers is based on the assumption that the original layers are decorrelated: that assumption is usually only an approximation. To illustrate the use of these criteria, we search for the optimal blur kernels to separate the images shown in the second row of Fig. 4. Here we simplified the calculations by restricting both kernels to be isotropic Gaussians of the same STD, as these were indeed the kernels used in the synthesis. Hence, the correlation and mutual information are functions of a single variable 4 . As seen in Fig. 6, using the correct kernel (with STD of 2.5 pixels) yields decorrelated basic solutions 1), with minimal mutual information (I n is plotted). The positive correlation for larger values of assumed STD, and the negative correlation for smaller values, is consistent with the visual appearance of positive and negative traces in Fig. 5. Observe that, as expected from the theory, in Fig. 5 the crosstalk was stronger for larger m. Indeed, in Fig. 6 the absolute correlation and mutual information are greater for when the wrong kernel is used. In a di#erent simulation, the focused slices corresponding to the original layers shown in the top of Fig. 4 were created using an exponential imaging kernel rather than a Gaussian, but the 4 The STD was sampled on a grid in our demonstrations. A practical implementation will preferably use e#cient search algorithms [28] to optimize the mutual information [56, 57]. SchechnerIJCVfig6.ps normalized information -0.4 -0.20.20.6correlation 2.5 4 -0.2 -0.4 m0.60.2 Mutual information Figure At the assumed kernel STD of 2.5 pixels the basic solutions are decorrelated and have minimal mutual information (shown normalized), in consistency with the true STD. [Dashed] The absolute correlation and the mutual information are larger for a large value of m. STD was still 2.5. The recovery was done with Gaussian kernels. The correlation and mutual information curves (as a function of the assumed STD) were similar to those seen in Fig. 6. The minimal mutual information was however at STD of r = 2.2 pixels. There was no visible crosstalk in the resulting images. The blurring along the sensor raster rows may be di#erent than the blurring along the columns. This is because blurring is caused not only by the optical processes, but also from interpixel crosstalk in the sensors, and the raster reading process in the CCD. Moreover, the inter-pixel spacing along the sensor rows is generally di#erent than along the columns, thus even the optical blur may a#ect them di#erently. We assigned a di#erent blur "radius" to each axis: r row and r column . When two slices are used, as in subsection. 2.1, there are two kernels, with a total of four parameters. Defining the parameter vector p # (r row the estimated vector - p is When a single focused slice is used in conjunction with a "pinhole" image, as described in subsection 2.3, the problem is much simpler. There are three parameters to determine: r row 2a and a. The parameter a is easier to obtain as it indicates the ratio of the light energy in the wide-aperture image relative to the pinhole image. Ideally, it is the square of the reciprocal of the ratio of the f-numbers of the camera, in the two states. If, however, the optical system is not calibrated, or if there is automatic gain control in the sensor, this ratio is not an adequate estimator of a. a can then be estimated by the ratio of the average values of the images, for example. Such an approximation may serve as a starting point for better estimates. When using the decorrelation criterion in the multi-parameter case, there may be numerous parameter combinations that lead to decorrelation, but will not all lead to the minimum mutual information, or to good separation. If p is N-dimensional, the zero-correlation constraint defines a dimensional hypersurface in the parameter space. It is possible to use this criterion to obtain initial estimates of p, and search for minimal mutual information within a lower dimensional manifold. For example, for each combination of r row and r column , a that leads to decorrelation can be found (near the rough estimate based on intensity ratios). Then the search for minimum mutual information can be limited to a subspace of only two parameters. 4 Recovery experiments 4.1 Recovery from two focused slices A print of the "Portrait of Doctor Gachet" (by van-Gogh) was positioned closely behind a glass window. The window partly reflected a more distant picture, a part of a print of the "Parasol" (by Goya). The f# was 5.6. The two focused slices 5 are shown at the top of Fig. 7. The cross correlation between the raw (focused) images is 0.98. The normalized mutual information is I n # 0.5 indicating that significant separation is achieved by the focusing process, but that substantial crosstalk remains. The optimal parameter vector - p in the sense of minimum mutual information is [1.9, 1.5, 1.5, 1.9] pixels, where r 1b corresponds to the blur of the close layer, and r 2a corresponds to the blur of the far layer. With these parameters, the basic solution shown at the middle row of Fig. 7 has I n # 0.006 (two orders of magnitude better than the raw images). Using better balance between the low and high frequency components, but I n increased to about 0.02. We believe that this is due to the error in the PSF model, as discussed above. In another example, a print of the "Portrait of Armand Roulin" (by van-Gogh) was positioned closely behind a glass window. The window partly reflected a more distant picture, a print of a part of the "Miracle of San Antonio" (by Goya). As seen in Fig. 8, the "Portrait" is hardly visible in the raw images. The cross correlation between the raw (focused) images is 0.99, and the normalized mutual information is I n # 0.6. The optimal parameter vector - 5 The system was not telecentric, so there was slight magnification with change of focus settings. This was compensated for manually by resizing one of the images. solution Basic Close layer Far layer slices Focused Figure 7: [Top] The slices in which either of the transparent layers is focused. [Middle row] The basic solution 1). [Bottom row]: Recovery with Close layer Far layer Focused slices Basic solution Figure 8: [Top] The slices in which either of the transparent layers is focused. [Middle row] The basic solution. [Bottom row]: Recovery with here is [1.7, 2.4, 1.9, 2.1] pixels. With these parameters I n # 0.004 at the basic solution, rising to about In a third example, the scene consisted of a distant "vase" picture that was partly-reflected from the glass-cover of a closer "crab" picture. The imaging system was telecentric [33, 59], so no magnification corrections were needed. The focused slices and the recovered layers are shown in Fig. 9. For the focused slices I n # 0.4, and the cross correlation is 0.95. The Far layer Close layer Figure 9: [Top] The slices in which either of the transparent layers is focused. [Bottom] The basic solution for the recovery of the "crab" (left) and "vase" (right) layers. optimal parameter vector - p in the sense of minimum mutual information is [4,4,11,1] pixels. The basic recovery, using - B(1), are shown in the bottom of Fig. 9. The crosstalk is significantly reduced. The mutual information I n and correlation decreased dramatically to 0.009 and 0.01, respectively. 4.2 Recovery from a focused slice and a pinhole image The scene consisted of a print of the "Portrait of Armand Roulin" as the close layer and a print of a part of the "Miracle of San Antonio" as the far layer. The imaging system was not telecentric, leading to magnification changes during focusing. Thus, in such a system it may be preferable to use a fixed focus setting, and change the aperture between image acquisitions. The "pinhole" image was acquired using the state corresponding to the mark on the lens, layer focused layer defocused layer defocused layer focused Figure 10: [Top left] The slice in far layer is focused, when viewed with the wide aperture. [Top right] The "pinhole" image. [Middle row]: The basic recovery. [Bottom row]: Recovery with while the wide aperture image was acquired using the state corresponding to the We stress that we have not calibrated the lens, so these marks do not necessarily correspond to the true values. The slice in which the far layer is focused (using the wide aperture) is shown in the top left of Fig. 10. In the "pinhole" image (top right), the presence of the "Portrait" layer is more noticeable. According to the ratio of the f#'s, the wide aperture image should have been brighter than the "pinhole" image by (11/4) 2 # 7.6. However, the ratio between the mean intensity of the wide aperture image to that of the pinhole image was 4.17, not 7.6. This could be due to poor calibration of the lens by its manufacturer, or because of some automatic gain control in the sensor. We added a to the set of parameters to be searched in the optimization process. In order to get additional cues for a, we calculated ratios of other statistical measures: the ratios of the STD, median, and mean absolute deviation were 4.07, 4.35 and 4.22, respectively. We thus let a assume values between 4.07 and 4.95. In this example we demonstrate the possibility of using decorrelation to limit the minimum mutual information search. First, for each hypothesized pair of blur diameters, the parameter a that led to decorrelation of the basic solution was sought. Then, the mutual information was calculated over the parameters that cause decorrelation. The blur diameters that led to minimal mutual information at pixels, with the best parameter a being 4.28. The reconstruction results are shown in the middle row of Fig. 10. Their mutual information (normalized) is 0.004. Using a larger m with these parameters increased the mutual information, so we looked for a better estimate, minimizing the mutual information after the application of - B(m). For the resulting parameters were di#erent: r row = r pixels, with a = 4.24. The recovered layers are shown in the bottom row of Fig. 10. Their mutual information (normalized) is 0.04. As discussed before, the increase is probably due to inaccurate modeling of the blur kernel. 5 Obtaining the focused slices 5.1 Using a standard focusing technique We have so far assumed that the focused slices are known. We now consider their acquisition using focusing as in Depth from Focus (DFF) algorithms. Depth is sampled by changing the focus settings, particularly the sensor plane. According to Refs. [1, 26, 38, 39], the sampling should be at depth of field intervals, for which d #x, where #x is the inter-pixel period (similar to stereo [38]). An imaging system telecentric on the image side [33, 59] is a preferred configuration, since it ensures constant magnification as the sensor is put out of focus. For such a system it is easy to show that the geometrical-optics blur-kernel diameter is where D is the aperture width, F is the focal length (see Fig. 1), and #v is the distance of the sensor plane from the plane of best focus. The axial sampling period is therefore #v # F#x/D. The sampling period requirement can also be analyzed in the frequency domain, as in [54]. Focus calculations are applied to the image slices acquired. The basic requirement from the focus criterion is that it will reach a maximum when the slice is in focus. Most criteria suggested in the literature [23, 25, 32, 34, 52, 55, 62] are sensitive to two dimensional variations in the slice 6 . Local focus operators yield "slices of local focus-measure", FOCUS (x, y, - v), where - v is the axial position of the sensor (see Fig. 1). If we want to find the depth at a certain region (patch) [31], and the scene is composed of a single layer, we can average FOCUS (x, y, - v) over the region, to obtain FOCUS (-v) from which a single valued depth can be estimated. This approach is inadequate in the presence of multiple layers. Ideally, each of them alone would lead to a main peak 7 in FOCUS (-v). But, due to mutual interference, the peaks can move from their original positions, or even merge into a single peak in some "average" position, thus spoiling focus detection. This phenomenon can be observed in experimental results. The scene, the focused slices of which are shown in Fig. 9, had the "crab"and the "vase" objects at distances of 2.8m and 5.3m from the lens, respectively. The details of the experimental imaging system are described in [40]. Depth variations within these objects were negligible with respect to the depth of field. 6 It is interesting to note that a mathematical proof exists [21] for the validity of a focus criterion that is completely based on local calculations which do not depend on transversal neighbors: As a function of axial position, the intensity at each transversal point has an extremum at the plane of best focus. 7 There are secondary maxima, though, due to the unmonotonicity of the frequency response of the blur operator, and due to edge bleeding. However, the misleading maxima are usually much smaller than the maximum associated with the focusing on feature-dense regions, as edges. Extension of the STD of the PSF by about 0.5 pixels was accomplished by moving the sensor array 0.338mm from the plane of best focus 8 . This extended the e#ective width of the kernel by about 1 pixel (#d # 1pixel), and was also consistent with our subjective sensation of DOF. The results of the focus search, shown by the dashed-dotted line in Fig. 11, indicate that the focus measure failed to detect the layers, as it yielded a single (merged) peak, somewhere between the focused states of the individual layers. This demonstrates the confusion of conventional autofocusing devices when applied to transparent scenes. 5.2 A voting scheme Towards solving the merging problem, observe that the layers are generally unrelated and that edges are usually sparse. Thus, the positions of brightness edges in the two layers will only sporadically coincide. Since edges (and other feature-dense regions) are dominant contributors to the focus criterion, it would be wise not to mix them by brute averaging of the local focus measurements over the entire region. If point (x, y) is on an edge in one layer, but on a smooth region in the other layer, then the peak in FOCUS (x, y, - v) corresponding to the edge will not be greatly a#ected by the contribution of the other layer. The following approach is proposed. For each pixel (x, y) in the slices, the focus measure FOCUS (x, y, - v) is analyzed as a function of - v, to find its local maxima. The result is expressed as a binary vector of local maxima positions. Then, a vote table analogous to a histogram of maxima locations over all pixels is formed by summing all the "hits" in each slice-index. Each vote is given a weight that depends on the corresponding value of FOCUS (x, y, - v), to enhance the contribution of high focus-measure values, such as those arising from edges, while reducing the random contribution of featureless areas. The results of the voting method are shown as a solid line in Fig. 11, and demonstrate its success in creating significant, separate peaks 8 Near the plane of best focus, the measured rate of increase of the STD as a function of defocus was much lower than expected from geometric considerations. We believe that this is due to noticeable di#raction and spherical aberration e#ects in that regime. SchechnerIJCVfig11.ps Slice index votes Weighted Traditional measure focus Figure 11: Experimental results. [Dashed-dotted line]: The conventional focus measure as a function of the slice index. It mistakenly detects a single focused state at the 6th slice. [Solid line]: The locations histogram of detected local maxima of the focus measure (the same scene). The highest numbers of votes (positions of local maxima) are correctly accumulated at the 4th and 7th slices - the true focused slices. corresponding to the focused layers. Additional details can be found in [40]. The estimated depths were correct, within the uncertainty imposed by the depth of field of the system. Optimal design and rigorous performance evaluation of DFF methods in the presence of transparencies remains an open research problem. 6 Conclusions This paper presents an approach based on focusing to separate transparent layers, as appear in semi-reflected scenes. This approach is more stable with respect to perturbations [38] and occlusions than separation methods that rely on stereo or motion. We also presented a method for self calibration of the defocus blur kernels given the raw images. It is based on minimizing the mutual information of the recovered layers. Note that defocus blur, motion blur, and stereo disparity have similar origins [38] and di#er mainly in the scale and shape of the kernels. Therefore, the method described here could possibly be adapted to finding the motion PSFs or stereo disparities in transparent scenes. In some cases the methods presented here are also applicable to multiplicative layers [49]: If the opacity variations within the close layer are small (a "weak" object), the transparency e#ect may be approximated as a linear superposition of the layers, as done in microscopy [2, 10, 29, 37]. In microscopy and in tomography, the suggested method for self calibration of the PSF can improve the removal of crosstalk between adjacent slices. In the analysis and experiments, depth variations within each layer have been neglected. This approximation holds as long as these depth variations are small with respect to the depth of field. Extending our analysis and recovery methods to deal with space-varying depth and blur is an interesting topic for future research. A simplified interim approach could be based on application of the filtering to small domains in which the depth variations are su#ciently small. Note that the mutual information recovery criterion can still be applied globally, leading to a higher-dimensional optimization problem. We believe that fundamental properties, such as the inability to recover the DC of each layer, will hold in the general case. Other obvious improvements in the performance of the approach can be achieved by incorporating e#cient search algorithms to solve the optimization problem [28], with e#cient ways to estimate the mutual information [56, 57]. Semi-reflections can also be separated using polarization cues [18, 41, 42, 43, 45]. It is interesting to note that polarization based recovery is typically sensitive to high frequency noise at low angles of incidence [45]. On the other hand, DC recovery is generally possible and there are no particular di#culties at the low frequencies. This nicely complements the characteristics of focus-based layer separation, where the recovery of the high frequencies is stable but problems arise in the low frequencies. Fusion of focus and polarization cues for separating semi-reflections is thus a promising research direction. The ability to separate transparent layers can be utilized to generate special e#ects. For example, in Ref. [5] images were rendered with each of the occluding (opaque) layers defocused, moved and enhanced arbitrarily. The same e#ects, and possibly other interesting ones can now be generated in scenes containing semireflections. Acknowledgments This work was conducted while Yoav Y. Schechner was at the Department of Electrical Engi- neering, Technion - Israel Institute of Technology, Haifa. We thank Joseph Shamir and Alex Bekker for their advice, support, and significant help, and for making the facilities of the Elec- trooptics Laboratory of the Electrical Engineering Department, Technion, available to us. We thank Bonnie Lorimer for the permission to use her photograph of Mt. Shuksan. This research was supported in part by the Eshkol Fellowship of the Israeli Ministry of Science, by the Ol- lendor# Center of the Department of Electrical Engineering, Technion, and by the Tel-Aviv University Research Fund. The vision group at the Weizmann Institute is supported in part by the Israeli Ministry of Science, Grant No. 8504. Ronen Basri is an incumbent of Arye Dissentshik Career Development Chair at the Weizmann Institute. --R Active stereo: integrating disparity Optical sectioning microscopy: Cellular architecture in three dimensions. Reduction of boundary artifacts in image restoration. Producing object-based special e#ects by fusing multiple di#erently focused images A three-frame algorithm for estimating two-component image motion Estimating multiple depths in semi-transparent stereo images Digital image processing. Three dimensional radiographic imaging with a restricted view angle. Enhanced 3-D reconstruction from confocal scanning microscope images Elements of information theory. Separation of transparent motion into layers using velocity-tuned mechanisms 'Nulling' filters and the separation of transparent mo- tions Pyramid based depth from focus. 3D representation of biostructures imaged with an optical microscope. Acquisition of 3-D data by focus sensing Reconstructing 3-D light-microscopic images by digital image processing Separating reflections from images by use of independent components analysis. Distribution of actinin in single isolated smooth muscle cells Simple focusing criterion. Computing occluding and transparent motions. Digitized optical microscopy with extended depth of field. Resolution enhancement of spectra. A perspective on range-finding techniques for computer vision Panoramic image acquisition. Registration and blur estimation methods for multiple di Linear and nonlinear programming The missing cone problem and low-pass distortion in optical serial sectioning microscopy Artifacts in computational optical-sectioning microscopy Robust focus ranging. Shape from focus system. Real time focus range sensor. Microscopic shape from focus using active illumination. A theory of specular surface geometry. Regularized linear method for reconstruction of three-dimensional microscopic objects from optical sections Depth from defocus vs. Stereo: How di The optimal axial interval in estimating depth from defocus. Separation of transparent layers using focus. Separation of transparent layers by polarization analysis. Vision through semireflecting media: Polarization analysis. Blind recovery of transparent and semireflected scenes. Polarization and statistical analysis of scenes containing a semireflector. Three dimensional optical transfer function for an annular lens. On visual ambiguities due to transparency in motion and stereo. Direct estimation of multiple disparities for transparent multiple surfaces in binocular stereo. Simultaneous multiple optical flow estimation. Fundamental restrictions for 3-D light distributions The optimal focus measure for passive autofocusing and depth from focus. Digital composition of images with increased depth of focus considering depth information. Are textureless scenes recoverable? Defocus detection using a visibility criterion. III, <Year>1997</Year>. Alignment by maximization of mutual information. Layered representation for motion analysis. "Telecentric optics for computational vision" Perception of multiple transparent planes in stereo vision. Depth from focusing and defocusing. Jayasooriah and Sinniah --TR --CTR Thanda Oo , Hiroshi Kawasaki , Yutaka Ohsawa , Katsushi Ikeuchi, The separation of reflected and transparent layers from real-world image sequence, Machine Vision and Applications, v.18 n.1, p.17-24, January 2007 Javier Toro , Frank Owens , Rubn Medina, Using known motion fields for image separation in transparency, Pattern Recognition Letters, v.24 n.1-3, p.597-605, January D.-M. Tsai , C.-C. Chou, A fast focus measure for video display inspection, Machine Vision and Applications, v.14 n.3, p.192-196, July Amit Agrawal , Ramesh Raskar , Shree K. Nayar , Yuanzhen Li, Removing photography artifacts using gradient projection and flash-exposure sampling, ACM Transactions on Graphics (TOG), v.24 n.3, July 2005 Morgan McGuire , Wojciech Matusik , Hanspeter Pfister , John F. Hughes , Frdo Durand, Defocus video matting, ACM Transactions on Graphics (TOG), v.24 n.3, July 2005 Sarit Shwartz , Michael Zibulevsky , Yoav Y. Schechner, Fast kernel entropy estimation and optimization, Signal Processing, v.85 n.5, p.1045-1058, May 2005 Zhang , Shree Nayar, Projection defocus analysis for scene capture and image display, ACM Transactions on Graphics (TOG), v.25 n.3, July 2006 Marc Levoy , Ren Ng , Andrew Adams , Matthew Footer , Mark Horowitz, Light field microscopy, ACM Transactions on Graphics (TOG), v.25 n.3, July 2006

blur estimation;semireflections;enhancement;optical sectioning;blind deconvolution;depth from focus;inverse problems;signal separation;image reconstruction and recovery;decorrelation

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The effect of reconfigurable units in superscalar processors.

This paper describes OneChip, a third generation reconfigurable processor architecture that integrates a Reconfigurable Functional Unit (RFU) into a superscalar Reduced Instruction Set Computer (RISC) processor's pipeline. The architecture allows dynamic scheduling and dynamic reconfiguration. It also provides support for pre-loading configurations and for Least Recently Used (LRU) configuration management.To evaluate the performance of the OneChip architecture, several off-the-shelf software applications were compiled and executed on Sim-OneChip, an architecture simulator for OneChip that includes a software environment for programming the system. The architecture is compared to a similar one but without dynamic scheduling and without an RFU. OneChip achieves a performance improvement and shows a speedup range from 2.16 up to 32 for the different applications and data sizes used. The results show that dynamic scheduling helps performance the most on average, and that the RFU will always improve performance the best when most of the execution is in the RFU.

INTRODUCTION Recently, the idea of using reconfigurable resources along with a conventional processor has led to research in the area Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. FPGA 2001, February 11-13, 2001, Monterey, CA, USA. of reconfigurable computing. The main goal is to take advantage of the capabilities and features of both resources. While the processor takes care of all the general-purpose computation, the reconfigurable hardware acts as a specialized coprocessor that takes care of specialized applica- tions. With such platforms, specific properties of applica- tions, such as parallelism, regularity of computation, and data granularity can be exploited by creating custom oper- ators, pipelines, and interconnection pathways. There has been research done in the Department of Electrical and Computer Engineering at the University of Toronto on such reconfigurable processors, namely, the OneChip processor model has been developed. At first, this model tightly integrated reconfigurable logic resources and memory into a fixed-logic processor core. By using the re-configurable units of this architecture, the execution time of specialized applications was reduced. The model was mapped into the Transmogrifier-1 field-programmable sys- tem. This work was done by Ralph Wittig [22]. A follow-on model, called OneChip-98, then integrated a memory-consistent interface. It is a hardware implementation that allows the processor and the reconfigurable array to operate concurrently. It also provides a scheme for specifying reconfigurable instructions that are suitable for typical programming models. This model was partially mapped into the Transmogrifier-2 field-programmable system. This work was done by Je# Jacob [11]. OneChip's architecture has now been extended to a super-scalar processor that allows multiple instructions to issue simultaneously and perform out-of-order execution. This leads to much better performance, since the processor and the reconfigurable logic can execute several instructions in parallel. Most of the performance improvement that this architecture shows comes from memory streaming applica- tions, that is, those applications that read in a block of data from memory, perform some computation on it, and write it back to memory. Multimedia applications have this characteristic and are used to evaluate the architecture. Previous subsets of the OneChip architecture 1 have been modeled by implementing them in hardware. The purpose of this work is to properly determine the feasibility of the architecture by building a full software model capable of simulating the execution of real applications. We will use the term OneChip from now on to refer to the latest version of the OneChip architecture. 1.1 Related Work In general, a system that combines a general-purpose processor with reconfigurable logic is known as a Field-Programmable Custom Computing Machine (FCCM). Re-search on FCCMs done by other groups [2, 5, 7, 14, 16, 18, 19] has reported speedup obtained by combining these two techniques, however, most of the research in these groups is focused on aspects of the reconfigurable fabric and the compilation system. Much of the OneChip work is focused toward the interface between the two technologies. As a re- sult, the applications are modified by hand; no modification was done to the compiler; and our simulations model only the functionality and latency of the reconfigurable fabric, not the specifics of the fabric architecture. In our work, we study the e#ect of combining reconfigurability with an advanced technique to speedup processors, a superscalar pipeline, by focusing on the interplay between them. With the use of out-of-order issue and execution, one can further exploit instruction-level parallelism in ap- plications, without incurring the overheads involved in re-configuring a specialized hardware. Previously, performance reports by other groups were done using application kernels such as the DCT, FIR filters, or some small kernel-oriented applications. Only recently have some groups [2, 18, 23] reported on the performance using complete applications, which give more meaningful results. In this work, we are focused on the architecture's performance with full applications 2. ONECHIP ARCHITECTURE In this section we give a brief overview of the OneChip architecture, including the more recently added features. The processor's main features, as proposed in [22, 11] are: . MIPS-like RISC architecture - simple instruction encoding and pipelining. . Dynamic scheduling - allows out-of-order issue and completion. . Dynamic reconfiguration - can be reconfigured at run-time. . Reconfigurable Functional Unit (RFU) integration - programmable logic in the processor's pipeline. In addition, OneChip has now been extended to include: . a Superscalar pipeline - allows multiple instructions to issue per cycle. . Configuration pre-loading support - allows loading configurations ahead of time. . Configuration compression support - reduces configuration size. . LRU configuration management support - reduces number of reconfigurations. 2.1 Processor pipeline The original OneChip pipeline described in [11] is based on the DLX RISC processor described by Hennessy & Patterson [9]. It consists of five stages: Instruction Fetch (IF), Instruction Decode (ID), Execute (EX), Memory Access (MEM) RFU Figure 1: OneChip's Pipeline FPGA Instruction Buffer RFU-RS Memory Interface RBT Controller Storage Context Memory FPGA Storage Context Memory Figure 2: RFU Architecture and Writeback (WB). A diagram of the pipeline is shown in Figure 1. The RFU is integrated in parallel with the EX and MEM stages. It performs computations as the EX stage does and has direct access to memory as the MEM stage does. The RFU contains structures such as the Memory Interface, an Instruction Bu#er, a Reconfiguration Bits Table (RBT) and Reservation Stations. OneChip is now capable of executing multiple instructions in parallel. The EX stage consists of multiple functional units of di#erent types, such as integer units, floating point units and a reconfigurable unit. Due to the flexibility of the reconfigurable unit to implement a custom instruction, a programmer or a compiler can generate a configuration for the reconfigurable unit to be internally pipelined, parallelized or both. Dynamic scheduling of RFU instructions is implemented in OneChip. Data dependencies between RFU and CPU instructions are handled using RFU Reservation Stations. 2.2 RFU Architecture The RFU in OneChip contains one or more FPGAs and an FPGA Controller as shown in Figure 2. The FPGAs have multiple contexts and are capable of holding more than one configuration for the programmable logic [4]. These configurations are stored in the Context Memory, which makes the FPGA capable of rapidly switching among configura- tions. Each context of the FPGAs is configured independently from the others and acts as a cache for configurations. Only one context may be active at any given time. Instructions that target the RFU in OneChip are forwarded to the FPGA Controller, which contains the reservation stations and a Reconfiguration Bits Table (RBT). The FPGA Controller is responsible for programming the FPGAs, the context switching and selecting configurations to be replaced when necessary. The FPGA Controller also contains a bu#er for instructions and the memory interface. The RBT acts as the configuration manager that will keep track of where the FPGA configurations are located. The memory interface in the FPGA Controller consists of a DMA controller that is responsible for transferring configurations from memory into the context memory according to the values in the RBT. It also transfers the data that an FPGA will operate on into the local storage. The local storage may be considered as the FPGA data cache memory. The multiple FPGAs in the RFU share the same FPGA Controller and each FPGA has its own context memory and local storage. OneChip has been enhanced to support configuration compression and reduce the overhead involved in configuring the FPGA. An algorithm for compressing configurations is proposed by Hauck et al. [8]. This feature has not been modeled in our simulator for these results since the internal architecture of the FPGA fabric is not yet defined, therefore the actual size of the configuration bitstreams is unknown. Futhermore, our benchmarks only use one configuration and the e#ect of the overhead can be easily managed by pre-loading the configuration. The architecture has also been extended to support configuration management. Although the FPGAs can hold multiple configurations, there is a hardware limit on the number of configurations it can hold. OneChip uses the Least Recently Used (LRU) algorithm as a mechanism for swapping configurations in and out of the FPGA. LRU is implemented in OneChip by using a table of configuration reference bits. The approach is similar to the Additional-Reference-Bits Algorithm described by Silberschatz & Galvin in [20]. A fixed-width shift register is used to keep track of each loaded configuration's history. On every context switch, all shift registers are shifted 1 bit to the right. On the high-order bit of each register, a 0 is placed for all inactive configurations and a 1 for the active one. If the shift register contains 00000000, it means that it hasn't been used in a long time. If it contains 10101010, it means that it has been used every other context switch. A configuration with a history register value of 01010000 has been used more recently than another with the value of 00101010, and this later one was used more recently than one with a value of 00000100. Therefore, the configuration that should be selected for replacement is the one that has the smallest value in the history register. Notice that the overall behavior of these registers is to keep track of the location of configurations in a queue, where a recently used configuration will come to the front and the last one will be the one to be replaced. Our simulator does not have this feature implemented at this time as it was not required in the benchmarks. The Reconfiguration Bits Table (RBT) acts as the configuration manager that will keep track of where the FPGA configurations are located. The information in this table includes the address of each configuration and flags that keep track of loaded and active configurations. The RBT described in [11] has been enhanced to support the algorithm for configuration management [3]. The history of each configuration is also stored in the table to allow LRU configuration management and to select configurations to be replaced. Table 1: Memory Consistency Scheme Hazard Hazard Number Type Actions Taken 1 RFU rd 1. Flush RFU source addresses from CPU cache when after instruction issues. CPU wr 2. Prevent RFU reads while pending CPU store instructions are outstanding. 2 CPU rd 3. Invalidate RFU destination addresses in CPU cache after when RFU instruction issues. Prevent CPU reads from RFU destination addresses until RFU writes its destination block. 3 RFU wr 5. Prevent RFU writes while pending CPU load after instructions are outstanding. CPU rd 4 CPU wr 6. Prevent CPU writes to RFU source addresses until after RFU reads its source block. RFU rd 5 RFU wr 7. Prevent RFU writes while pending CPU store after instructions are outstanding. 6 CPU wr 8. Prevent CPU writes to RFU destination addresses until after RFU writes its destination block. 7 RFU rd 9. Prevent RFU reads from locked RFU destination after addresses. 8 RFU wr 10. Prevent RFU writes to locked RFU source addresses. after RFU rd 9 RFU wr 11. Prevent RFU writes to locked RFU destination after addresses 2.3 Instruction specification OneChip is designed to obtain speedup mainly from memory streaming applications in the same way vector coprocessors do. In general, RFU instructions take a block of data that is stored in memory, perform a custom operation on the data and store it back to memory. Previously, OneChip supported only a two-operand RFU instruction. To have more flexibility for a wider range of applications, it has now been extended to support a three- operand RFU instruction. In the two-operand instruction, one can specify the opcode, the FPGA function, one source and one destination register that hold the respective memory addresses, and the block sizes. In this instruction, the source and destination block sizes can be di#erent. In the three- operand instruction, one of the block sizes is replaced by another source register. This allows the RFU to get source data from two di#erent memory locations, which need not be continuous. In this instruction, all three blocks should be the same size. In OneChip, there are two configuration instructions. One of them is the Configure Address instruction, which is used for assigning memory addresses in the RBT. The other configuration instruction is the Pre-load instruction, which is used for pre-fetching instructions into the FPGA and reducing configuration overhead. Some compiler prefetching techniques have been previously published for other reconfigurable systems [6, 21]. 2.4 Memory controller OneChip allows superscalar dynamic scheduling, hence instructions with di#erent latencies may be executed in paral- lel. The RFU in OneChip has direct access to memory and is also allowed to execute in parallel with the CPU. When there are no data dependencies between the RFU and the CPU, the system will act as a multiprocessor system, providing speed up. However, when data dependencies exist between them, there is a potential for memory inconsistency that must be prevented. The memory consistency scheme previously proposed for OneChip, as described in [11], allows parallel execution between one FPGA and the CPU. The scheme has now been extended to support more than one FPGA in the RFU. The nine possible hazards that OneChip may experience along with the actions taken to prevent them, are listed in Table 1. This scheme preserves memory consistency when the CPU and an FPGA, or when two or more FPGAs, are allowed to execute concurrently. OneChip implements the memory consistency scheme by using a Block Lock Table (BLT). The BLT is a structure that contains four fields for each entry and locks memory blocks to prevent undesired accesses. The information stored in the table includes the block address, block size, instruction tag and a src/dst flag. 3. SIM-ONECHIP This section will describe the implementation of sim- onechip, the simulator that models the architecture of OneChip. It is a functional, execution-driven simulator derived from sim-outorder from the simplescalar tool set [1]. To model the behaviour of OneChip, we needed an already existing simulator capable of doing out-of-order execution and that was easily cutomizable to be used a basis to add OneChip's features. Two existing architecture simulators [1, were considered for modification and SimpleScalar was the chosen platform. Besides being a complete set of tools, the annotations capability was an attractive feature, since it would allow the addition of new instructions in a very simple manner. 3.1 Modifications to sim-outorder Modifications were done to sim-outorder to model OneChip's reservation stations, reconfiguration bits table, block lock table and the reconfigurable unit. The overall functionality of sim-outorder was preserved. The reservation stations for Sim-OneChip were implemented as a queue. Besides the already existing scheduler queues for Basic Functional Unit (BFU) instructions and for Memory (MEM) instructions, a third scheduler queue was implemented to hold RFU instructions. This queue is referred to as the Reconfigurable Instructions Queue (RecQ). The dispatch stage detects instructions that target the RFU and places them in the RecQ for future issuing. The RBT is implemented as a linked list. The RBT models the FGPA controller by performing dynamic reconfiguration and configuration management. Functions are provided for assigning configuration addresses, loading configurations and to perform context switching. The BLT is implemented as a linked list. Each entry holds the fields for the two sources and the destination memory blocks for each RFU instruction. It ensures the OneChip memory consistency scheme by modeling the actions taken for each of the hazards presented. By keeping track of the memory locations currently blocked, conflicting instructions are properly stalled. The RFU was included with the rest of the functional units and in the resource pool in the functional unit resource configuration. 3.2 Pipeline description To be able to adapt OneChip to the SimpleScalar ar- chitecture, several modifications were done to the original pipeline. Sim-OneChip's pipeline, as in sim-outorder, consists of six stages: fetch, dispatch, issue, execute, writeback and commit. This section will describe the modifications I-Cache D-Cache Fetch Dispatch Issue BFU Writeback Commit Mem Main Memory RFU Execute Figure 3: Sim-OneChip's Pipeline done to each stage in sim-outorder and the places where each of OneChip's structures were included. Sim-OneChip's pipeline is shown in Figure 3. The fetch stage remained unmodified and fetches instructions from the I-cache into the dispatch queue. The dispatch stage decodes instructions and performs register renaming. It moves instructions from the dispatch queue into the reservation stations in the three scheduler queues: the Register Update Unit (RUU), the Load Store Queue (LSQ) and the Reconfigurable Instructions Queue (RecQ). This stage adds entries in the BLT to lock memory blocks when RFU instructions are dispatched. The issue stage identifies ready instructions from the scheduler queues (RUU, LSQ and RecQ) and allows them to proceed in the pipeline. This stage also checks the BLT to keep memory consistency and stalls the corresponding instructions. The execute stage is where instructions are executed in the corresponding functional units. Completed instructions are scheduled on the event queue as writeback events. This stage is divided into three parallel stages: BFU stage, MEM stage and RFU stage. The BFU stage is where all operations that require basic functional units, such as integer and floating point are executed; the MEM stage is where all memory access operations are executed and has access to the D-cache, and; the RFU stage is where RFU instructions are executed. The writeback stage remained unmodified and moves completed operation results from the functional units to the RUU. Dependency chains of completing instructions are also scanned to wake up any dependent instructions. The commit stage retires instructions in-order and frees up the resources used by the instructions. It commits the results of completed instructions in the RUU to the register file and stores in the LSQ will commit their result data to the data cache. This stage clears BLT entries to remove memory locks once the corresponding RFU instruction is committed. The BLT is accessed by the dispatch, issue and commit stages. The memory consistency scheme requires that instructions are entered in the BLT and removed from it in program order. In the pipeline, the issue, execute and writeback stages do not necessarily follow program order since out-of order issue, execution and completion is allowed. Hence, memory block locks and the corresponding entries in the BLT need to be entered when an RFU instruction is dis- patched, since dispatching is done in program order. Like- sim-onechip simulator oc-lib.h fpga.conf ssgcc compiler source code (*.c) onechip binary (*.oc) gcc compiler sim-onechip source code simulation statistics Figure 4: Sim-OneChip's Simulation Process wise, entries from the BLT need to be removed when RFU instructions commit, since committing is also performed in program order. All actions in the memory consistency scheme are taken in the issue stage. The issue stage is allowed to probe the BLT for memory locks. Instructions that conflict with locked memory blocks are prevented from issuing at this point. All others are allowed to proceed provided there are no dependencies 3.3 RFU instructions Annotation of instructions in SimpleScalar are useful for creating new instructions. They are attached to the opcode in assembly files for the assembler to translate them and append them in the annotation field of assembled instructions. Taking advantage of this feature, new instructions can be created without the need to modify the assembler. OneChip's RFU instructions will be disguised as already existing, but annotated instructions that the simulator will recognize as an RFU instruction and model the corresponding operation. Without the annotation, instructions are treated as regular ones; with the annotation they become instructions that target the reconfigurable unit. The four instructions defined for OneChip (i.e. two RFU operation instructions and two configuration instructions) were created for Sim-OneChip. Macros are used to translate from a C specification to the corresponding annotated assembly instruction. 3.4 Programming model Currently, the programming model for OneChip is the use of circuit libraries. Programming for Sim-OneChip is done in C. The user may use existing configurations from a library of configurations, or create custom ones. Configurations are defined in C and several macros are available for accessing memory or instruction fields. The complete simulation process is shown in Figure 4. A C program that includes calls to RFU instructions is compiled by the simplescalar gcc compiler ssgcc along with the OneChip Library oc-lib.h. This will produce a binary file that can be executed by the simulator sim-onechip. All the program configurations specified in fpga.conf must be previously compiled by gcc along with the simulator source code to produce the simulator. Once both binaries are ready, the simulator can simulate the execution of the binary and produce the corresponding statistics. Sim-OneChip's processor specification can be defined as command-line arguments. One can specify the processor core parameters, such as fetch and decode bandwidth, internal queues sizes and number of execution units. The memory hierarchy and the branch predictor can also be modified. 3.4.1 OneChip library The library defines the following five macros: oc configAddress(func, addr) is used for specifying the configuration address for a specified function. It will associate the function func with the address addr where the FPGA configuration bits will be taken from and will enter the corresponding entry in the BLT. oc preLoad(func) is used for pre-loading the configuration associated with the specified function func into an available FPGA context. rec 2addr(func, src addr, dst addr, src size, dst size) is the two-operand reconfigurable instruction. func is the FPGA function number, src addr and dst addr are the source and destination block addresses, src size and dst size are source and destination block sizes encoded. rec 3addr(func, src1 addr, src2 addr, dst addr, blk size) is the three-operand reconfigurable instruction. func is the FPGA function number, src1 addr, src2 addr and dst addr are the source-1, source-2 and destination block addresses, and blk size is the block size encoded. Both reconfigurable instructions will perform the context switch to activate function func and will execute the corresponding operation associated with it. They will also lock their respective source and destination blocks of memory by entering the corresponding fields in the BLT for as long as the function takes to execute. When finished, the BLT entries corresponding to the instruction will be cleared. oc encodeSize(size) is a macro used for encoding the size of memory blocks. It obtains the encoded value from a table that is defined by the function log2 (size) - 1. This macro should be used to encode block sizes in reconfigurable instructions. For example, where a, b and c are defined as unsigned char a[16], b[16], c[16]; will activate function 2 and perform the operation with arrays a and b as source data and array c as destination data. The encoded size passed to the reconfigurable instruction will be log2(16) 3.4.2 Configuration definition The behavior of the RFU is modeled with a high-level functional simulation. It is given some inputs, and a function produces the corresponding outputs without performing a detailed micro-architecture simulation of the programmable logic. Configurations are defined as follows, { where Configuration address (i.e. the location of the configuration bits in memory). Operation latency (i.e. the number of cycles until result is ready for use). Issue latency (i.e. the number of cycles before another operation can be issued on the same resource). Expression that describes the reconfigurable function. The separation of the instruction latency into operation and issue latencies, allows the specification of pipelined con- figurations. For example, assume one configuration takes 20 cycles to complete one instruction, but the configuration is pipelined and one instruction can be started every 4 cycles. In this case, the operation latency will be 20 and the issue latency will be 4. Hence, the configuration can have 20 executing instructions at a time and the throughput for the configuration is implied as 20 per instruction. The expression field is where the semantics of the configuration will be specified. It is a C expression that implements the configuration being defined, the expression must modify all the processor state a#ected by the instruction's execution All memory accesses in the DEFCONF() expression must be done through the memory interface. There are macros available for doing memory reads and writes; for accessing general purpose registers, floating point and miscellaneous registers; for accessing the value of the RFU instruction operand field values, and; for creating a block mask and decoding a block size. Some configuration examples are included in [3]. 4. PROGRAMMING FOR SIM-ONECHIP This section will present an example of how to port an application so that it uses the RFU in OneChip to get speedup. The application to be implemented is an 8-tap FIR filter. Consider that you have this C code in a file called fir.c. 1: /* FILE: fir.c */ 2: 3: #include <stdio.h> 4: 5: #define TAPS 8 #define MAX_INPUTS 1024 7: 8: int 9: int inputs[MAX_INPUTS]; 10: 11: void main(){ 12: int i, j; 13: int *x; 14: int y[MAX_INPUTS]; 15: the inputs to some random numbers */ 17: for 23: /* FIR Filter kernel */ 24: for 25: for (j 27: } 28: x++; 30: The inner loop in the FIR filter kernel on lines 25-27 of fir.c can be ported to be executed entirely on the OneChip RFU. For that, we need to do some modifications to the C code. The file fir.oc.c that reflects this changes is shown below. 1: /* FILE: fir.oc.c */ 2: 3: #include <stdio.h> 4: #include "oc-lib.h" 5: 7: #define MAX_INPUTS 1024 8: 9: int 10: int inputs[MAX_INPUTS]; 12: void main(){ 13: int i, j; 14: int *x; 15: int y[MAX_INPUTS]; 17: oc_configAddress(0, 0x7FFFC000); 19: 20: /* Set the inputs to some random numbers */ 21: for 27: /* FIR Filter kernel */ 28: for 29: rec_3addr(0, x, coef, &y[i], oc_encodeSize(8)); 30: x++; 33: printf("\nFIR filter done!\n"); 34: } The first step was including the OneChip library in the code as shown on line 4 in fir.oc.c. The second step was defining the address of the configuration bitstream for the FIR filter. In this case, we are using configuration #0 and the memory address is 0x7FFFC000, as shown on line 17. As a third step, notice that lines 25-27 on fir.c have been removed and replaced by a 3-operand RFU instruction in line 29 on fir.oc.c. This instruction is using configuration #0 and is passing the address of the two source memory blocks, x and coef, which are pointers, as well as the address of destination memory block, which for each iteration will be &y[i]. The block size, 8, is passed using the function oc encodeSize. The previous three changes are necessary. Furthermore, if we want to reduce configuration overhead, we would introduce a pre-load instruction as in line 18. This instruction tells the processor that configuration #0 will be used soon. This way, by the time it gets to execute the RFU instruc- tion, the configuration is already loaded and no time is spent waiting for the configuration to be loaded. This instruction is not necessary, because if the configuration is not loaded in the FPGA, the processor will automatically load it. Now that the C code has been modified to use the RFU, we need to define the FPGA configuration that will perform the FIR filter. Configurations are defined in fpga.conf. The fir filter definition used is shown below. 1: /* This configuration is for a 3-operand instruction. 2: It is used for a fir filter program. */ 3: 4: DEFCONF(0x7FFFC000, 17, 17, 5: { int oc_index; /* for indexing */ 7: unsigned int oc_word; /* for storing words */ 8: unsigned int oc_result; /* for storing result */ 9: 12: oc_index <= OC_MASK(OC_3A_BS); 13: oc_index++) 14: { 17: oc_result += oc_word; 19: WRITE_WORD(oc_result, GPR(OC_3A_DR)); This configuration is the equivalent of the inner loop in the FIR filter kernel on lines 25-27 in fir.c. Note that in the configuration, each memory access is done through the memory interface. Line 4 defines the configuration address 0x7FFFC000 and the operation and issue latencies of 17. Lines 11-13 define the iteration loop for the FIR fil- ter. Line 15 reads a word from the memory location defined by the address stored in the general purpose register that contains one source address plus the corresponding memory o#set. In the same way, line 16 reads a word from the other source block and multiplies it with the data previously read and stored in the oc word variable. Line 17 simply accumulates the multiplied values across loop iterations. When the loop is finished, line 19 writes the result into the memory location defined by the address stored in the general purpose register that has the destination block address. The simulator will generate statistics for the number of instructions executed in each program. The speedup obtained with Sim-OneChip can be verified. 5. APPLICATIONS To evaluate the performance of the OneChip architecture, several benchmark applications were compiled and executed on Sim-OneChip. 5.1 Experimental Setup To do the experiments, four steps were performed for each application. Step one is the identification of which parts of each application are suitable for implementation in hard- ware. Step two is modeling the hardware implementation of the identified parts of the code. Step three is the replacement of the identified code in the application, with the corresponding hardware function call. And step four is the execution and verification of both the original and the ported versions of the application. The pipeline configuration used for both simulations was the default used in SimpleScalar. Among the most relevant characteristics are an instruction fetch queue size of 4 instructions; instruction decode, issue and commit bandwidths of 4 instructions per cycle; a 16-entry register update unit (RUU) and an 8-entry load/store queue (LSQ). The number of execution units available in the pipeline are 4 integer ALU's, 1 integer multiplier/divider, 2 memory system ports available (to CPU), 4 floating-point ALU's, 1 floating-point multiplier/divider. Also, in the case of sim-onechip, 1 reconfigurable functional unit (RFU), an 8-entry RBT and a 32-entry BLT were used. The branch predictor and cache configuration remained unmodified as well. 5.2 Benchmark applications There is currently no standard benchmark suite for reconfigurable processors. C. Lee et al.[13] from the University of California at Los Angeles have proposed a set of benchmarks for evaluating multimedia and communication sys- tems, which is called MediaBench. Since current reconfigurable processors available are used mostly for communications applications, MediaBench was taken as the suite for evaluating OneChip. Not all of the applications were used for the evaluation. Some of them could not be ported to SimpleScalar, due to the complexity of the makefiles or due to some missing libraries. However, the rest of the applications can provide good feedback on the architecture's performance 5.3 Profiles Profiling the execution of an application helps to identify the parts of the application take a lot of time to execute and hence, being candidates for rewriting to make it execute faster. Profiling of the applications was performed using GNU's profiler gprof included in GNU's binutils 2.9.1 package. From the profiling information, we identified specific functions in each application that are worth improving by executing them in specialized hardware implemented in the OneChip reconfigurable unit. To port an application to OneChip, a piece of code must have a long execution time and perform memory accesses in a regular manner, as in applications suitable for vector processors. In general, any application that can be sped up by a vector processor, will be also suitable for OneChip. 5.4 Analysis and modifications Four applications met our requirements and were ported to OneChip [3]. JPEG Image compression, ADPCM Audio coding, PEGWIT Data encryption and MPEG2 Video encoding. The encoder and the decoder for each one was ported. The modifications to the applications are done by hand (i.e. no compiler technologies are used). For the RFU timing in each of the applications, we assume that memory accesses dominate the computational logic and that our bottleneck is the memory bandwidth. If we also assume that one memory access is perfomed in one cycle, the latency of an operation will be obtained from counting the total number of memory accesses performed by the opera- tion. This timing approach may not be precise for highly compute intensive operations, but it is not the case on these applications. 6. RESULTS The original and the modified versions of the eight chosen applications were executed on the simulator. Each application was tested with three di#erent sizes of data, one small, one medium and one large. Four experiments were done for each application. Our first experiment was executing the original applications with in-order issue (A) to verify how many cycles each one takes to execute. As a second experiment, we executed the OneChip version of each Table 2: Speedup Application Data size Onechip inorder OneChip outorder (C/D) Outorder original Outorder OneChip Total JPEG encode Small 1.37X 1.34X 2.29X 2.25X 3.08X Medium 1.36X 1.33X 2.29X 2.24X 3.05X Large 1.38X 1.35X 2.33X 2.29X 3.15X JPEG decode Small 1.29X 1.20X 2.47X 2.29X 2.96X Medium 1.29X 1.19X 2.52X 2.34X 3.01X Large 1.25X 1.16X 2.53X 2.35X 2.93X ADPCM encode Small 22.38X 17.04X 1.54X 1.18X 26.31X Medium 26.25X 17.85X 1.62X 1.10X 28.94X Large 29.92X 20.57X 1.56X 1.07X 32.02X Large 24.43X 16.27X 1.61X 1.07X 26.13X PEGWIT encrypt Small 1.46X 1.43X 2.09X 2.06X 3.00X Medium 1.33X 1.36X 2.20X 2.26X 3.00X Large 1.16X 1.24X 2.48X 2.65X 3.07X PEGWIT decrypt Small 1.40X 1.42X 2.08X 2.11X 2.95X Medium 1.28X 1.32X 2.27X 2.35X 3.00X Large 1.13X 1.18X 2.62X 2.72X 3.08X Medium 5.07X 5.70X 2.07X 2.33X 11.82X Large 5.23X 5.91X 2.08X 2.36X 12.33X MPEG2 encode Small 1.16X 1.14X 1.90X 1.87X 2.16X Large 1.28X 1.24X 1.87X 1.81X 2.32X application also with in-order issue (B). The third experiment was executing the original version of the applications with out-of-order issue (C). And the fourth and last experiment was executing again the OneChip version, but now with out-of-order issue (D). This way we could verify the speedup obtained by using both features, the reconfigurable unit and the out-of order issue, in the OneChip pipeline. All output files were verified to have the correct data after being encoded and decoded with the simulator. The speedup obtained from the experiments is shown in Table 2. The first column (A/B) shows the speedup obtained by only using the reconfigurable unit. The second column (C/D) shows the speedup obtained by introducing a reconfigurable unit to an out-of-order issue pipeline. The third column (A/C) shows the speedup obtained by only using out-of-order issue. The fourth column (B/D) shows the speedup obtained by introducing out-of-order issue to a pipeline with a reconfigurable unit as OneChip. The fifth column (A/D) shows the total speedup obtained by using the reconfigurable unit and out-of order issue at the same time. Further analyzing the simulation statistics, we note that there are no BLT instruction stalls (i.e. instructions stalled due to to memory locks) in the applications, except for JPEG. This means that either the RFU is fast enough to keep up with the program execution or there are no memory accesses performed in the proximity of the RFU instruction execution. The second one is the actual case. It is important not to confuse BLT stalls, which prevent data hazards, with stalls due to unavailable resources, which are structural hazards. If there are two consecutive RFU instructions with no reads or writes in between, there will most likely be a structural hazard. Since there is only one RFU, the trailing RFU instruction will be stalled until the RFU is available. This is not considered a BLT stall. In the case of JPEG, there are CPU reads and writes performed in the proximity of RFU instructions. These are shown in Table 3. RFU instructions shows the total dynamic Table 3: JPEG RFU instructions Application Data size RFU instructions (X) instruction stalls Stalls per RFU instruction RFU Overlapping JPEG encode Small 851 99531 116.96 11.04 Large 18432 2156508 117.00 11.00 Large 18432 2264375 122.85 5.15 count of RFU instructions in the program, BLT instruction stalls is the number of CPU reads and writes stalled after an RFU write is executing (this was the only type of hazard present). The next column shows the Stalls per RFU instruction and the last one shows the average RFU instruction overlap with CPU execution. Note that 128 is the operation latency for JPEG. We can see that for the JPEG encoder there is an overlap of approximately 11 instructions, and for the JPEG decoder an overlap of approximately 5 instructions. 6.1 Discussion In Table 3 we can see that there is an approximate overlap of 11 instructions for the JPEG decoder. This means that when an RFU instruction is issued, 11 following instructions are also allowed to issue out-of-order because there are no data dependencies. Then, even if the RFU issue and operation latencies are improved (i.e. reduced) by new hardware technologies, the maximum improvement for this application will be observed if the configuration has a latency of 11 cycles. That is, any latency lower than 11 will not improve performance because the other 11 overlapping instructions will still need to be executed and the RFU will need to wait for them. The same will be observed for the JPEG decoder with a latency of 5 cycles. For the rest of the applications there is no overlap, so any improvement in the RFU latency will be reflected in the overall performance. ADPCM shows a fairly large speedup from OneChip. This is because the application does not perform any data validation or other operations besides calling the encoder kernel. The data is simply read from standard input, encoded on blocks of 1000 bytes at a time, and written to standard out- put, so the behaviour of the application is more like that of a kernel. It is expected that ADPCM is the application with the most speedup due to Amdahl's Law [9], which states that the performance improvement to be gained from using a faster mode of execution is limited by the fraction of time the faster mode can be used. ADPCM's performance clearly depends on the size of the data. The larger the data, the less time the application reads and writes data, and most of the time the RFU executes instructions. There are no BLT instruction stalls in this application. PEGWIT's performance also shows a dependence on the data size. However, di#erent behavior is observed for the RFU and the out-of-order issue features. The RFU shows better performance with small data, while out-of-order issue shows a better performance improvement with larger data. The overall speedup with both features is greater as the input data is larger. This is because for the decoder, the application makes a number of RFU instruction calls independent of the data size, and even if the data size for each call is di#erent, the latency is the same for every call. With the encoder, almost the same thing happens. No BLT instruction stalls were originated in this application. MPEG2's performance is also data size dependent. In the case of the decoder, the larger the data, the greater the performance improvement. This applies for both, the RFU feature and the out-of-order issue feature. In the case of the encoder, the performance improvement is shown to be larger, as the frame sizes get larger. There is a a higher performance improvement between the tests with small and medium input data, which have a di#erent frame size and almost the same number of frames, than between the tests with medium and large data, which have the same frame size and di#erent number of frames. In the application, there are no BLT instruction stalls. For all the applications, we can see that out-of-order issue by itself produces a big gain (A/C). Using an RFU still adds more speedup to the application. Speedup obtained from dynamic scheduling ranges from 1.60 up to 2.53. Speedup obtained from an RFU (A/B)ranges from 1.13 up to 29. When using both at the same time, even when each technique limits the potential gain that the other can produce, the overall speedup is increased. Dynamic scheduling seems to be more e#ective with the applications, except for AD- PCM, where the biggest gain comes from using the RFU. This leads us to think that for kernel-oriented applications, it is better to use an RFU without the complexity of out-of- order issue and for the other applications it is better to use dynamic scheduling possibly augmented with an RFU. 7. CONCLUSION AND FUTURE WORK In this work, the behavior of the OneChip architecture model was studied. Its performance was measured by executing several o#-the-shelf software applications on a software model of the system. The results obtained confirm the performance improvement by the architecture on DSP-type applications. From the work, a question arises whether the additional hardware cost of a complex structure, such as the Block Lock Table, is really necessary in reconfigurable processors. It has been shown that the concept of the BLT does accomplish its purpose, which is maintaining memory consistency when closely linking reconfigurable logic with memory and when parallel execution is desired between the CPU and the PFU. However, considering that only one of the four applications (i.e. JPEG) used in this research actually uses the BLT and takes advantage of it, we conclude that by removing it and simply making the CPU stall when any memory access occurs while the RFU is executing, will not degrade performance significantly on the types of benchmarks studied. In JPEG there is an average of 11 overlapping instructions, which is only 8.6% of the configuration operation latency slot of 128. If the RFU is used approximately 20% of the time in the JPEG encoder, the performance improvement by the overlapping is only 1.72%. This is a small amount compared to the performance improvement of dynamic schedul- ing, which is approximately 56% (i.e. 2.29 speedup). Hence, dynamic scheduling improves performance significantly only when used with relatively short operation delay instructions, as opposed to OneChip's RFU instructions, which have large operation delays. Based on the four applications in this work, it appears to be that the number of contexts does not need to be large to achieve good performance improvement with an RFU. In these applications, only one context was used for each application and a considerable speedup was obtained. For some applications, a second context could have provided an increase in this improvement, but not as much as for the first context. This is because, based on our profiles, we could implement a di#erent routine in a second context in the same way the first one was, but it would not be so frequently used. Another question that arises from this work is whether the configurations are small enough to fit on today's re-configurable hardware, or if they can be even implemented. Hardware implementations of DSP structures done in our and other groups [22, 11, 12], and which have even been shown to outperform digital signal processors, have been proven to fit on existing Altera and Xilinx devices with a maximum of 36,000 logic gates. Today's FPGAs have more than 1 million system gates available. To estimate the silicon area of this version of OneChip, we can start with the area of the processor that is required. It will be much larger than the simple processor used in the previous version of OneChip. A similar processor is the MIPS R10000 processor core [15], which is a 4-way super-scalar processor that supports out-of-order execution and includes a 32KB instruction cache and a 32KB data cache. Using a CMOS 0.35-m process, the die area is approximately . We can estimate that as fabrication technology approaches a 0.13-m process, the size of the processor core would be approximately 41mm 2 . The OneChip-98 processor [10] includes a small processor core of insignificant area, an eight-context FPGA structure with about 85K gates of logic, and 8 MBytes of SRAM. In a 0.18-m process, this was estimated to take about 550mm 2 . Scaling to 0.13-m brings this to 287mm 2 to which we can add the 41mm 2 for the processor. The complete OneChip device would be about which is quite manufacturable. Obviously, it would be desirable to add more gates of FPGA logic and as the process technology continues to shrink, this would be easy to do. We also conclude that dynamic scheduling is important to achieve good performance. By itself it produces a big gain for a number of applications. With kernel oriented applications, the gain obtained by an RFU is bigger, but with complete applications, the biggest gain is obtained from out-of-order issue and execution. Further investigation is necessary in the area of compilers for reconfigurable processors. Specifically, a compiler designed for the OneChip architecture is needed to fully exploit it and to better estimate the advantages and disadvantages of the architecture's features. Developing a compilation system that allows automatic detection of structures suitable for the OneChip RFU, as well as generating the corresponding configuration and replacing the structure in the program, will allow further investigation of the optimal number of contexts for the RFU. The compiler should be able to pre-load configurations to reduce delays, and should also make an optimal use of the BLT by scheduling as many instructions in the RFU delay slot. Also, future work should investigate the architecture of the RFU. In our work we have assumed an optimal RFU. No work has been done on what type of logic blocks or interconnection resources should be used in the FPGA. The simulator should be extended to properly simulate the FPGA fabric and any configuration latency issues. Ye et. al. [23] have modeled RFU execution latencies using simple instruction-level and transistor-level models. However, their architecture target fine-grain instructions, while OneChip targets coarse-grain instructions. At this point, it becomes di#cult to make a detailed comparison between OneChip's performance and other current reconfigurable systems. This is because there are no standard application benchmarks available for reconfigurable processors. However, other groups have reported performance improvement results similar to the ones presented in this paper, using Mediabench and SPEC benchmarks [2, 23]. Although Onechip shares certain similarities with other systems [2, 19] that target memory-streaming applications and focus on loop-level code optimizations, a standardized set of benchmarks and metrics for reconfigurable processors is needed to properly evaluate the di#erences between them. 8. ACKNOWLEDGEMENTS We would like to akcnowlegde Chameleon Systems Inc. for financially supporting the OneChip project. Jorge Carrillo was also supported by a UofT Open Fellowship. We would also like to thank the reviewers for their helpful comments. 9. --R The SimpleScalar tool set The Garp architecture and C compiler. A reconfigurable architecture and compiler. Configuration prefetch for single context reconfigurable coprocessors. The Chimaera reconfigurable functional unit. Configuration compression for the Xilinx XC6200 FPGA. Computer Architecture A Quantitative Approach. Memory interfacing for the OneChip reconfigurable processor. Memory interfacing and instruction specification for reconfigurable processors. A tool for evaluating and synthesizing multimedia and communications systems. The MorphoSys parallel reconfigurable system. A quantitative analysis of reconfigurable coprocessors for multimedia applications. RSIM: An execution-driven simulator for ILP-based shared-memory multiprocessors and uniprocessors The NAPA adaptive processing architecture. Operating System Concepts. A compiler directed approach to hiding configuration latency in chameleon processors. OneChip: An FPGA processor with reconfigurable logic. --TR A high-performance microarchitecture with hardware-programmable functional units MediaBench Configuration prefetch for single context reconfigurable coprocessors Computer architecture (2nd ed.) Memory interfacing and instruction specification for reconfigurable processors CHIMAERA The Garp Architecture and C Compiler PipeRench The MorphoSys Parallel Reconfigurable System A Compiler Directed Approach to Hiding Configuration Latency in Chameleon Processors The Chimaera reconfigurable functional unit Configuration Compression for the Xilinx XC6200 FPGA The NAPA Adaptive Processing Architecture A Quantitative Analysis of Reconfigurable Coprocessors for Multimedia Applications FPGA-Based Structures for On-Line FFT and DCT --CTR Hamid Noori , Farhad Mehdipour , Kazuaki Murakami , Koji Inoue , Maziar Goudarzi, Interactive presentation: Generating and executing multi-exit custom instructions for an adaptive extensible processor, Proceedings of the conference on Design, automation and test in Europe, April 16-20, 2007, Nice, France Scott Hauck , Thomas W. Fry , Matthew M. Hosler , Jeffrey P. Kao, The chimaera reconfigurable functional unit, IEEE Transactions on Very Large Scale Integration (VLSI) Systems, v.12 n.2, p.206-217, February 2004 Paul Beckett , Andrew Jennings, Towards nanocomputer architecture, Australian Computer Science Communications, v.24 n.3, p.141-150, January-February 2002 Nathan Clark , Manjunath Kudlur , Hyunchul Park , Scott Mahlke , Krisztian Flautner, Application-Specific Processing on a General-Purpose Core via Transparent Instruction Set Customization, Proceedings of the 37th annual IEEE/ACM International Symposium on Microarchitecture, p.30-40, December 04-08, 2004, Portland, Oregon Exploring the design space of LUT-based transparent accelerators, Proceedings of the 2005 international conference on Compilers, architectures and synthesis for embedded systems, September 24-27, 2005, San Francisco, California, USA Shobana Padmanabhan , Phillip Jones , David V. Schuehler , Scott J. Friedman , Praveen Krishnamurthy , Huakai Zhang , Roger Chamberlain , Ron K. Cytron , Jason Fritts , John W. Lockwood, Extracting and improving microarchitecture performance on reconfigurable architectures, International Journal of Parallel Programming, v.33 n.2, p.115-136, June 2005 Philip Garcia , Katherine Compton , Michael Schulte , Emily Blem , Wenyin Fu, An overview of reconfigurable hardware in embedded systems, EURASIP Journal on Embedded Systems, v.2006 n.1, p.13-13, January 2006

reconfigurable processors;onechip;superscalar processors

360393

Validation of an Augmented Lagrangian Algorithm with a Gauss-Newton Hessian Approximation Using a Set of Hard-Spheres Problems.

An Augmented Lagrangian algorithm that uses Gauss-Newton approximations of the Hessian at each inner iteration is introduced and tested using a family of Hard-Spheres problems. The Gauss-Newton model convexifies the quadratic approximations of the Augmented Lagrangian function thus increasing the efficiency of the iterative quadratic solver. The resulting method is considerably more efficient than the corresponding algorithm that uses true Hessians. A comparative study using the well-known package LANCELOT is presented.

Introduction In recent years we have been involved with the development of algorithms based on sequential quadratic programming [11] and inexact restoration [16, 18] for minimization problems with nonlinear equality constraints and bounded variables. The validation of these algorithms require their comparison with well established computer methods for the same type of problems, which include methods of the same family (as other SQP methods in the first case and GRG like methods in the second) as well as methods that adopt a completely di#erent point of view, as is the case of Penalty and Augmented Lagrangian algorithms. The most consolidated practical Augmented Lagrangian method currently available is the one implemented in the package LANCELOT, described in [4]. This was the method used, for example, in [11], to test the reliability of a new large-scale sequential quadratic programming algorithm. * This author was supported by FAPESP (Grant 96/8163-9). * These authors were supported by PRONEX, FAPESP (Grant 90-3724-6), CNPq and FAEP- UNICAMP. In the course of the above mentioned experimental studies we felt the necessity of intervening in the Augmented Lagrangian code in a more active way than the one permitted to users of LANCELOT. As a result of this practical necessity, we became involved with the development of a di#erent Augmented Lagrangian code, which preserves most of the principles of the LANCELOT philosophy, but also has some important di#erences. Following the lines of [4], a modern Augmented Lagrangian method is essentially composed by three nested algorithms: . The external algorithm updates the Lagrange multipliers and the penalty pa- rameters, decides stopping criteria for the internal algorithm and the rules for declaring convergence or failure of the overall procedure. . An internal algorithm minimizes the augmented Lagrangian function with bounds on the variables. Trust region methods, where the subproblem consists of the minimization of a quadratic model on the intersection of two boxes, the one that defines the problem and the trust-region box, are used both in [4] and in our implementation. . A third algorithm deals with the resolution of the quadratic subproblem. While LANCELOT restricts its search to the face determined by an approximate Cauchy point, our code explores the domain of the subproblem as a whole. The second item, specifically where it deals with the formulation of the quadratic subproblem, is the one in which we felt more strongly the desire to intervene. On one hand, we tried many alternative sparse quasi-Newton schemes (without success, up to now). On the other hand, we used a surprisingly e#ective simplification of the true Hessian of the Lagrangian, called, in this paper, "the Gauss-Newton Hessian approximation" by analogy with the Gauss-Newton method for nonlinear least- squares, which can be interpreted as the result of excluding from the Hessian of a sum of squares those terms involving Hessian of individual components. In order to validate our augmented Lagrangian implementation we selected a family of problems in which we have particular interest, known as the family of Hard-Spheres problems. The Hard-Spheres Problem belongs to a family of sphere packing problems, a class of challenging problems dating from the beginning of the seventeenth cen- tury. In the tradition of famous problems in mathematics, the statements of these problems are elusively simple, and have withstood the attacks of many worthy mathematicians (e.g. Newton, Hilbert, Gregory), while most of its instances remain open problems. Furthermore, it is related to practical problems in chemistry , biology and physics, see, for instance, the list of examples in [19], concerning mainly three-dimensional problems, or peruse the 1550-item-long bibliography in [5]. The Hard-Spheres Problem is to maximize the minimum pairwise distance between p points on a sphere in R n . This problem may be reduced to a nonlinear optimization problem that turns out, as might be expected from the mentioned history, to be a particularly hard, nonconvex problem, with a potentially large number of (nonoptimal) points satisfying KKT conditions. We have thus a class of problems indexed by the parameters n and p, that provides a suitable set of test problems for evaluating Nonlinear Programming codes. Very convenient is the fact that the Hard-Spheres Problem may be regarded as the feasibility problem associated with another famous problem in the area, the Kissing Number Problem, which seeks to determine the maximum number K n of nonoverlapping spheres of given radius in R n that can simultaneously touch (kiss) a central sphere of same radius. Thus, if the distance obtained in the solution of the Hard-Spheres Problem, for given n and p, is greater than or equal to the radius of the sphere on which the points lie, one may conclude that K n # p. We use the known solution of the three-dimensional Kissing Number Problem to calibrate our code, described below, and choose for testing the code values of n, p that might bring forth new knowledge about the problem, or strengthen existing conjectures about the true (but, alas, not rigorously established) value of K n , from the following table of known values/bounds of K n given in [5]: Table 1. Known values and bounds for Kn . 9 306-380 This paper is organized as follows. In Section 2 we formulate the Hard-Spheres Problem as a nonlinear programming problem and we relate the main characteristics of ALBOX, our Augmented Lagrangian Algorithm. In Section 3 we explain how the main algorithmic parameters of ALBOX were chosen. (Here we follow a previous study in [15].) In Section 4 we introduce the Gauss-Newton Hessian approximation and discuss the e#ect of its use in comparison with the use of true Hessians of the Lagrangians. In Section 5 we describe the parameters used with LANCELOT. The numerical experiments, obtained by running ALBOX and LANCELOT for a large number of Hard-Spheres problems, are presented in Section 6. Finally, some conclusions are drawn in Section 7. 2. ALBOX The straightforward formulation of the Hard-Spheres Problem leads to the following maxmin problem, where r is the radius of the sphere, centered at the origin, on which the points lie: s.t. #y k (1) The vectors y k belong to R n and # is the Euclidean norm. Since the answer to the problem is invariant under the choice of positive r, we let using the definition of #, the standard inner product in R n , and the constraints, it is easy to see that (1) is equivalent to s.t. #y k (2) Applying the classical trick for transforming minimax problems into constrained minimization problems, we reduce (2) to the nonlinear program min z s.t. z #y i , y j Adding slack variables to the first set of constraints and squaring the second set of equations in order to avoid nonsmoothness in the first derivatives, we obtain min z which is of the general form min f(x) ALBOX, the augmented Lagrangian code developed, approximately solves min L(x, #) at each Outer Iteration, where is the augmented Lagrangian function associated with (5), # is the current approximation to the Lagrange multipliers and # 0) is the current vector of penalty parameters. These are updated at the end of the Outer Iteration. Subproblem (6) is solved using BOX, the box-constrained solver described in [10]. This iterative method minimizes a quadratic approximation to the objective function on the intersection of the original feasible set, the box # x # u, and the trust region (also a box), at each iteration. If the original objective function is sufficiently reduced at the approximate minimizer of the quadratic, the corresponding trial point is accepted as the new iterate. Otherwise, the trust region is reduced. The main algorithmic di#erence between BOX and the method used in [2] is that in BOX the quadratic is explored on the whole intersection of the original box and the trust region whereas in [2] only the face determined by an "approximate Cauchy point" is examined. ALBOX is a Double Precision FORTRAN 77 code that aims to cope with large-scale problems. For this reason, factorization of matrices is not used at all. The quadratic solver used to solve the subproblems of the box-constraint algorithm, QUACAN, visits the di#erent faces of its domain using conjugate gradients on the interior of each face and "chopped gradients" as search directions to leave the faces. We refer the reader to [1], [9] and [10], for details on the actual implementation of QUACAN. In most iterations of this quadratic solver, a matrix-vector product of the Hessian approximation and a vector is computed. Occasionally, an additional matrix-vector product may be neccessary. The performance of ALBOX, and, in fact, of most sophisticated algorithms, depends on the choice of many parameters. The most sensitive parameters were adjusted using the Kissing Problem with We discuss these choices in the next section. A similar analysis was carried out for LANCELOT, and is described in section 5. 3. Choice of parameters for ALBOX 3.1. Penalty parameters and Lagrange multipliers The vector # of penalty parameters associated with the equality constraints are updated after each Outer Iteration. We considered two possibilities: to update each component according to the decrease of the corresponding component of h(x) or using a global criterion based on h(x). The specific alternatives contemplated were, assuming x to be the initial point at some Outer Iteration and - x the final one: 1. increase # i only if |h(-x) i | is not su#ciently smaller than |h(x) i |; 2. increase # i only if #h(-x)# is not su#ciently smaller than #h(x)# . Preliminary experiments revealed, perhaps surprisingly, that the "global strat- egy" 2 is better than the first. In fact, when # i is not updated, but the other components of # are, the feasibility level |h(-x) i | tends to deteriorate at the next iteration and, consequentely, a large number of Outer Iterations becomes necessary. In other words, it seems that a strategy based on 1 encourages a zigzagging be- havior, with successive iterates alternatingly satisfying one constraint or another. Thus, although the original formulation allows for one penalty parameter for each equality constraint, in practice it is as if we worked with one parameter for all of them, since they are all initialized at the same value (tests indicate that 10 is an adequate initial value) and are all updated according to the same rule (once again based on tests, they are increased by a factor of 10 when su#cient improvement of feasibility is not detected). Here we considered that "a su#ciently smaller than b" means that a # 0.01b. It must be pointed out that the behavior of penalty parameters is not independent of the strategy for updating the Lagrange multipliers. With algorithmic simplicity in mind, we adopted a "first order formula". Letting - # be the Lagrange multiplier at the start of a new Outer Iteration and # be the Lagrange multipliers and penalty parameters at the previous iteration, we set for all 3.2. Stopping criteria for box-constraint solver Each Outer Iteration ends when one of the several stopping criteria for the algorithm that solves the augmented Lagrangian box-constrained minimization problem (6) is reached. There is the usual maximum number of iterations safeguard, which is set at 100 for QUACAN calls. Other than that, we consider that the box-constraint algorithm BOX converges when is the "continuous projected gradient" of the objective function of (6) at the point x. This vector is defined as the di#erence between the projection of x- #L(x, #) on the box and the point x. The tolerance # may change at each Outer Iteration. We tested two strategies for #: one that defines # dynamically depending on the degree of feasibility of the current iterate and another that fixes # at 10 -5 . Althought not conclusive, results for the Icosahedron Problem were better when the constant # strategy was used. This was, therefore, the strategy adopted for further tests. Incidentally, the opposite was adopted in [8], where a similar Augmented Lagrangian Algorithm was used to solve linearly constrained problems derived from physical applications. Theoretical justifications for the inexact minimization of subproblems in the augmented Lagrangian context can also be found in [12, 13]. The box-constraint code admits other stopping criteria. For instance, execution may stop if the progress during some number of consecutive iterations is not good enough or if the the radius of the trust region becomes too small. Nevertheless, best results were obtained inhibiting these alternative stopping criteria. 3.3. Parameters for the quadratic solver QUACAN is the code called to minimize quadratic functions (augmented Lagrangians in this case) subject to box constraints. Its e#ciency, or lack thereof, plays a crucial role in the overall behavior of the Augmented Lagrangian Algorithm. Its parameters must therefore be carefully chosen. Firstly we examine the convergence criterion. If the projected gradient of the quadratic is null, the corresponding point is stationary. Accordingly, convergence is considered achieved when the norm of this projected gradient is less than a fraction of the corresponding norm at the initial point. In this case, we use "non- continuous projected gradients," in which the projections are not computed on the feasible box but on the active constraints. Fractions 1/10, 1/100 and 1/100000 were tested on the Icosahedron Problem, and the first choice provided the best behavior, being the one employed subsequently. The maximum number of iterations allowed is also an important parameter, since otherwise we may invest too much e#ort solving problems only distantly related to the original one. We found that the number of variables of the problem, np+ - p is a suitable delimiter in this case. Other non-convergence stopping criteria were inhibited. The radius of the trust region determines the size of the auxiliary box used in QUACAN. The nonlinear programming algorithm is sensitive to the choice of #, the first trust region radius. After testing di#erent values, we selected as an appropriate choice. Another important parameter is # (0, 1), the parameter that determines whether the next iterate must belong to the same face as the current one, or not. Roughly speaking, if # is small, the algorithm tends to leave the current face as soon as a mild decrease of the quadratic is detected. On the other hand, if # 1, the algorithm only abandons the current face when the current point is close to a stationary point of the quadratic on that face. A rather surprising result was that, for the Icosahedron Problem, the conservative value better than smaller values. Finally, when the quadratic solver hits the boundary of its feasible region, an extrapolation step may be tried, depending on the value of the extrapolation parameter # 1. If # is large, new points will be tried at which the number of active bounds may be considerably increased. No extrapolation is tried when indicated that convenient choice for the Hard-Spheres Problem. 4. Approximate Hessian The nonlinear optimization problem (4) obtained in section 2 is the version of the Hard-Spheres Problem that was chosen for our tests. It was pointed out that (4) is of the general form min f(x) whose associated augmented Lagrangian is . Thus and Although # 2 L(x, #) tends to be positive definite when # is large, # is close to the correct Lagrange multipliers and x is close to a solution, this is not the case at the early stages of augmented Lagrangian calculations. On the other hand, the simplified matrix obtained by neglecting the term involving second order derivatives of the constraint functions is always positive semidefinite in our case, independently of # and x. Of course, this is always the case when f is a convex function. Another insight into B(x, #) is provided by examining the problem min f(x) where z is the current point being used in a BOX iteration. Problem (8) is obtained by replacing the original constraints with its first order (linear) approx- imation. But B(z, #) happens to be the Hessian of the augmented Lagrangian associated with (8) at z! Furthermore, both the augmented Lagrangian associated with (8) and its gradient evaluated at z coincide with their counterparts associated with the original problem (4), evaluated at z. The matrix vector products # 2 L(x, #)v and B(x, #)v seem cumbersome to compute at a first glance. But taking advantadge of their structure enables the computation to be done in O(np) time. In principle, using the true Hessian of the Lagrangian should be the best possible choice, since it represents better the structure of the true problem. However, available algorithms for minimizing quadratics in convex sets are much more e#cient when the quadratic is convex than otherwise. QUACAN is not an exception to this rule. Therefore, in the interest of improving the overall performance of the augmented Lagrangian algorithm, we decided to use B(x, #) as Hessian Lagrangian approximation. The results were indeed impressive. Table 2 lists the average statistics obtained for four of the eighteen test sets, where each (n, p) pair was run for fifty random starting points. The average number of Outer iterations, BOX iterations, Function evaluations, Matrix Vector Products, CPU time in seconds and minimum distance are given for the runs using the exact Hessian (first row of each set) and the ones using the approximate Hessian (second row). The minimum distances obtained were very close and on some instances the minimum distance obtained using the approximate Hessian was smaller than the one obtained using the exact Hessian. While the number of Outer iterations does not di#er very much from one choice to the other, the number of BOX iterations and, consequently, the number of Matrix Vector Products sensibly decreases. The overall result is a marked decrease in CPU time. In Figure 1 we plot the average CPU times, for all eighteen tests, using the exact Hessian versus times using the approximate Hessian. Also shown is the line that gives the best fit of the data by a linear (not a#ne) function, namely that is, the approximate Hessian option implies in a decrease of almost two thirds in CPU times. Table 2. Running ALBOX with exact (first row) and approximate Hessian (second row). Problem size Outer Box Funct. MVP CPU Min it. it. eval. time dist. 4.86 37.06 52.14 1564.36 0.765 1.086487225412 22 4.56 160.02 193.14 67020.22 373.141 0.998675348042 -CPU times using exact Hessian CPU times using approx. Hessian 500 1000 1500 2000 2500 3000400800- Figure 1. CPU times using exact Hessian (x-axis) versus using approximate Hessian (y-axis). 5. Choice of parameters for LANCELOT LANCELOT allows for the choice of exact or approximate first and second order derivatives. However, LANCELOT's manual [3] (p.111) "strongly recommends the use of exact second derivatives whenever they are available", and, on the other hand, there is no provision for a user supplied Hessian approximation. In fact we ran a few tests with the default approximation (SR1) but the results were worse than those obtained using exact second derivatives, and thus this was the option adopted for all further tests. In the light of the experiments described in the previous section, this provides corroborating evidence to the e#ect that general purpose, consolidated packages, designed to provide a good performance with little interference from the user, may be more convenient to use than open ended, low-level interface codes, such as ALBOX; but, for the user willing to "get his hands dirty" the latter rawer code might not only prove to be competitive, it may actually outperform the former code, with its more polished though restrictive finish. We also experimented with several di#erent options for solving the linear equation solver, namely, without preconditioner, with diagonal preconditioner and with a band matrix preconditioner. The best results were obtained with the first option (no preconditioner). Another choice that slowed the algorithm, without noticeable improve the quality of solution, was requiring that the exact Cauchy point be com- puted. We settled to use the inexact Cauchy point option. The maximum number of iterations allowed was 1000. Finally, the gradient and constraints tolerances were the same chosen for ALBOX, namely 10 -8 . The FORTRAN compiler option adopted for LANCELOT and ALBOX was "-O". 6. Numerical experiments Tests were run on a Sun SparcStation 20, with the following main characteristics: 128Mbytes of RAM, 70MHz, 204.7 mips, 44.4 Mflops. Results for the fifty runs for each (n, p) pair are summarized in the following tables. Table 3 summarizes the statistics that are "machine independent," typically involving number of iterations, number of function evaluations, with the exception of the optimal distances found. Quotes are needed because this is not completely accurate, since these numbers will in fact depend on factors such as machine precision, compiler manufacturer, and so on. Nevertheless, they certainly provide more independent grounds for comparison than CPU times, presented in Table 4, along with optimal distances. Table 3 presents the minimum, maximum and average amounts (the number triple in each box) of Outer and BOX iterations, function evaluations, Quacan iterations and matrix-vector products/conjugate-gradient iterations (for Box and LANCELOT, respectively). First row of each set corresponds to ALBOX and second to LANCELOT. Unfortunately the only statistics available for both are the number of function evaluations and BOX iterations/Derivative evaluations. We paired the number of matrix-vector products (MVP) output by ALBOX with the number of conjugate-gradient iterations (CGI) produced by LANCELOT, since each conjugate-gradient iteration involves a matrix-vector-product. Table 3. ALBOX - LANCELOT test results Problem size Outer BOX Function Quacan MVP Derivative eval. iter. CGI 4,5,4.6 21, 55, 34.7 25, 77, 45.5 309, 2343, 1064 340, 2702, 1195 15, 61, 38.1 16, 71, 43.3 377, 1949, 992 20, 62, 38.0 21, 80, 43.4 511, 2709, 1032 22, 58, 39.7 24, 66, 45.3 553, 1776, 1069 4,5,4.8 27, 75, 47.9 34,104, 64.2 933, 4923, 2708 1017, 5382, 2963 27, 84, 52.3 29, 96, 60.1 967, 4248, 2313 4,5,4.6 32,110, 60.2 41,140, 77.8 1625, 8385, 4130 1751, 8742, 4444 30, 91, 56.8 33,112, 65.1 1107, 5652, 3015 22 4,5,4.3 52,115, 78.0 62,148, 97.4 5688, 16767, 10502 6097, 17871, 11222 45,225, 104.1 49,262, 120.0 5122, 37546, 12381 37,176, 108.9 39,208, 124.6 4799, 29367, 14607 2,5,4.1 45,141, 86.4 58,183, 107.9 6282, 28049, 14077 6769, 29825, 15009 4,5,4.2 63,180, 97.8 75,226, 120.6 10492, 35660, 17105 11034, 37639, 18143 54,225, 119.7 60,262, 137.5 6870, 38419, 18736 26 4,6,4.2 51,176, 95.4 63,216, 117.2 6765, 38932, 17185 7317, 40932, 18237 53,266, 131.4 59,311, 150.5 5094, 77233, 21796 4,5,4.3 62,206, 99.5 76,254, 122.1 11480, 45129, 19490 12169, 47121, 20616 62,215, 128.9 68,253, 147.6 9420, 41799, 21534 4,8,4.6 80,800, 160.0 102,984, 193.1 27836,471751, 63778 29476,497038, 67020 85,334,190.42 95,381, 218.6 9119, 96036, 56899 4,6,4.6 89,600, 166.2 107,717, 200.0 29224,326333, 67969 30804,340424, 71261 4,7,4.9 78,700, 195.8 89,815, 231.3 26692,448509, 88566 27892,472730, 92422 91,385,231.24 99,453,263.44 24178,160611, 85972 4,7,4.9 90,700, 202.9 106,880, 242.9 34936,463883, 98266 36614,485784,102816 4,8,4.9 93,800, 225.1 117,954, 271.6 36194,547421,117417 38311,577662,122924 4,7,4.6 109,700, 212.3 132,887, 256.1 47402,502810,109630 49993,529036,114749 102,440, 246.4 115,499, 281.3 34730,200558, Although the algorithms behave very di#erently timewise, as we will shortly see, this is not a direct consequence of the number of function evaluations each performs. The best least-squares fit by a first degree polynomial gives where y is the number of function evaluations of ALBOX and x is the corresponding amount for LANCELOT, whereas a similar fit involving CPU times will give a coe#cient of less than a third. On Figure 2 we plot the function evaluation pairs for all eighteen instances along with the best fit obtained. ALBOX Figure 2. Number of function evaluations of LANCELOT versus ALBOX. Further still from providing an explanation for the higher e#ciency of ALBOX is the comparison of MVP versus CGI. In this case the best fit gives 1.10655x, where y is the number of MVP and x is the number of CGI. This suggests that, although both iterations involve a matrix-vector-product, a CGI is substantially costlier, timewise, than the MVP performed in ALBOX. A main factor for this is that the matrix-vector-product in LANCELOT's conjugate gradient iteration deals with the true Hessian, whereas the one in ALBOX involves the approximate (and simpler) Hessian. Figure 3 contains the line corresponding to the best linear fit and the position of the (CGI, MVP) pairs. Next we have Table 4, that presents similar statistics involving the optimal distances encountered and the CPU times, in seconds. The first (resp., second) row for each (n, p) pair gives the numbers obtained by ALBOX (resp., LANCELOT). ALBOX Figure 3. Number of CGIs of LANCELOT versus number of MVPs of ALBOX. Table 4. Minimum distances and CPU times for tests. Problem size minimum distance between 2 points CPU time (seconds) 1.0514622 1.0914262 1.08323633 0.170 1.010 0.476 1.0514622 1.0514622 1.05146223 0.170 1.420 0.636 0.9463817 1.0514622 1.04515739 0.290 1.870 0.906 22 0.9529038 0.9619429 0.95809771 25.150 86.570 41.290 26 0.9606935 0.9779378 0.96928704 420.170 4527.652 1000.608 0.9599791 0.9798367 0.97025160 807.570 4664. The information contained in Table 4 is depicted graphically below. The intervals (min., max) of distances/CPU times are represented by vertical segments, the averages are indicated with a diamond symbol for ALBOX and a bullet for LANCELOT. Graphs on the left refer to distances whereas graphs on the right refer to CPU times.- min. dist. #- time Figure 4. ALBOX (#) and LANCELOT (.) results for 22 - 4 26 - 4 min. dist. #- 22 - 4 26 - 4 CPU time Figure 5. ALBOX (#) and LANCELOT (.) results for The graphs in Figures 4-6 evidence the qualitative relative behavior of both codes. Notice that the diamonds and bullets are always close together in the graphs on the left, indicating that the quality of the optimal solutions obtained by both codes min. dist. #- CPU time Figure 6. ALBOX (#) and LANCELOT (.) results for is similar. On the other hand, the bullets rise faster than the diamonds on the graphs on the right, which means that the CPU times for LANCELOT tend to be higher than those for ALBOX. The linear fit of ALBOX CPU times versus x-the coe#cient is less than one third-, ploted in Figure 7 confirms this. -CPU times CPU times ALBOX 500 1000 1500 2000 2500400800- Figure 7. CPU times of LANCELOT versus those of ALBOX. Finally, it should be noted that CPU times increase sharply as a function of problem size (represented, for instance, by the number of constraints). We tried several fits (linear, quadratic, exponential) and, though none seemed to provide a very good model for the data, the quadratic fit was the best one. 7. Conclusions The main aspects of the Augmented Lagrangian methodology for solving large-scale nonlinear programming problems have been consolidated after the works of Conn, Gould and Toint which gave origin to the LANCELOT package. This algorithmic framework has been very useful in the last ten years for solving practical problems and for comparison purposes with innovative nonlinear programming methods. Very likely, this tendency will be maintained in the near future. The present research was born as a result of our need to have more freedom in the formulation and resolution of the quadratic subproblems that arise in the LANCELOT-like approach to the Augmented Lagrangian philosophy. On one hand, we decided to exploit more deeply the whole trust region by means of the use of a box-constraint quadratic solver. On the other hand, perhaps more importantly, we tested a Gauss-Newton convex simplification of the quadratic model which turned out to be much more e#cient than the straight Newton-like version of this model. Behind this gain of e#ciency is the fact that the quadratic solver, though able to deal with nonconvex models, is far more e#cient when the underlying quadratic has a positive semidefinite Hessian. It is usual, in Numerical Analysis, that a decision on the implementation of a high level algorithm depends on the current technology for solving low-level subproblems. It must only be warned that such a decision could change if new more e#cient algorithms for solving the subproblems (nonconvex quadratic programming in our case) are found. Our main objective is to use ALBOX, not only for solving real-life problems, but also for testing alternative nonlinear programming methods against it. We feel that having a deep knowledge of the implementation details of the code will enable us to be much more exacting when testing new codes, since it will be possible to fine tune the standard against which the new code is tested. The present study, apart from calling the reader's attention to convex simplified Gauss-Newton like subproblems, had the objective of validating our code, by means of its comparison with LANCELOT, using a set of problems that have an independent interest. The result of this comparison seems to indicate that ALBOX can be used as a competitive tool for nonlinear programming calculations. --R "An adaptive algorithm for bound constrained quadratic minimization," "A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds," "Global convergence of a class of trust region algorithms for optimization with simple bounds," Lattices and Groups Mathematics: The Science of Patterns "Comparing the numerical performance of two trust-region algorithms for large-scale bound-constrained minimization," "Augmented Lagrangians with adaptive precision control for quadratic programming with equality constraints," "On the maximization of a concave quadratic function with box constraints," "A new trust-region algorithm for bound constrained minimization," "Nonlinear programming algorithms using trust regions and augmented Lagrangians with nonmonotone penalty parameters," "Analysis and implementation of a dual algorithm for constraint optimization," "Dual techniques for constraint optimization," "Bounds on the kissing numbers in R n : mathematical programming formulations," "Augmented Lagrangians and the resolution of packing problems," "A two-phase model algorithm with global convergence for nonlinear pro- gramming," "Preconditioning of truncated-newton methods," "Linearly constrained spectral gradient methods and inexact restoration sub-problems for nonlinear programming," "Distributing many points on a sphere," --TR Dual techniques for constrained optimization Global convergence of a class of trust region algorithms for optimization with simple bounds A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds Analysis and implementation of a dual algorithm for constrained optimization Two-phase model algorithm with global convergence for nonlinear programming Lancelot Augmented Lagrangians with Adaptive Precision Control for Quadratic Programming with Equality Constraints --CTR Graciela M. Croceri , Graciela N. Sottosanto , Mara Cristina Maciel, Augmented penalty algorithms based on BFGS secant approximations and trust regions, Applied Numerical Mathematics, v.57 n.3, p.320-334, March, 2007 R. Andreani , A. Friedlander , M. P. Mello , S. A. Santos, Box-constrained minimization reformulations of complementarity problems in second-order cones, Journal of Global Optimization, v.40 n.4, p.505-527, April 2008 G. Birgin , Jos Mario Martnez, Large-Scale Active-Set Box-Constrained Optimization Method with Spectral Projected Gradients, Computational Optimization and Applications, v.23 n.1, p.101-125, October 2002 Nikhil Arora , Lorenz T. Biegler, A Trust Region SQP Algorithm for Equality Constrained Parameter Estimation with Simple Parameter Bounds, Computational Optimization and Applications, v.28 n.1, p.51-86, April 2004 G. Birgin , J. M. Martnez, Structured minimal-memory inexact quasi-Newton method and secant preconditioners for augmented Lagrangian optimization, Computational Optimization and Applications, v.39 n.1, p.1-16, January 2008

numerical methods;augmented Lagrangians;nonlinear programming

360547

Selecting Examples for Partial Memory Learning.

This paper describes a method for selecting training examples for a partial memory learning system. The method selects extreme examples that lie at the boundaries of concept descriptions and uses these examples with new training examples to induce new concept descriptions. Forgetting mechanisms also may be active to remove examples from partial memory that are irrelevant or outdated for the learning task. Using an implementation of the method, we conducted a lesion study and a direct comparison to examine the effects of partial memory learning on predictive accuracy and on the number of training examples maintained during learning. These experiments involved the STAGGER Concepts, a synthetic problem, and two real-world problems: a blasting cap detection problem and a computer intrusion detection problem. Experimental results suggest that the partial memory learner notably reduced memory requirements at the slight expense of predictive accuracy, and tracked concept drift as well as other learners designed for this task.

Introduction Partial memory learners are on-line systems that select and maintain a portion of the past training examples, which they use together with new examples in subsequent training episodes. Such systems can learn by memorizing selected new facts, or by using selected facts to improve the current concept descriptions or to derive new concept descriptions. Researchers have developed partial memory systems because they can be less susceptible to overtraining when learning concepts that change or drift, as compared to learners that use other memory models (Salganicoff, 1993; Maloof, 1996; Widmer & Kubat, 1996; Widmer, 1997). The key issues for partial memory learning systems are how they select the most relevant examples from the input stream, maintain them, and use them in future learning episodes. These decisions affect the system's predictive accuracy, memory requirements, and ability to cope with changing concepts. A selection policy might keep each training example that arrives, while the maintenance policy forgets examples after a fixed period of time. These policies strongly bias the learner toward recent events, and, as a consequence, the system may forget about important but rarely occurring events. Alternatively, the system may attempt to select proto- A. MALOOF AND RYSZARD S. MICHALSKI typical examples and keep these indefinitely. In this case, the learner is strongly anchored to the past and may perform poorly if concepts change or drift. This paper presents a method for selecting training examples for a partial memory learner. Our approach extends previous work by using induced concept descriptions to select training examples that lie at the extremities of concept boundaries, thus enforcing these boundaries. The system retains and uses these examples during subsequent learning episodes. This approach stores a nonconsecutive collection of past training examples, which is needed for situations in which important training events occur but do not necessarily reoccur in the input stream. Forgetting mechanisms may be active to remove examples from partial memory that no longer enforce concept boundaries or that become irrelevant for the learning task. As new training examples arrive, the boundaries of the current concept descriptions may change, in which case the training examples that lie on those boundaries will change. As a result, the contents of partial memory will change. This continues throughout the learning process. After surveying relevant work, we present a general framework for partial memory learning and describe an implementation of such a learner, called AQ-Partial Memory (AQ-PM), which is based on the AQ-15c inductive learning system (Wnek, Kaufman, Bloedorn, & Michalski, 1995). We then present results from a lesion study (Kibler & Langley, 1990) that examined the effects of partial memory learning on predictive accuracy and on memory requirements. We also make a direct comparison to IB2 (Aha, Kibler, & Albert, 1991), since it is similar in spirit to AQ- PM. In applying the method to the STAGGER Concepts (Schlimmer & Granger, 1986), a synthetic problem, and two real-world problems-the problems of blasting cap detection in X-ray images (Maloof & Michalski, 1997) and computer intrusion detection (Maloof & Michalski, 1995)-experimental results showed a significant reduction in the number of examples maintained during learning at the expense of predictive accuracy on unseen test cases. AQ-PM also tracks drifting concepts comparably to STAGGER (Schlimmer & Granger, 1986) and the FLORA systems (Widmer & Kubat, 1996). 2. Partial Memory Learning On-line learning systems must have a memory model that dictates how to treat past training examples. Three possibilities exist (Reinke & Michalski, 1988): 1. full instance memory, in which the learner retains all past training examples 2. partial instance memory, in which it retains some of the past training examples, and 3. no instance memory, in which it retains none. Researchers have studied and described learning systems with each type of memory model. For example, STAGGER (Schlimmer & Granger, 1986) and Winnow (Lit- tlestone, 1991) are learning systems with no instance memory, while GEM (Reinke & Michalski, 1988) and IB1 (Aha et al., 1991) are learners with full instance mem- ory. Systems with partial instance memory appear to be the least studied, but examples include LAIR (Elio & Watanabe, 1991), HILLARY (Iba, Woogulis, & Langley, 1988), IB2 (Aha et al., 1991), DARLING (Salganicoff, 1993), AQ-PM (Maloof & Michalski, 1995), FLORA2 (Widmer & Kubat, 1996), and MetaL(B) (Widmer, 1997). On-line learning systems must also have policies that deal with concept memory, which refers to the store of concept descriptions. Researchers have investigated a variety of strategies in conjunction with different models of instance memory. For example, IB1 (Aha et al., 1991) maintains all past training examples but does not store generalized concept descriptions. In contrast, GEM (Reinke & Michalski, 1988) keeps all past training examples in addition to a set of concept descriptions in the form of rules. ID5 (Utgoff, 1988) and ITI (Utgoff, Berkman, & Clouse, 1997) store training examples at the leaf nodes of decision trees, so they are also examples of systems with full instance memory. Actually, ID5 stores a subset of an example's attribute values at the leaves and is an interesting special case of full instance memory. Finally, as an example of a system with no instance memory, ID4 (Schlimmer & Fisher, 1986) uses a new training example to incrementally refine a decision tree before discarding the instance. For systems with concept memory, learning can occur either in an incremental mode or in a temporal batch mode. Incremental learners modify or adjust their current concept descriptions using new examples in the input stream. If the learner also maintains instance memory, then it uses these examples to augment those arriving from the environment. FLORA2, FLORA3, and FLORA4 (Widmer & Kubat, 1996) are examples of systems that learn incrementally with the aid of partial instance memory. Temporal batch learners, on the other hand, replace concept descriptions with new ones induced from new training examples in the input stream and any held in instance memory. DARLING (Salganicoff, 1993) and AQ-PM are examples of temporal batch learners with partial instance memory. Any batch learning algorithm, such as C4.5 (Quinlan, 1993) or CN2 (Clark & Niblett, 1989), can be used in conjunction with full or no instance memory. However, this choice depends greatly on the problem at hand. Figure 1 displays a classification of selected learning systems based on concept memory and the various types of instance memory. Having described the role of instance and concept memory in learning, we will now discuss partial instance memory learning systems that have appeared in the literature. In the following sections, we will focus on learning systems with instance memory. Thus, for the sake of brevity, we will drop the term instance when referring to such systems. For example, we will use the term partial memory to mean "partial instance memory." 2.1. A Survey of Partial Memory Learning Systems appears to be one of the first partial memory systems. In some sense, it has a minimal partial memory model 4 MARCUS A. MALOOF AND RYSZARD S. MICHALSKI partial instance memory full instance memory memory no instance partial instance memory full instance memory memory no instance partial instance memory full instance memory concept memory no concept memory On-line Learning Systems temporal batch incremental GEM AQ-15c AQ-15c Winnow HILLARY ID4 ITI Figure 1. Learning systems classified by concept and instance memory. because the system keeps only the first positive example. Consequently, it always learns from the one positive example in partial memory and the arriving training examples. HILLARY (Iba et al., 1988) maintains a collection of recent negative examples in partial instance memory. Positive examples may be added to a concept description as disjuncts but are generalized in subsequent learning steps. HILLARY retains negative examples if no concept description covers them; otherwise, it specializes the concept description. Negative examples that are retained may be forgotten later if they are covered by a positive concept description. IB2 (Aha et al., 1991), an instance-based learning method, uses a scheme that, like AQ-PM, keeps a nonconsecutive sequence of training examples in memory. When IB2 receives a new instance, it classifies the instance using the examples currently held in memory. If the classification is correct, the instance is discarded. Conversely, if the classification is incorrect, the instance is retained. The intuition behind this is that if an instance is correctly classified, then we gain nothing by keeping it. This scheme tends to retain training examples that lie at the boundaries of concepts. IB3 extends IB2 by adding mechanisms that cope with noise. DARLING (Salganicoff, 1993) uses a proximity-based forgetting function, as opposed to a time-based or frequency-based function, in which the algorithm initializes the weight of a new example to one and decays the weights of examples within a neighborhood of the new example. When an example's weight falls below a thresh- old, it is removed. DARLING is also an example of a partial memory learning system, since it forgets examples and maintains only a portion of the past training examples. The FLORA2 system (Widmer & Kubat, 1996) selects a consecutive sequence of training examples from the input stream and and uses a time-based scheme to forget those examples in partial memory that are older than a threshold, which is set adaptively. This system was designed to track drifting concepts, so during periods when the system is performing well, it increases the size of the window and keeps more examples. If there is a change in performance, presumably due to some change in the target concepts, the system reduces the size of the window and forgets the old examples to accommodate the new examples from the new target concept. As the system's concept descriptions begin to converge toward the target concepts, the size of the window increases, as does the number of training examples maintained in partial memory. FLORA3 extends FLORA2 by adding mechanisms to cope with changes in con- text. The change of seasons, for instance, is a changing context, and the concept of warm is different for summer and for winter. Temperature is the contextual variable that governs which warm concept is appropriate. FLORA4 extends FLORA3 by adding mechanisms for coping with noise, similar to those used in IB3 (Aha et al., 1991). Finally, the MetaL(B) and MetaL(IB) systems (Widmer, 1997) are based on the naive Bayes and instance-based learning algorithms, respectively. These systems, like FLORA3 (Widmer & Kubat, 1996), can cope with changes in context and use partial memory mechanisms that maintain a linear sequence of training examples, but over a fixed window of time. When the algorithm identifies the context, it uses only those examples in the window relevant for that context. MetaL(IB) uses additional mechanisms, such as exemplar selection and exemplar weighting, to further concentrate on the relevant examples in the window. The FAVORIT system (Krizakova & Kubat, 1992; Kubat & Krizakova, 1992), which extends UNIMEM (Lebowitz, 1987), uses mechanisms for aging and forgetting nodes in a decision tree. Although FAVORIT has no instance memory, we include this discussion because aging and forgetting mechanisms are important for partial memory learners, and because this system uses a third type of forgetting: frequency-based forgetting. FAVORIT uses incoming training examples to add nodes and to strengthen existing nodes in a decision tree. Over time, aging mechanisms gradually weaken the strengths of nodes. If incoming training examples do not reinforce a node's presence in the tree, then the node's score decays until it falls below a threshold. At this point, the algorithm forgets, or removes, the node. Conversely, if incoming training examples continue to reinforce and revise the node, its score increases. If the score surpasses an upper threshold, then the node's score is fixed and remains so. 2.2. A General Framework for Partial Memory Learning Based on an analysis of these systems and on our design of AQ-PM, we developed a general algorithm for inductive learning with partial instance memory, presented in 6 MARCUS A. MALOOF AND RYSZARD S. MICHALSKI 1. Learn-Partial-Memory(Data t 2. Concepts 0 3. PartialMemory 4. for to n do 5. Missed t Find-Missed-Examples(Concepts , Data t PartialMemory Missed t 7. Concepts 8. TrainingSet 0 Concepts t 9. PartialMemory Concepts t ); 11. end . /* Learn-Partial-Memory */ Table 1. Algorithm for partial memory learning. table 1. The algorithm begins with a data source that supplies training examples distributed over time, represented by Data t , where t is a temporal counter. We generalize the usual assumption that a single instance arrives at a time by placing no restrictions on the cardinality of Data t , allowing it to consist of zero or more training examples. This criterion is important because it ultimately determines the structure of time for the learner. 1 By allowing Data t to be empty, the learner can track the passage of time, since the passage of time is no longer associated with the explicit arrival of training examples. By allowing Data t to consist of one or more training examples, the learner can model arbitrary periods of time (e.g., days and weeks) without requiring that a specific number of training examples arrive during that interval. Intuitively, there may be a day when the system learns one thing, but simply because it learns something else does not mean that another day passed. Initially, the learner begins with no concepts and no training examples in partial memory (steps 2 and 3), although it may possess an arbitrary amount of background knowledge. For the first learning step 1), the partial memory learner behaves like a batch learning system. Since it has no concepts and no examples in partial memory, the training set consists of all examples in Data 1 . It uses this set to induce the initial concept descriptions (step 7). Subsequently, the system must determine which of the training examples to retain in partial memory (steps 8 and 9). In subsequent time steps (t ? 1), the system begins by determining which of the new training examples it misclassifies (step 5). The system uses these examples and the examples in partial instance memory to learn new concept descriptions (step 7). As we have seen in the review of related systems, there are several ways to accomplish this. The system could simply memorize the new examples in the training set. It could also induce new concept descriptions from these examples. And finally, it could use the examples in the training set to modify or alter its existing concept descriptions to form new concept descriptions. The precise way in which a particular learner determines misclassified examples (step 5), learns (step 7), selects examples to retain (step 8), and maintains those examples (step depends on the concept description language, the learning meth- ods employed, and the task at hand. Therefore, to ground further discussion, we will describe the AQ-PM learning system. 3. Description of the AQ-PM Learning System AQ-PM is an on-line learning system that maintains a partial memory of past training examples. To implement AQ-PM, we extended the AQ-15c inductive learning system (Wnek et al., 1995), so we will begin by describing this system before delving into the details of AQ-PM. AQ-15c represents training examples using a restricted version of the attributional language VL 1 (Michalski, 1980). Rule conditions are of the form '[' !attribute? `=' !reference? ']', where !attribute? is an attribute used to represent domain objects, and !reference? is a list of attribute values. A rule condition is true if the attribute value of the instance to which the condition is matched is in the !reference?. Decision rules are of the form where D is an expression in the form of a rule condition that assigns a decision to the decision variable, ( is an implication operator, and C is a conjunction of rule conditions. If all of the conditions in the conjunction are true, then the expression D is evaluated and returned as the decision. We can also represent training instances in VL 1 by restricting the cardinality of each reference to one. That is, we can view training instances as VL 1 rules in which all conditions have references consisting of single values. The performance element of AQ-15c consists of a routine that flexibly matches instances with VL 1 decision rules. Decision rules carve out regions in the representation space, leaving some of the space uncovered. If an instance falls into an uncovered region of the space, then, using strict matching technique, the system would classify the instance as unknown, which is important for some applications. Flexible matching involves computing the degree of match between the instance and each decision rule. We can compute this metric in a variety of ways, but, for the experiments discussed here, we computed the degree of match as follows. 2 For each decision class D i , consisting of n conjunctions C j , the degree of match, for an instance is given by (1) is the number of conditions in C j satisfied by the instance, and fi ij is the total number of conditions in C j . yields a number in the range [0, 1] and expresses the proportion of the conditions of a rule an instance matches. A value of zero means there is no match, and a value of one means there is a complete match. The flexible matching routine 8 MARCUS A. MALOOF AND RYSZARD S. MICHALSKI returns as the decision the label of the class with the highest degree of match. If the degree of match falls below a certain threshold, then the routine may report "unknown" or "no match". To learn a set of decision rules, AQ-15c uses the AQ algorithm (Michalski, 1969), a covering algorithm. Briefly, AQ randomly selects a positive training example, known as the seed. The algorithm generalizes the seed as much as possible, given the constraints imposed by the negative examples, producing a decision rule. In the default mode of operation, the positive training examples covered by the rule are removed from further consideration, and this process repeats using the remaining positive examples until all are covered. To implement AQ-PM, we extended AQ-15c by incorporating the features outlined in the partial memory algorithm in table 1. AQ-PM finds misclassified training examples by flexibly matching the current set of decision rules with the examples in Data t (step 5). These "missed" examples are grouped with the examples currently held in partial memory (step and passed to the learning algorithm (step 7). Like AQ-15c, AQ-PM uses the AQ algorithm to induce a set of decision rules from training examples, meaning that AQ-PM operates in a temporal batch mode. To form the new contents of partial memory (step 8), AQ-PM selects examples from the current training set using syntactically modified characteristic decision rules derived from the new concept descriptions, which we discuss further in section 3.1. Finally, AQ-PM may use a variety of maintenance policies (step 9), like time-based forgetting, aging, and inductive support, which are activated by setting parameters. 3.1. Selecting Examples One of the key issues for partial memory learners is deciding which of the new training examples to select and retain. Mechanisms that maintain these examples are also important because some of the examples held in partial memory may no longer be useful. This could be due to the fact that concepts changed or drifted, or that what the system initially thought was crucial about a concept is no longer important to represent explicitly, since the current concept descriptions implicitly capture this information. Returning to AQ-PM, we used a scheme that selects the training examples that lie on the boundaries of generalized concept descriptions. We will call these examples extreme examples. Each AQ-PM decision rule is an axis-parallel hyper-rectangle in discrete n-dimensional space, where n is the number of attributes used to represent domain objects. Therefore, the extreme examples could be those that lie on the surfaces, the edges, or the corners of the hyper-rectangle covering them. For this study, we chose the middle ground and retained those examples that lay on the edges of the hyper-rectangle, although we have considered and implemented the other schemes for retaining examples (Maloof, 1996). Referring to figure 2, we see a portion of a discrete version of the iris data set 1936). We took the original data set from the UCI Machine Learning Repository (Blake, Keogh, & Merz, 1998) and produced a discrete version using the SCALE implementation (Bloedorn, Wnek, Michalski, & Kaufman, 1993) of setosa example versicolor example Figure 2. Visualization of the setosa and versicolor training examples.plsetosa example versicolor example010101010101010101010101010101010101010101010101010101010101010101010101010101010100110101010101 sw setosa concept versicolor concept Figure 3. Visualization of the setosa and versicolor concept descriptions with overlain training examples. the ChiMerge algorithm (Kerber, 1992). Shown are examples of the versicolor and setosa classes with each example represented by four attributes: petal length (pl), petal width (pw), sepal length (sl), and sepal width (sw). To find extreme or boundary training examples, AQ-PM uses characteristic decision rules, which specify the common attributes of domain objects from the same class (Michalski, 1980). These rules consist of all the domain attributes and their A. MALOOF AND RYSZARD S. MICHALSKIplsw setosa example versicolor example Figure 4. Visualization of the setosa and versicolor extreme examples. values for the objects represented in the training set, and form the tightest possible hyper-rectangle around a cluster of examples. Returning to our example, figure 3 shows the characteristic rules induced from the training examples pictured in figure 2. AQ-PM syntactically modifies the set of characteristic rules so they will match examples that lie on their boundaries and then uses a strict matching technique to select the extreme examples. Although AQ-PM uses characteristic rules to select extreme examples, it can use other types of decision rules (e.g., discriminant rules) for classification. Figure 4 shows the examples retained by the selection algorithm, which are those examples that lie on the edges of the hyper-rectangles expressed by characteristic decision rules. Theorem 1 states the upper bound for the number of examples retained by AQ- PM and its lesioned counterpart. The lesioned version of AQ-PM, which we describe formally in the next section, is equivalent to a temporal batch learning system with full instance memory. For the best case, the partial memory learner will retain fewer training examples than the lesioned counterpart by a multiplicative factor. For the worst case, or the lower bound, the number of examples maintained by the partial memory learner will be equal to that of the lesioned learner. This follows from the proof of Theorem 1 and occurs when the training set consists only of examples that lie on the edges of a characteristic concept description. Theorem 1 For the characteristic decision rule D ( C induced from training examples drawn from an n-dimensional discrete representation space, the number of training examples retained by the partial memory learner is (jreference k while its lesioned counterpart will retain Y jreference k j: Proof: Let D ( C be a characteristic decision rule induced from training examples drawn from an n-dimensional discrete representation space !. Let c k be the kth condition in C. By definition, the following three are numerically equivalent: 1. The dimensionality n of !. 2. The number of conditions c 2 C. 3. The number of attributes forming !. For the partial memory learner, c k expresses the kth dimension in the hyper-rectangle and will match jreference k j training examples along each edge of the kth dimension. Furthermore, c k corresponds to 2 n\Gamma1 edges in the kth dimension of the hyper-rectangle realized by D ( C. Therefore, the number of training examples matched by c k is If we were to compute this number for would overcount the training examples that lie at the corners of the hyper-rectangle. Therefore, we must subtract the two training examples that lie at the endpoints of each edge of the hyper-rectangle, yielding (jreference k But now this undercounts the number of training examples because it excludes all of the training examples that lie at the corners. Since there are 2 n corners in an n-dimensional hyper-rectangle, the total number of examples matched is For the lesioned learner, each attribute value of a training example will map to a corresponding value in a condition c k , by definition of a characteristic concept description. For a set of training examples, each attribute will result in a condition c k such that the number of attribute values in the condition's reference is equal to the number of unique values the attribute takes. Therefore, the number of training examples maintained by the lesioned learner is equal to Y jreference k j: 12 MARCUS A. MALOOF AND RYSZARD S. MICHALSKI 1. AQ-BL(Data t 2. Concepts 0 3. TrainingSet 4. for to n do 5. TrainingSet t 6. Concepts t 7. 8. end . /* AQ-BL */ Table 2. Algorithm for the lesioned version of AQ-PM, AQ-Baseline (AQ-BL). 3.2. Forgetting Mechanisms Forgetting mechanisms are important for partial memory learners for two reasons. First, if the learner selects examples that lie on the boundaries of concept descrip- tions, as AQ-PM does, and these boundaries change, then there is no reason to retain the old boundary examples. The new extreme examples do the important work of enforcing the concept boundary, so the learner can forget the old ones. Second, if the learner must deal with concept drift, then forgetting mechanisms are crucial for removing irrelevant and outdated examples held in partial memory. As we will see in the experimental section, when concepts change suddenly, the learner must cope with the examples held in partial memory from the previous target concept. In the context of the new target concept, many of these examples will be contradictory, and forgetting them is imperative. In AQ-PM, there are two types of forgetting: explicit and implicit. Explicit forgetting occurs when examples in partial memory meet specific, user-defined cri- teria. In the current implementation, AQ-PM uses a time-based forgetting function to remove examples from partial memory that are older than a certain age. Implicit forgetting occurs when examples in partial memory are evaluated and deemed useless because they no longer enforce concept boundaries. When computing partial memory, the basic algorithm (table 1) evaluates the training examples currently held in partial memory and those misclassified by the current concept descriptions (step 8). Consequently, it repeatedly evaluates the extreme examples and determines if they still fall on a concept boundary, which gives rise to an implicit forgetting process. That is, if the learning algorithm generalizes a concept description such that a particular extreme example no longer lies on the concept boundary, then it forgets the example. We call this an implicit forgetting process because there is no explicit criterion for removing examples (e.g., remove examples older than fifty time steps). R R R Size Red Green Blue Shape Color R R R Size Red Green Blue Shape Color R R R Size Red Green Blue Shape Color a. Target concept for time steps 1-39. b. Target concept for time steps 40-79. c. Target concept for time steps 80-120. Figure 5. Visualization of the STAGGER Concepts. 4. Experimental Results In this section, we present a series of experimental results from a lesion study (Ki- bler & Langley, 1990), in which we used AQ-PM for three problems. To produce the lesioned version of AQ-PM, we simply disabled its partial memory mechanisms, resulting in a system equivalent to a temporal batch learner with full instance mem- ory. We present this learner formally in table 2 and will refer to it as AQ-Baseline (AQ-BL). We also included IB2 (Aha et al., 1991) for the sake of comparison, which is a instance-based learner with a partial memory model. The first problem, a synthetic problem, is referred to as the "STAGGER Con- cepts" (Schlimmer & Granger, 1986). It has become a standard benchmark for testing learning algorithms that track concept drift. We derived the remaining two data sets from real-world problems. The first problem entails detecting blasting caps in X-ray images of airport luggage (Maloof & Michalski, 1997), and the second involves using learned profiles of computing behavior for intrusion detection (Maloof & Michalski, 1995). We chose these real-world problems because they require on-line learning and likely involve concepts that change over time. For ex- ample, computing behavior changes as individuals move from project to project or from semester to semester. The appearance of visual objects can also change due to deformations of the objects or to changes in the environment. For these ex- periments, the independent variable is the learning algorithm, and the dependent variables are predictive accuracy and the number of training examples maintained. For both of these measures, we computed 95% confidence intervals, which are also presented. Detailed results for learning time and concept complexity for these and other problems can be found elsewhere (Maloof, 1996). 14 MARCUS A. MALOOF AND RYSZARD S. MICHALSKI406080100 Predictive Accuracy Time Step AQ-BL Figure 6. Predictive accuracy for AQ-PM, AQ-BL, and IB2 for the STAGGER Concepts. 4.1. The STAGGER Concepts The STAGGER Concepts (Schlimmer & Granger, 1986) is a synthetic problem in which the target concept changes over time. Three attributes describe domain ob- jects: size, taking on values small, medium, and large; color, taking on values red, green, and blue; and, shape, taking on values circle, triangle, and rectangle. Conse- quently, there are 27 possible object descriptions (i.e., events) in the representation space. The presentation of training examples lasted for 120 time steps with the target concept changing every 40 steps. The target concept for the first 39 steps was red]. For the next 40 time steps, the target concept was And for the final 40 time steps, the target concept was [size = medium - large]. The visualization of these target concepts appears in figure 5. At each time step, a single training example and 100 testing examples were generated randomly. 3 For the results presented, we conducted 60 learning runs using IB2, AQ-PM, and AQ-BL, the lesioned version of AQ-PM. Referring to figure 6, we see the predictive accuracy results for IB2, AQ-PM, and AQ-BL for the STAGGER Concepts. IB2 performed poorly on the first target concept and worse on the final two (53\Sigma2.7% and 62\Sigma4.0%, respectively). Conversely, AQ-PM and AQ-BL achieved high predictive accuracies for the first target concept (99\Sigma1.0% and 100\Sigma0.0%, respectively). However, once the target concept changed at time step 40, AQ-BL was never able to match the partial memory learner's predictive accuracy because the former was burdened with examples irrelevant to the new target concept. This experiment illustrates the importance of forgetting mechanisms. AQ-PM was less burdened by past examples because it kept fewer examples in memory and forgot those held in memory after a fixed period of time. AQ-PM's predictive accuracy on the second target concept was Examples Maintained Time Step AQ-BL Figure 7. Memory requirements for AQ-PM, AQ-BL, and IB2 for the STAGGER Concepts. 89\Sigma3.38%, while AQ-BL's was 69\Sigma3.0%. For the third target concept, AQ-PM achieved 96\Sigma1.8% predictive accuracy, while AQ-BL achieved 71\Sigma3.82%. The predictive accuracy results for AQ-PM are comparable to those of STAGGER (Schlimmer & Granger, 1986) and of the FLORA systems (Widmer & Kubat, 1996) with the following exceptions. On the first target concept, AQ-PM did not converge as quickly as the FLORA systems but ultimately achieved similar predictive accuracy. On the second target concept, AQ-PM's convergence was like that of the FLORA systems, but it performed about 10% worse on the test cases. Performance (i.e., slope and asymptote) on the third target concept was similar. Turning to memory requirements, shown in figure 7, we see that the partial memory learners, AQ-PM and IB2, maintained far fewer training examples than AQ-BL. Without the partial memory mechanisms, the baseline learner, AQ-BL, simply accumulated more and more examples. Intuitively, this is an inefficient and inadequate policy when learning changing concepts. Yet, as IB2's predictive accuracy showed, selection mechanisms alone are not enough. Taking a closer look at the memory requirements for AQ-PM and IB2 (figure 8), we see that the number of examples each learner maintained increases because of example selection mechanisms. Overall, IB2 maintained fewer training examples than AQ-PM, but this savings cannot mitigate its poor predictive accuracy. During the first 40 time steps, for instance, both learners accumulated examples. As each achieved acceptable predictive accuracies, the number of examples maintained stabilized. Once the concept changed at time step 40, both learners increased the number of examples held in partial memory to retain more information about the new concept. The increases in IB2's memory requirements occurred because it adds new examples only if they are misclassified by the examples currently held in mem- ory. When the target concept changed, most of the new examples were misclassified and, consequently, added to memory. Because IB2 kept all of the examples related to the previous target concept, predictive accuracy suffered on this and the final Examples Maintained Time Step Figure 8. Memory requirements for AQ-PM and IB2 for the STAGGER Concepts. Blasting Caps Figure 9. Example of X-ray image used for experimentation. target concept. Although AQ-PM also increased the number of examples held in memory, it used an explicit forgetting process to remove outdated and irrelevant training examples after a fixed period of fifty time steps, which proved crucial for learning these concepts. We cannot compare AQ-PM's memory requirements to STAGGER's, since the latter does not maintain past training examples, but we can indirectly compare it to one run of FLORA2 (Widmer & Kubat, 1996). Recall that the size of the representation space for the STAGGER problem is only 27 examples. At time step 50, FLORA2 maintained about 24 examples, which is 89% of the representation space. At the same time step, AQ-PM maintained only 10.11 examples, on average, which is only 37% of the representation space. Over the entire learning run, FLORA2 kept an average of 15 examples, which is 56% of representation space. AQ-PM, on the other hand, maintained only 6.6 examples, on average, which is only 24% of the space. 4.2. Blasting Cap Detection Problem The blasting cap detection problem involves detecting blasting caps in X-ray images of airport luggage (Maloof & Michalski, 1997). The 66 training examples for this experiment were derived from 5 images that varied in the amount of clutter in the luggage and in the position of the bag relative to the X-ray source. Figure 9 shows a typical X-ray image from the collection. Positive and negative examples of blasting caps were represented using 27 intensity, shape, and positional attributes (Maloof, Duric, Michalski, & Rosenfeld, 1996). We computed eleven attributes for the blob produced by the heavy metal explosive near the center of the blasting cap. We also computed these same eleven attributes for the rectangular region produced by the blasting cap's metal tube. Finally, we used five attributes to capture the spatial relationship between the blob and the rectangular region. These real-valued attributes were scaled and discretized using the SCALE implementation (Bloedorn et al., 1993) of the ChiMerge algorithm (Kerber, 1992). 4 The 15 most relevant attributes were then selected using the PROMISE measure (Baim, 1988). The resulting attributes for the blob were maximum intensity, average intensity, length of a bounding rectangle, and three measures of compactness. For the rectangular region, the selected attributes were length, width, area, standard deviation of the intensity, and three measures of compactness. And finally, the remaining spatial attributes were the distance between the centroids of the blob and rectangle, and the component of this distance that was parallel to the major axis of a fitted ellipse. We randomly set aside 10% of the original data as a testing set. The remaining 90% was partitioned randomly and evenly into 10 data sets (i.e., Data t , for We then conducted an experimental comparison between IB2, AQ-PM, and the lesioned version of the system, AQ-BL. For each learning run, we presented the learners with Data t and tested the resulting concept descriptions on the testing set, making note of predictive accuracy and memory requirements. We conducted 100 learning runs, in which we randomly generated a new test set and new data sets Data t , for averaging the performance metrics over these 100 runs. Figure shows the predictive accuracy results for the blasting cap detection problem. AQ-PM's predictive accuracy was consistently lower than AQ-BL's, which learned from all of the available training data at each time step. When learning stops at time step 10, AQ-PM's predictive accuracy was 7% less than that of the lesioned learner, AQ-BL (81\Sigma3.4% vs. 88\Sigma2.8%). IB2 did not perform well on this task and ultimately achieved a predictive accuracy of 73\Sigma3.8%. A notable decrease in memory requirements has to be measured against AQ-PM's loss in predictive accuracy, as shown by figure 11. When learning ceased at time step 10, the baseline learner maintained the entire training set of 61\Sigma0.0 examples, while the partial memory learner kept 18\Sigma0.5 training examples, on average, which is roughly 30% of the total number of examples. IB2 retained slightly more examples in partial memory than AQ-PM: 25\Sigma0.6. A. MALOOF AND RYSZARD S. MICHALSKI5565758595 Predictive Accuracy Time Step (t) AQ-BL Figure 10. Predictive accuracy for AQ-PM, AQ-BL, and IB2 for the blasting cap detection Examples Maintained Time Step (t) AQ-BL Figure 11. Memory requirements for AQ-PM, AQ-BL, and IB2 for the blasting caps detection problem. 4.3. Computer Intrusion Detection Problem For the computer intrusion detection problem, we must learn profiles of users' computing behavior and use these profiles to authenticate future behavior. Learning descriptions of intrusion behavior is problematic, since adequate training data is difficult, if not impossible, to collect. Consequently, we chose to learn profiles for each user, assuming that misclassification means that a user's profile is inadequate or that an unauthorized person is masquerading as the user in question. Most existing intrusion detection systems make this assumption. The data for this experiment were derived from over 11,200 audit records collected for 9 users over a 3 week period. We first parsed each user's computing activity Predictive Accuracy Time Step (t) AQ-BL Figure 12. Predictive accuracy for AQ-PM, AQ-BL, and IB2 for the intrusion detection problem. from the output of the UNIX acctcom command (Frisch, 1995) into sessions by segmenting at logouts and at periods of idle time of twenty minutes or longer. This resulted in 239 training examples. We then selected seven numeric audit metrics: CPU time, real time, user time, characters transferred, blocks read and CPU factor, and hog factor. Next, we represented each of the seven numeric measures for a session, which is a time series, using the maximum, minimum, and average values, following Davis (Davis, 1981). These 21 real and integer attributes were scaled and discretized using the SCALE implementation (Bloedorn et al., 1993) of the ChiMerge algorithm (Kerber, 1992). Finally, using the PROMISE measure (Baim, 1988), we selected the 13 most relevant attributes: average and maximum real time, average and maximum system time, average and maximum user time, minimum and average characters transferred, average blocks transferred, average and maximum CPU factor, and average and maximum hog factor. The experimental design for this problem was identical to the one we used for the blasting cap problem. Referring to figure 12, we can see the predictive accuracy results for AQ-PM, AQ-BL, and IB2 for the intrusion detection problem. AQ-PM's predictive accuracy was again slightly lower than AQ-BL's. When learning stopped at time step 10, AQ-PM's accuracy was 88\Sigma1.6%, while AQ-BL's was 93\Sigma1.2%, a difference of 5%. IB2 fared much better on this problem than on previous ones. When learning ceased, IB2's predictive accuracy was a slightly better than AQ- PM's: 89\Sigma1.3%, although this result was not statistically significant (p ! :05). Figure 13 shows the memory requirements for each learner for this problem. AQ-PM maintained notably fewer training examples than its lesioned counterpart. When learning ceased at time step 10, the baseline learner, AQ-BL, maintained examples, while AQ-PM maintained 64\Sigma1.0 training examples, which is roughly 29% of the total number of examples. IB2 maintained even fewer examples than AQ-PM. When learning stopped, IB2 held roughly 52\Sigma0.8 examples in partial memory, which was slightly fewer than the number held by AQ-PM. A. MALOOF AND RYSZARD S. MICHALSKI50150250 Examples Maintained Time Step (t) AQ-BL Figure 13. Memory requirements for AQ-PM, AQ-BL, and IB2 for the intrusion detection problem. 4.4. Summary The lesion study comparing AQ-PM and AQ-BL suggested that the mechanisms for selecting extreme examples notably reduced the number of instances maintained in partial memory at the expense of predictive accuracy. When concepts changed, AQ- PM relied on forgetting mechanisms to remove outdated and irrelevant examples held in memory. Recall that AQ-PM can use two types of forgetting: implicit and explicit. Explicit mechanisms proved crucial for the STAGGER Concepts, but the implicit forgetting mechanisms, in general, had little effect, an issue we explore further in the next section. The direct comparison to IB2 using the STAGGER Concepts further illustrated the importance of forgetting policies, as it was apparent that the example selection mechanisms alone did not guarantee acceptable predictive accuracy when concepts changed. On the other hand, when concepts were stable, as was the case with the computer intrusion detection and blasting cap detection problems, forgetting mechanisms played a less important role than the selection mechanisms. Moreover, we predict that the differences in performance between AQ-PM and IB2 on these problems were due to inductive bias rather than a limitation of IB2's example selection method. This would explain why IB2 performed well on the intrusion detection problem but performed poorly on the blasting cap detection problem. Indeed, AQ-PM and IB2 used similar selection methods, and experimental results showed that each maintained roughly the same number of examples in memory. Regarding the indirect comparison to the FLORA systems, AQ-PM performed as well on two of the three STAGGER Concepts, and it appears to have maintained fewer training examples in partial memory. The difference in memory requirements is due to how the learners selected examples from the input stream. The FLORA systems kept a sequence of examples of varying length from the input stream, and, as a result, partial memory likely contained duplicate examples. This would be PARTIAL MEMORY LEARNING 21 especially true for a problem like the STAGGER Concepts in which we randomly draw 120 examples from a representation space consisting of 27 domain objects. Conversely, AQ-PM retained only those examples that lay on the boundaries of concept descriptions and, consequently, would not retain duplicate examples or examples from the interior of the concept. We claim that AQ-PM was able to achieve comparable accuracy while maintaining fewer examples in partial memory because the selected examples enforced concept boundaries and, hence, were of high utility. The two systems do use different concept description languages: AQ-PM uses VL 1 , which is capable of representing DNF concepts, whereas the FLORA systems use a conjunctive description language. However, upon analyzing the STAGGER Concepts, we concluded that it is unlikely that representational bias accounted for the differences or similarities in predictive accuracy. As we noted, AQ-PM did not fare as well as the FLORA systems on the second of the three STAGGER Concepts. Transitioning concept descriptions from the first target concept to the second is the most difficult because it is here that there is the most change in the representation space. It is here that there is the most overlap between the old negative concept and the new positive concept, as depicted in figure 5. AQ-PM should have had an advantage over an incremental learning system in this situation because it operates in a temporal batch mode. Since AQ-PM replaces old concept descriptions with new ones, it would not be burdened by the information about the old concepts encoded in the concept descriptions. But, because AQ-PM operates in a temporal batch mode, the only cause for its fair performance on the second concept is the examples held in partial instance memory. As we have discussed, AQ-PM used a simple forgetting policy that removed examples older than fifty time steps. The FLORA systems, on the other hand, used an adaptive forgetting window, which, in this case, more efficiently discarded examples after the concept changed and may account for the difference in performance on the second concept. Making the transition between the second and third STAGGER Concepts is easier than transitioning between the first and second because there is more overlap between the old positive concept description and the new positive concept description (see figure 5). AQ-PM's static forgetting policy worked better during this transition than during the previous one, and the learner achieved predictive accuracies that were comparable to the FLORA systems. 5. Discussion Intelligent systems need induced hypotheses for reasoning because they generalize the system's experiences. We anticipate that manipulating some concept descriptions to cope with changing concepts will slow a system's reactivity. By keeping extreme examples in addition to concept descriptions, the learner maintains a rough approximation of the current concept descriptions and, consequently, is able to both reason and react efficiently. 22 MARCUS A. MALOOF AND RYSZARD S. MICHALSKI When learning stable concepts, we expect slight changes in the positions of concept boundaries. The extreme examples, in this case, document the past and provide stability. On the other hand, when examples arrive that radically change concept boundaries, then the examples held in memory that no longer fall on concept boundaries are removed and replaced with examples that do. This process is actually happening in both situations, but to different degrees. The extreme examples provide stability when it is needed. Yet, they do not hinder the learner because forgetting mechanisms ensure that stability does not result in low reac- tivity. For systems to succeed in nonstationary environments, they must find a balance between stability and reactivity. In the sections that follow, we examine a variety of issues related to this study and, more globally, to partial memory learning and nonstationary concepts. In particular, we examine experimental results from other aspects of our study (Mal- oof, 1996), such as learning time, concept complexity, other methods of example selection, and incremental learning. Then, after discussing the some of the current limitations of this work, we consider directions for the future. 5.1. Learning Time The experimental results from the lesion study showed that the example selection method greatly reduced the number of training examples maintained when compared to the baseline learner. Because the number of training examples affects run time of the algorithms investigated, reducing the number of training examples maintained resulted in notable decreases in learning time. For the intrusion detection problem, as an example, at time step 10, AQ-PM's learning time was 36.7 seconds, while AQ-BL's was 55.6 seconds, 5 meaning that AQ-BL was 52% slower than AQ-PM for this problem. 5.2. Complexity of Concept Descriptions We also examined complexity of induced concept descriptions in terms of conditions and rules. AQ-PM produced concepts descriptions that were as complex or simpler than those produced by AQ-BL. The degree to which descriptions induced by AQ- PM were simpler was not as notable as other measures, such as learning time and memory requirements. Table 3 shows decision rules from the intrusion detection problem that AQ-PM induced for two computer users, daffy and coyote. 6 The first rule, for daffy, consists of one condition involving the average system time attribute, which must fall in the high range of [25352.53: : :63914.66]. 7 The class label "daffy" is assigned to the decision variable if the average system time for one of daffy's sessions falls into this range. Therefore, daffy's computing use is characterized by a considerable consumption of system time. The weights appearing at the end of the rules are strength measures. The t- weight indicates how many total training examples the rule covers. The u-weight indicates how many unique training examples the rule covers. Rules may overlap, Table 3. Examples of AQ-PM rules for daffy and coyote's computing behavior. (t-weight: 10, u-weight: 10) (t-weight: 7, u-weight: (t-weight: 4, u-weight: so two rules can cover the same training example. The rule for daffy's computing use is strong, since it alone covers all of the available training examples. The next two rules characterize coyote's behavior, whose use of computing resources is low, especially compared to daffy's. 5.3. Other Example Selection Methods The selection method used for this study retained the examples that lay on the edges of the hyper-rectangle expressed by a decision rule. We alluded to similar methods that keep the examples that lie on the corners and surfaces of these hyper-rectangles. Experimental results from a previous study (Maloof, 1996) for the blasting caps and intrusion detection problems showed that keeping the examples that lie on the corners of the hyper-rectangle, as opposed to those on the edges, resulted in slightly lower predictive accuracy and slightly reduced memory requirements. We anticipate that a method retaining the examples that lie on surfaces of the hyper-rectangle will slightly improve predictive accuracy and slightly increase memory requirements when compared to the edges method. From these results, we can conclude that as AQ-PM keeps more and more examples in partial memory, its predictive accuracy will converge to that of the full memory learner. 5.4. Adding Examples to Partial Memory In this paper, we have discussed a reevaluation strategy for maintaining examples in partial memory. Using this scheme, AQ-PM uses new concept descriptions to test if the misclassified examples and all of the examples in partial memory lie on concept boundaries. It retains those examples that do and removes those that do not. And, as we discussed, this gives rise to an implicit forgetting process. An alternative scheme is to accumulate examples by computing partial memory using only the misclassified examples and by adding the resulting extreme examples to those already in partial memory. Therefore, once an example is placed in partial memory, it remains there until removed by an explicit forgetting process. For the problems discussed here and elsewhere (Maloof, 1996), we did not see notable differences in performance between the reevaluation policy and the accumulation policy. For example, one would expect that the reevaluation policy would work best A. MALOOF AND RYSZARD S. MICHALSKI for dynamic problems, like the STAGGER Concepts, and the accumulation policy would work best for more stable problems, like the blasting cap problem. To date, our experimental results have not supported this intuition. 5.5. Incremental Learning In the basic algorithm, we used a temporal batch learning method (table 1, step 7). We have also examined variants using incremental learning algorithms (Maloof, 1996), meaning that the system learns new concept descriptions by modifying the current set of descriptions using new training examples and those examples in partial memory. We have investigated this notion using two incremental algorithms: the GEM algorithm (Reinke & Michalski, 1988), a full instance memory technique, and the AQ-11 algorithm (Michalski & Larson, 1983), a no instance memory tech- nique. We chose these algorithms because their inductive biases are most similar to that of AQ-PM: both use the VL 1 representation language (Michalski, 1980) and the AQ induction algorithm (Michalski, 1969). Experimental results for the computer intrusion detection and the blasting cap detection problems show evidence that the incremental learning variants of AQ-PM lose less predictive accuracy than AQ-PM using a temporal batch learning method. We can infer that the incremental learning variants perform better because the concepts themselves encode information that is lost when using a temporal batch method. The incremental learning methods take advantage of this information, whereas the temporal batch method does not. Moreover, we intend to evaluate these incremental learning methods using the STAGGER problem to determine how they perform. We may find that the incremental methods perform worse because they encode too much information about the past and reduce the learner's ability to react to changing environments. 5.6. Current Limitations Many of the current limitations of the approach stem from assumptions the system makes. For example, the system assumes the given representation space is adequate for learning. That is, it is currently incapable of constructive induction (Michalski, 1983). Also, it assumes that the context in which training examples are presented is stationary. Hence, it cannot learn contextual cues (Widmer, 1997). Although we did not implement explicit mechanisms to handle noise, there has been work on such mechanisms in contexts similar to these (Widmer & Kubat, 1996). In general, the selection methodology works best for ordered attributes, taking advantage of their inherent structure. Consequently, for purely nominal domains, the method selects all training examples, since, for each training example, there exists a projection of the representation space in which the example lies on a concept boundary. 5.7. Future Work Much of the current research assumes that the representation space in which concepts drift or contexts change is adequate for learning. If an environment is nonsta- tionary, then the representation space itself could also change. Learners typically detect concept change by a sudden drop in predictive accuracy. If the learner is subsequently unable to achieve acceptable performance, then it may need to apply constructive induction operators in an effort to improve the representation space for learning. To this end, one may use a program that automatically invokes constructive induction, like AQ-18 (Bloedorn & Michalski, 1998; Kaufman & Michalski, 1998). Another interesting problem for future research is how to detect good and bad types of change. Consider the problem of intrusion detection. We need systems that are flexible enough to track changes in a user's behavior; otherwise, when changes do occur, the system's false negative rate will increase. Yet, if intrusion detection systems are too flexible, then they may perceive a cracker's behavior as a change in the true user's behavior and adapt accordingly. We envision a two-layer system that learns a historical profile of a user's computing behavior and learns how that historical profile has changed over time. If a user's computing behavior no longer matches the historical profile, then the system would determine if the type of change that occurred is plausible for that user. If it is not, then the system would issue an alert. Such systems should prove to be more robust and should perform with lower false negative rates. From the standpoint of the methodology, we would like to investigate policies that let the learner function when instances arrive without feedback. Producing decisions without feedback is not necessarily problematic, but, when feedback does arrive after a period of time, the system may realize that many of its past decisions were wrong. Naturally, the simplest policy is to forget past decisions, in which case the learner would never realize that it had made mistakes. Certain applications, like intrusion detection, require systems to be more accountable. However, even though a system may remember past decisions, when it realizes that some were wrong, perhaps it should only issue an alert. Alternatively, the system may seek feedback for the events that led to the incorrect decisions and relearn from them. From the perspective of the implementation, a fruitful exercise would be to implement the example selection method using another concept representation, like decision trees. There is nothing inherent to the method that limits it to decision rules. We could apply the method to any symbolic representation that uses linear attributes. We could also implement other example selection and maintenance schemes as well as mechanisms for coping with noise and contextual changes, but these latter areas, as we have commented, have been well-studied elsewhere. Finally, there are several opportunities for additional experimental studies. Here, we investigated concepts that change suddenly. Changes in concepts could also occur more gradually. If we think of concepts as geometric objects in a space, then they could change in shape, position, and size. Consequently, concepts could grow (i.e., change in size, but not in position and shape), move (i.e., change in position, 26 MARCUS A. MALOOF AND RYSZARD S. MICHALSKI but not in shape and size), deform (i.e., change in shape, but not in position and size), and so on. Although synthetic data sets like these provide opportunities to investigate specific research hypotheses, we are also interested in concept drift in real-world applications like intrusion detection and agent applications (e.g., an agent for prioritizing e-mail). 6. Conclusion Partial memory learning systems select and maintain a portion of the past training examples and use these examples for future learning episodes. In this paper, we presented a selection method that uses extreme examples to enforce concept bound- aries. The method extends previous work by using induced concept descriptions to select a nonconsecutive sequence of examples from the input stream. Reevaluating examples held in partial memory and removing them if they no longer enforce concept boundaries results in an implicit forgetting process. This can be used in conjunction with explicit forgetting mechanisms that remove examples satisfying user-defined criteria. Experimental results from a lesion study suggested that the method notably reduces memory requirements with small decreases in predictive accuracy for two real-world problems, those of computer intrusion detection and blasting cap detection in X-ray images. For the STAGGER problem, AQ-PM performed comparably to STAGGER and the FLORA systems. Finally, a direct comparison to IB2 revealed that AQ-PM provided comparable memory requirements and often higher predictive accuracy for the problems considered. Acknowledgments We would like to thank Eric Bloedorn, Ren'ee Elio, Doug Fisher, Wayne Iba, Pat Langley, and the anonymous reviewers, all of whom provided suggestions that improved this work and earlier versions of this paper. We would also like to thank the Department of Computer Science at Georgetown University, the Institute for the Study of Learning and Expertise, and the Center for the Study of Language and Information at Stanford University for their support of this work. This research was conducted in the Machine Learning and Inference Laboratory at George Mason University. The laboratory's research has been supported in part by the National Science Foundation under grants IIS-9904078 and IRI-9510644, in part by the Advanced Research Projects Agency under grant N00014-91-J-1854, administered by the Office of Naval Research, under grant F49620-92-J-0549, administered by the Air Force Office of Scientific Research, and in part by the Office of Naval Research under grant N00014-91-J-1351. Notes 1. The structure of time is not crucially important for this paper, but we do feel that this issue warrants a more sophisticated treatment. 2. AQ-15c has other methods for computing the degree of match, but, based on empirical analysis, we found that this method worked best for the problems in this study. 3. For the first time step, we generated two random examples, one for each class. 4. We ran IB2 using the unscaled, continuous data. 5. We conducted these experiments using a C implementation of AQ-PM running on a Sun 2. 6. 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Selective induction learning system AQ15c: the method and user's guide (Reports of the Machine Learning and Inference Laboratory No. --TR A Method for Attribute Selection in Inductive Learning Systems Incremental learning of concept descriptions: A method and experimental results Instance-Based Learning Algorithms Redundant noisy attributes, attribute errors, and linear-threshold learning using winnow Essential system administration An Incremental Deductive Strategy for Controlling Constructive Induction in Learning from Examples <italic>FAVORIT</italic> Forgetting and aging of knowledge in concept formation C4.5: programs for machine learning Learning in the presence of concept drift and hidden contexts Tracking Context Changes through Meta-Learning Progressive partial memory learning Data-Driven Constructive Induction The CN2 Induction Algorithm Experiments with Incremental Concept Formation A Method for Partial-Memory Incremental Learning and its Application to Computer Intrusion Detection --CTR Chi-Chun Huang, A novel gray-based reduced NN classification method, Pattern Recognition, v.39 n.11, p.1979-1986, November, 2006 Steffen Lange , Gunter Grieser, Variants of iterative learning, Theoretical Computer Science, v.292 n.2, p.359-376, 27 January Steffen Lange , Gunter Grieser, On the power of incremental learning, Theoretical Computer Science, v.288 n.2, p.277-307, 16 October 2002 Antonin Rozsypal , Miroslav Kubat, Association mining in time-varying domains, Intelligent Data Analysis, v.9 n.3, p.273-288, May 2005 Marcus A. 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on-line concept learning;partial memory models;concept drift

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Diamond Quorum Consensus for High Capacity and Efficiency in a Replicated Database System.

Many quorum consensus protocols have been proposed for the management of replicated data in a distributed environment. The advantages of a replicated database system over a non-replicated one include high availability and low response time. We note further that the multiple sites can act as multiple agents so that at any time, multiple requests can be handled in parallel. This feature leads to the desirable consequence of high workload capacity. In this paper, we define a new metric of read-capacity for this feature. We propose a new protocol called diamond quorum consensus which has two major properties that are superior to the previous protocols of majority, tree, grid, and hierarchical quorum consensus: (1) it has the highest read-capacity, (2) it has the smallest optimal read quorum size of 2. We show that these two features are achievable without jeopardizing the availability. The small quorum size is a significant feature because it relates to the messaging cost. Few previous work on quorum consensus has discussed the handling of partition failure, which in many cases will depend on the quorum consensus protocol, we show how we can use the generalized virtual partition protocol to handle partition failure in the case of diamond quorum consensus.

Introduction A replicated database system is built in a distributed environment with multiple sites, in which copies of data are stored at multiple sites. The main motivations of a replicated database are to improve the reliability and performance. By storing data at multiple sites, the database system can continue to operate even if some sites have failed. Also, multiple sites make it possible to support multiple concurrent operations at different sets of sites. A major problem with replicated data is that we need to ensure data consistency in spite of concurrent operations. A reliable concurrency control protocol is necessary to synchronize the user transactions in order to maintain data validity. For example, two write operations from two different user transactions must not be allowed to simultaneously update different copies of a data object. In order to achieve this kind of synchronization among multiple copies, additional communication and processing costs are incurred. We have a problem of how to do the synchronization in such a way that the cost can be minimized. One well-studied approach for this management of replicated data is to use certain sets of replicas, called read and write quorums, for read and write operations, respectively. Any write quorum has at least one copy in common with every read quorum and every write quorum. For example, write quorums could be all sets containing a majority of copies [18]. For better performance, some logical structure is imposed on the network, and the quorums are chosen under the consideration of such structures. Such logical structures include the tree [2, 1, 3, 20] and grid [4, 11] structures. A geometric approach for dealing with logical structures is proposed in [12]. A number of metrics have been used for evaluating such a protocol, they include the following: 1. Availability 2. Quorum size (best case and worst case) 3. Is the algorithm fully distributed? 4. Cost of one node failure in the worst case 5. Message overhead 6. Communication delay The first metric on availability has been studied extensively but its significance has decreased as systems tend to be more and more reliable, a protocol needs only to have high availabilities under a reasonably reliable environment. Metrics (2) to are listed in [10]. The quorum size is also a well-studied issue because it can be related to the messaging cost. From [10], the N algorithm in [13] is a fully distributed algorithm that achieves the smallest quorum size. The protocol in [16] has a bigger quorum size but remedies the availability and other problems of [13]. The third metric is whether all copies assume equal burden for synchronization, for example, the tree quorum is not fully distributed because the nodes higher in the tree are given more share of the burden in order to achieve a small quorum size. However, we shall argue below that this metric is problematic. The fourth metric is the impact that a single node failure has on the size of a quorum. For example, for the N algorithm of [13], if one of the sites in a chosen quorum fails, then in the worst case, a new quorum must be formed with a totally different set of nodes. The average message overhead of replica control protocols is studied in [17]. Communication delay is studied in [7, 8]. For communication overhead, both the quorum size and the communication protocol are examined [9]. Other than availability, the above metrics try to ensure two important performance criteria of a database system: fast response time and high throughput, even when the system is under heavy workload. However, we argue here that while these metrics are useful, they are not sufficient. Let us look at the second metric of quorum size. A quorum size can be small, but if each request is targeted at the same set of sites, these sites will become a bottleneck and when the system is busy, both the throughput and response time of the system will become poor. The third metric of even workload distribution is relevant, it is set to be a yes/no metric in [10]. An even distribution of workload may not necessarily lead to good performance under heavy workload. For example, the majority quorum consensus has even workload distribution, but at any one time, only one operation can proceed, which means DIAMOND QUORUM CONSENSUS 3 that the performance cannot be enhanced by concurrent operations. Therefore, we would need another metric for the performance under heavy workload. In this paper, we introduce such a metric, which we shall call the read-capacity. Let us consider a replicated database system, it is in a distributed environment with multiple sites, where copies of a data are kept at different sites. For quorum consensus, when a read (write) operation is executed, it must obtain the consensus from a read (write) quorum. A write quorum intersects each other write quorum and each read quorum, therefore at any time only one transaction can be writing a logical data. A read quorum needs only to intersect a write quorum, therefore, as long as two read quorums do not intersect each other, two read operations on the same logical data can be executed concurrently. One may think that the read quorums of two concurrent read operations can intersect each other, but when we measure the throughput or response time, we must consider the amount of work involved in a read operation for each copy. In particular, the messaging to and from each site in a quorum will be a major overhead, and an intersection in read quorums will create a bottleneck. This is why the metric of even workload distribution is important. If two read quorums are disjoint, there will not be any bottleneck for response time and throughput. In view of the above, we introduce a metric here that measures the maximal number of read operations that can be handled simultaneously, which is effectively the maximal number of disjoint read quorums that can be formed. This metric is the read-capacity. In [14, 15, 19], there is a metric called the load which roughly speaking measures the minimal load on the busiest site. It has a similar flavor to the metric of read-capacity. However, [15] considers only a collection of quorums every two of which intersect. A protocol with a high read-capacity can handle heavy workload without much deterioration in response time and system throughput. However, high read-capacity does not guarantee competitive performance under light workload conditions and does not guarantee high availability. Therefore, a good protocol should simultaneously satisfy all the criteria of high read-capacity, small best case quorum sizes, high availability, and low cost of one node failure. In this paper, we present such a new protocol, which is called diamond quorum consensus or simply the diamond protocol, for managing replicated data. In this protocol, the sites in the network are logically organized into a two-dimensional diamond structure. This protocol can be viewed as a specialized version of the grid protocol [10] because the logical structure can be seen as a grid with holes. It has been noted in [11] that grids with holes often produce a higher availability than solid grids. We show here that there is much more to the story. The diamond protocol also resembles the crumbling walls protocol [15]. However, [15] does not consider read and write operations. There are two main properties of the diamond quorum consensus that make it a good choice for replicated data management: 1. Compared with the majority quorum, tree quorum and grid quorum (without holes) protocols, the diamond protocol results in the greatest number of disjoint read quorums which shall lead to a better throughput and response time. 2. The protocol achieves the smallest optimum read quorum size and the second smallest write quorum size among the above protocols. Since the quorum size is a good indicator of messaging cost, and read operations are usually in higher proportion than write operations, this is a very desirable property. Other than the above, we also show that diamond quorum consensus has high availability when the probability of site failures is reasonably low, and it has a low cost of node failures. Partition failure can be a problem in a replicated database system, and handling partition failure has been a main motivation for the introduction of quorum consensus in the earlier work. However, few recent work in quorum consensus discuss partition failure handling, which in many cases will depend on the quorum consensus protocol. We shall examine the use of the generalized virtual partition protocol with the diamond protocol to handle partition failure. This paper is organized as follows. In the next section, we present the diamond quorum consensus. Analysis of our protocol and comparison with other protocols are given in Section 3. The handling of partition failures is discussed in Section 4. Section 5 is a conclusion. 2. Diamond Quorum Consensus In our protocol, the sites in a network are logically arranged in a two-dimensional diamond shape as in Figure 1. Often we shall refer to the physical sites of the network as sites, and refer to the nodes in the logical structure as nodes. When this distinction is not relevant, we shall use the terms sites and nodes interchangeably. In Figure 1, nodes are represented by circles. The top and bottom rows of the diamond shape contain two nodes each. In a diamond structure, suppose the number of rows is odd, we label the rows from the top row to the middle row by levels, so that the top row is at level 0, the second row is at level 1, etc. Similarly we label the rows from the bottom row to the middle row by levels, so that the bottom row is at level 0, the second row from the bottom is at level 1, etc. The number of nodes in the rows increases by a certain amount w i for every level i for i ? 0. If and each of the top and bottom rows contains 2 nodes, then we call the resulting diamond structure a regular diamond structure and denote it by d . In our basic model, w i is 2 for all i, therefore the resulting structure is a 2 structure. For example, in Figure 1, there are 32 nodes, or 32 sites, in the network, the number of nodes for the rows are 2, 4, 6, 8, 6, 4, 2. The table in Figure 2 shows some relevant figures for the diamond structure, k is any integer greater than 0. For example, the diamond structure in Figure 1 has a maximum level of 3, 7 rows and 32 nodes in total. In the diamond protocol, read and write quorums are formed in the following ways: DIAMOND QUORUM CONSENSUS 5 Figure 1. Example of a read quorum and a write quorum Max Level Configuration No. of rows No. of nodes Figure 2. Some figures for the diamond structure 6 FU, WONG AND WONG ffl Write Quorum To form a write quorum, we can choose all nodes of any one row plus an arbitrary node for each remaining row. ffl Read Quorum To form a read quorum, we can choose 1. any entire row of nodes, or 2. an arbitrary node of each row. Figure 1 and Figure 3 show some quorums forming instances. It is obvious that the above method ensures that each write quorum intersects each other write quorum and each read quorum. We shall refer to the above protocol as diamond quorum consensus or simply the diamond protocol. For the diamond protocol, the minimum read quorum size is obtained by choosing the whole top or bottom row of nodes. The minimal write quorum size is obtained by choosing the whole top or bottom row of nodes plus a node for each remaining row. Therefore, if k is the maximum level in a diamond structure, the minimal read quorum size is 2 and the minimal write quorum size is 2k 2. In the more general case, the number of rows in a diamond structure can be even or odd, the difference in the number of nodes between 2 adjacent rows in the diamond structure can be any integer, and the number of nodes in the top and bottom row can be other than 2. In the extreme case where w level 0 contains only one node, the diamond reduces to a single column of nodes, and we shall have the read-one-write-all protocol. In another generalization, the diamond logical structure can be adjusted according to the number of sites in a particular network. That is, it is not restricted only to the numbers as shown in the Table 2. For instance, if there are 40 sites in a network, we can use a structure with 4 levels, 9 rows and 50 nodes, with 10 of them not being occupied by physical sites. The idea is similar to that of a hollow grid structure [11]. If the total number of nodes in the network does not fit into the shape above, the protocol can also accommodate the change of the shape by the addition or deletion of a number of nodes in any row and also by the addition or deletion of rows. Therefore, we have the following definition of a general diamond structure. Definition 1. A general diamond structure (also called a G structure) is a stack S of n rows of nodes. Let us label the rows by numbers 1 to n. We say that a row A is above (below) another row B if the label of A is less (greater) than the label of B. For each row R in S such that no row contains more nodes than R, if we consider the rows in the stack S above and including R, each row has a size greater than or equal to the size of the row above it. If we consider the rows in the stack S below and including R, each row has a size greater than or equal to the size of the row below it. Figure 3. More examples of read and write quorums 2.1. Some Properties of the Diamond Protocol In the d structure, for the top k rows, a row at level i + 1 has w i more nodes than a row at level i. If w i is set to a constant d, then the sum of nodes for k rows from levels 0 to k \Gamma 1 is given by a finite arithmetic series: A s (k; d) = a 2: Then, let oe(k; d) be the number of nodes in a d structure with a maximum level of k, and with w We shall emphasize on the 2 structures because it will give us good performance characteristics. If k is the maximum level in 2 , then from the above, the number of nodes is given by oe(k; have 2N . The number of rows in a 2 structure is given by The number of nodes in the longest row in a 2 structure is given by 2(k 2N . We shall see that the above properties of a 2 structure lead to good performance in the next section. However, a 2 structure is not a general structure that can accommodate any number of sites. Therefore, we would like to discover general diamond structures that can preserve these desirable properties. This is given in the following theorem. Theorem 1 Given any integer N - 5, one can build a general diamond structure (a G structure) that contains N nodes, where the number of rows in the G structure a b c d e f r s A Figure 4. Hypotenuse of even length (even arrangement) is l p 1, the top and bottom rows have size 2, and the maximum row size is bounded from above by l p Proof: In Figure 4, the nodes in a diamond structure are represented by alphabets a to s. Each node occupies one square in the figure. Since the dark grey area is equal to the light grey area, the space occupied by all nodes is equal to the space occupied by the square ABCD, which has sides of length S. For the right-angled triangles ABC or ACD, the hypotenuse AC has length In Figure 5, the nodes are represented by alphabets a to x. Each node occupies one square in the figure. Let the area of each square be 1. The square WXY Z has sides of length S and a hypotenuse of length 7. Since the dark grey area is equal to the light grey area, the space occupied by all nodes is less than the space occupied by square WXY Z by an amount equal to the black area, which is given by 0.5. It is easy to see that the two figures can be extended to any values of H - 3, and the relationship between the space occupied by nodes and the space occupied by the diamond ABCD or WXY Z will not change. The arrangement of nodes for even H is as follows: the top and bottom rows have size 2, there is only one row of maximum length, difference in size between adjacent rows is 2, Let us call this the even arrangement. The arrangement of nodes for an odd H is similar, except that there are two rows of maximum length, so the difference in size between the middle two rows is 0. Let us call this the odd arrangement. By the Pythagoras Theorem, H Given any integer N - 2 (this corresponds to the N given sites), we can find the smallest integer H such that N - S 2N , and l p . If H is even (odd), DIAMOND QUORUM CONSENSUS 9 a b c d e f Y Z Figure 5. Hypotenuse of odd length (odd arrangement) then we have a case as shown in Figure 4 (5), and we can match the N sites into an even (odd) arrangement. In this matching, there may be nodes that are not matched to any site. One can see that there exists a matching such that each row contains at least 2 sites. To see this we note that N has at least S nodes. This is because 0:5), then we should have chosen looking for the smallest integer H as specified in the above paragraph. Therefore, there will be at most H \Gamma 1 holes in the G structure, which are the nodes not being matched to sites. For N - 5, one needs only to place at most one hole at each of the rows other than the top and bottom rows, and then place at most one more hole at the longest row. The resulting general diamond ( G ) structure has the same number of rows as in the full ( 2 ) structure in the even (odd) arrangement, which is l p and its top and bottom rows have size 2. Each row in the G structure has size less than or equal to the corresponding row in the 2 structure. Therefore, the longest row in the G structure contains at most l p nodes. 3. Performance Analysis Different logical structures and quorum forming methods result in different performance in different metrics. In this section, we first introduce the metric of read-capacity. Then we examine the performance of the diamond protocol under different metrics, including the read-capacity, quorum size and availability. We compare the protocol with known protocols of majority quorum consensus, the grid protocol, the tree protocol, and hierarchical quorum consensus. For the grid protocol, we shall examine the modified grid protocol in [11], since it is an improvement over the original gird protocol, we shall only examine the cases without holes in the grid. 3.1. Read-Capacity With the non-empty intersection property of read and write, write and write, quo- rums, only one write quorum can be formed at any instance in a replicated database system. Hence, the capacity analysis here is targeted only on the read operations. We define the read-capacity of a replicated database as: maximal number of concurrent read operations maximal number of disjoint read quorums Given a network with N nodes in which data is fully replicated, assume that any node of the network system can only handle one read operation at any time. In the following, we examine the read-capacity for each protocol. ffl majority quorum consensus: It can only handle one read operation as there is an intersection between each pair of read quorums. ffl hierarchical quorum consensus: We consider the case when the branching factor is set to 3, which is recommended in [10] as the structure that gives a minimal quorum size. In this case, each quorum is not disjoint from any other quorum, therefore it can only handle one read operation at a time. ffl tree quorum consensus: For the majority tree protocol, the nodes are logically organized into a ternary tree(i.e. degree of height h, i.e., each node has d children, and the maximum height is h. Each read or write quorum should have a length l and a width w, and we denote the quorum by dimensions hl; wi. The protocol tries to construct a quorum by selecting the root and w children of the root, and for each selected child, w of its children, and so on, for depth l. If some node is inaccessible at depth h 0 from the root while constructing this tree quorum, then the node is replaced recursively by w tree quorums of height l \Gamma h 0 starting from the children of the inaccessible node. The details can be found in [2]. Suppose the read quorums q r have dimensions hl r ; w r i and the write quorums have dimensions q [2]). When the quorums are constructed, the following constraints should be fulfilled so that the nonempty intersection of read and write, write and write, quorums holds: DIAMOND QUORUM CONSENSUS 11 our protocol . majority - tree quorum Number of nodes Number of read Figure 6. Comparison of read-capacities For the comparison, we used a ternary tree model. The maximal number of disjoint read quorums occurs when the length of read quorums is 1, and each level of the tree will give rise to one disjoint read quorum. Hence it can handle at most log 3 N read operations simultaneously. However, to achieve acceptable availability and overall performance, the quorum length should be set to a greater value, Hence, the maximum number of read operations that the tree quorum can handle in parallel will actually be smaller. ffl grid protocol: for the grid protocol, read quorums of size approximated by N can be constructed without intersection. So, it can handle, on average, N read operations simultaneously. ffl diamond quorum consensus: The maximal number of disjoint read quorums is obtained by taking a read quorum from each row of the diamond structure. From the proof of Theorem 1, there are l p rows in a general quorum structure that has the properties stated in the theorem, hence the diamond protocol can handle l p simultaneous read operations. The diamond protocol can handle the maximum number of simultaneous read operation among the above protocols. That is, it can achieve the best read-capacity among the majority, tree quorum and grid protocols. The comparison is also shown in Figure 6. Recall that the number of nodes in a diamond structure with a maximum level of k, and with w all i, is given by oe(k; d) d. Given a value of k, the value of oe(k; d) increases with the value of d. Therefore, to obtain a higher read-capacity for a fixed number of nodes (oe(k; d)), one should try to obtain a greater value for k, which implies a smaller value for d 1 . However, we must also consider other factors for quorum consensus. Therefore we have chosen a value of 2 for d because with this setting, the other properties of the diamond protocol are also satisfactory. 3.2. Quorum Size We have already discussed the importance of the quorum size. In this subsection we examine the optimal quorum sizes as well as the worst case quorum sizes. Again a number of quorum consensus are compared with the diamond quorum consensus method. ffl majority quorum consensus: the read and write quorum size is approximated by \Upsilon ffl hierarchical quorum consensus: As shown in [10], the minimal quorum size is given by N 0:63 in both the best case and the worst case. ffl tree quorum consensus: If read quorums have dimensions hl; wi (a tree quorum of length l and width w), then write quorums must have dimensions the height and d is the degree of the tree. Then, the read quorum size of tree quorum protocol varies from w l \Gamma 1 where the smaller size occurs when all copies of the quorum are from the upper levels (near the root) of the tree. The size of write quorums varies from to \Theta ffl grid quorum consensus: For the grid protocol, we assume that the grid structure is approximately a square, the read quorum size is approximated by N , and the write quorum size is approximated by 2 N . ffl diamond quorum consensus: The optimal read quorum size is 2 and is independent of the total number of sites. The worst case is when the longest row is chosen, or when a node from each row is chosen. From the proof of Theorem 1, the general diamond structure that satisfies the conditions in the theorem has a biggest quorum size bounded by l p , and has l p Therefore the worst case read quorum size is l p DIAMOND QUORUM CONSENSUS 13 our protocol . majority - tree quorum Number of nodes Optimal read quorum size our protocol . majority - tree quorum Number of nodes Read quorum size-worst case Figure 7. Read quorum size The smallest write quorum is a node from each row union either the top or bottom row, hence the optimal write quorum size in terms of N is l p \Gamma1+1. l p . In the worst case, a node from each row union the longest row forms the biggest write quorum, the size is bounded by l p l p is equal to 2 l p For the diamond protocol, we can choose the top or bottom row as the read quorum, provided that they are functional, to keep the read quorum size at 2. In the other protocols under comparison, only the tree protocol can attain a comparable optimal read quorum size of 1, which is by choosing only the root node. However, in such a case, the root node becomes a bottleneck and the corresponding write quorum size will be quite big. Also, from [2], the availability will be severely affected by this choice of read quorum. 2 Figure 7 shows the best case and worst case read quorum sizes of the tree quorum consensus, majority quorum consensus, grid quorum consensus, and our protocol. For the tree protocol, we set because this has been chosen for performance studies in [2]. We have set the dimensions of the read quorums to be hl r and those for write quorums to be hl w This is because from [2], h2; 2i is the dimensions for hl r ; w r i that lead to the smallest read quorum size while maintaining acceptable availabilities, for the cases studied in [2]. Figure 8 shows the best case and worst case write quorum sizes of the protocols. In both cases, the diamond protocol can achieve a quorum size that is the second best and very close to best. The tree quorum has the optimal write quorum size, but its worst case write quorum size is very large. The grid quorum protocol has the smallest worst case quorum size but its optimal read quorum size and optimal write quorum size are the second highest. The comparison leads us to the conclusion that 14 FU, WONG AND WONG our protocol . majority - tree quorum Number of nodes Optimal quorum size our protocol . majority - tree quorum Number of nodes Write quorum size-worst case Figure 8. Write quorum size in overall considerations of the quorum sizes, the diamond protocol is superior to the other protocols. 3.3. Availabilities As in most previous work, the availability of an algorithm is defined as the probability of forming a quorum successfully in that algorithm. In the following we denote the probability of X by P rob(X). Let p be the probability that a site is up and be the probability that a site is down. Let the number of sites be N . ffl majority quorum consensus: the availability of read and write operations are defined as: rob(majority copies are available) are available) rob(all copies are available) ffl tree quorum consensus: For the tree quorum protocol, the availability can be calculated by a recurrence relations for both read and write availabilities. Let A h [l; w; d] be the availability of operations with a tree quorum of dimensions hl; wi in a tree of height h and degree d (see [2] for the definition of dimensions). From [2], the availability in a tree of height h formulated as A h+1 [l; w; [Availability of w subtrees with A h [l \Gamma DIAMOND QUORUM CONSENSUS 15 +P rob(Root is down) \Theta [Availability of w subtrees with A h [l; w; d]]: The expansion of this formula can be found in [2]. ffl diamond quorum consensus: The calculation of the read and write availabilities for the diamond protocol is very similar to that for the grid protocol. The diamond structure can be treated as a hollow grid, some of the positions in the grid structure are not occupied by nodes. For instance, the structure shown in Figure 1 can be regarded as an 8 \Theta 7 grid structure model, with all the corner nodes being chopped off. In this particular case, 24 nodes, 6 in each corner, are being eliminated in the structure. The read and write availabilities of our model can be calculated in a similar way as for the modified grid protocol [11]. That is, a row is alive if at least one site in it is up, and is dead otherwise. A row is good if all sites in it are up, and is bad otherwise. Suppose there are n i columns that contain m i nodes each, for Read availability are bad) rob(all rows are bad and alive) Write availability are alive) \Gammaprob(all rows are bad and alive) where there are n k rows that contain m k sites in the diamond structure. ffl grid protocol The availabilities are also given by RA and WA, although we shall examine solid grids so that each row will have the same number of nodes. For the above protocols, we examine 3 cases, each one with a different number of sites, for comparison of the availabilities. They are 13, 40 and 121 sites. Figures 9 to 11 show the read availabilities and the write availabilities. For Figure 9, there are 13 sites in the network. and a 3 \Theta 4 grid structure is chosen for the modified grid protocol. The general diamond structure is chosen so that the row sizes are 2g. For Figure 10, there are 40 sites for Figure 10, and a 5 \Theta 8 grid structure is chosen. The general diamond structure is chosen so that the row sizes are f2, 4, 6, 8, 8, 6, 4, 2g. . tree -x- majority -o- our protocol probability that a site is up read availability nodes) -x- majority our protocol probability that a site is up availability nodes) Figure 9. Availability (13 sites) . tree -x- majority -o- our protocol probability that a site is up read availability nodes -x- majority our protocol probability that a site is up availability nodes Figure 10. Availability (40 sites) For Figure 11, there are 121 sites and a 10 \Theta 12 grid structure is chosen. The general diamond structure is chosen so that the row sizes are f2, 4, 6, 8, 9, 10, 12, 14, 14, 12, 10, 8, 6, 4, 2g. Note that in all the above diamond structures, we have the properties that if N is the number of sites, then the number of rows is given by l p 1, the top and bottom rows have size 2, and the maximum row size is bounded from the above by l p , as specified in Theorem 1. From the figures, we see that when the site reliability is reasonably high, the diamond protocol performs well both in the read and write availabilities. If one would like to achieve higher write availability for the diamond quorum consensus, it is possible to do so. For example, for the general diamond structure DIAMOND QUORUM CONSENSUS 17 -x- majority our protocol probability that a site is up read availability nodes -x- majority our protocol probability that a site is up availability nodes Figure 11. Availability (121 sites) . tree -x- majority -o- our protocol probability that a site is up read availability nodes -x- majority our protocol probability that a site is up availability nodes Figure 12. Improved availability (40 sites) can be chosen so that the row sizes are f 3, 3, 6, 8, 8, 6, 3, 3g. Figure 12 shows the resulting availabilities. For 121, the general diamond structure can be chosen so that the row sizes are f 3, 3, 6, 8, 9, 10, 12, 14, 14, 12, 10, 8, 6, 3, 3g. Figure 13 shows the results. Compared with Figures 10 and 11, we can see a big improvement in the write availability when the site availability is high. In both cases, the read-capacity remains the same after the diamond structure is modified, the optimal read quorum size is increased only by one, which remains to be smaller than that of the other protocols. -x- majority our protocol probability that a site is up read availability nodes -x- majority our protocol probability that a site is up availability nodes Figure 13. Improved Availability (121 sites) 4. Generalized Virtual Partition Protocol To handle partition failures, we use the generalized virtual partition protocol, GVP, which is presented in [6]. This is a generalization of the virtual partition protocol (VP) in [5]. We shall give a brief description of VP and then point out the differences of GVP from VP. We present VP in a slightly generalized form. In VP each transaction executes in a view. A view can be considered to correspond to the reachable part of the network as seen from a site after partition failure. Transactions executing in a view are controlled by a concurrency control protocol within the view as follows. For each data object X, there are two positive integers, A r [X] and Aw [X], called read and write accessibility thresholds, respectively, satisfying A r [X] where n[X] denotes the total number of copies of X. Thus, a set of copies of X of size Aw [X] has at least one copy in common with any set of copies of X of size A r [X]. Each site maintains a set of sites called its view. Views are totally ordered according to their unique view-id's, which are non-negative integers. Each copy of a data object has a version number = hV id; ki, indicating that it was last written in view V with view-id V id and that its value is the result of the kth update in that view, where indicates the initial value written by the "view-update transaction" (see below). A "less than" (or "larger than") relation is defined among version numbers by their lexicographical ordering (I.e. a version number In view V , each logical data object X is assigned, if possible, read and write quorum sizes q r [X; V ] and q w [X; V ], which specify, respectively, how many copies of X must be accessed to, respectively, read and write X in view V . (An access DIAMOND QUORUM CONSENSUS 19 operation on a copy may return only its version number, not its value.) In our terminology, a view read(write) quorum for data object X in view V , is a set of copies of X that can be accessed to perform logical read(write) on X in view V . denotes the set of all view read (write) quorums for X in . Let n[X; V ] be the number of copies of X that reside at sites in view V . The quorum sizes must satisfy the following conditions. For all X and V , These ensure that each view write quorum for X in V , if any, has at least one copy in common with each view read quorum for X in V (by Equation (2)) and with any view which has at least A r [X] copies of X (by Equations (1) and (4)). If there are at least A r [X] copies of X in view V , then we say that X is inheritable in V . then there is no choice for q w Equation (4) above; in this case both rq(X; V ) and wq(X; V ) will be empty. Consider a transaction T executing at a site s having view V with view-id V id. (In this case we say that T executes in V .) It can read or write copies at another site s 0 only if s 0 also has view V with the same view-id. (Ways to relax this restriction are discussed in [5].) If rq(X; V ) 6= OE then the logical read operation R T [X] by transaction T executing in V with view-id V id is implemented as follows: 1. Access all copies in a view read quorum in rq(X; V ) at sites having view V with 2. Determine ki, the maximum version number among the accessed copies, 3. If V id 6= V idmax, then abort T , else read a copy in rq(X; V ) with version number vnmax. Note that in [5], X cannot be read in V unless X is also inheritable in V . We relax this requirement by allowing a read operation on X in V once X has been "initialized" in V . This will make it possible for two concurrent partitions (under different views) to perform both read and write operations on the same logical data, provided that some transaction has performed a write without read on the data. then the logical write operation executing in V is implemented as follows: 1. Access all copies in a view write quorum in wq(X; V ) at sites having view V with view-id V id, 2. Determine ki, the maximum version number among the accessed copies, and 3. Update the copies in a view write quorum in wq(X; V ) and change their version numbers to hV id; k+1i, if V and to hV id; 1i if A site may change its view from time to time. For example, a site may want to change its view when it notices a difference between its current view and the sites it can actually communicate with. Whenever a site s changes its view to a new view, s must execute a view-update transaction that updates data object copies stored at site s. Site s may decide on the members of a new view V based on its own information, in which case s is called the initiator of V . It may also decide to use a view V initiated by another site, in which case, s adopts view V . Sites change their views automatically as follows. (For details, see [5]). An initiator s of a new view V first assigns to V a unique view-id, new view id, that is larger than any other view-id that s has encountered. (Uniqueness of the view-id can be achieved by using the initiator's unique site ID (identification number) to be the least significant digits of the view-id.) Site s then executes a view-update transaction to update the local copy of each data object inheritable in V . For each such data object X, the view-update transaction reads the copy of X at a site in V that has a copy of X with the largest version number among a set of A r [X] copies. The version number of X is set to be hnew view id; 0i. If the view-id of S 0 (X) is not larger than new view id, then the value of X is copied from S 0 (X); otherwise, the view-update transaction is aborted. When this is repeated for all inheritable data objects X, the new view is installed at s. If a site s 0 accessed by the view-update transaction has a view-id less than new view id, then s 0 immediately adopts new view id. If a site accessed by the view-update transaction has a view-id greater than new view id, then the view-update transaction is aborted, in which case site s immediately adopts the greater view-id and initiates a view-update transaction with that view-id. Some comments are in order for the case where X is not inheritable in V , since unlike [5], we may still allow reading of X in V . After V has been installed at some sites, but before any user transaction is executed in V , the copies of X at these sites, if any, have version numbers hV 0 id; ki such that V 0 id ! V id. At this time, no user transaction should be allowed to read X. However, X can be written if and the first write on X in V will initialize X in V , by changing the version number of the updated copies to hV id; li. Thereafter, X can be read in V . We now describe the Generalized Partition Protocol, GVP. The major difference of GVP from VP is that the definition of a quorum is in terms of a set of copies rather than by the size of the quorum. This gives us greater flexibility in defining the quorums. We shall not give the full description of the protocol here, instead, we shall list the main features that are dependent on the quorum consensus method, and explain how to incorporate the protocol in the case of diamond quorum consensus. As in VP, in GVP each transaction executes in a view which is a subset of the set of all replication sites. Each view has a unique view-id, V id. Each copy of a data object has a version number = hV id; ki, indicating that it was last written in view V with view-id V id, and that its value is the result of the th update in that view. Transactions executing in a view are controlled by a concurrency control protocol within the view. The following are some major features. DIAMOND QUORUM CONSENSUS 21 Figure 14. An example of global read, view read, and view write quorums 1. Global read quorums : For each data object X, a global read quorum set RQ(X) is defined. X is inheritable in view V if V contains a global read quorum belonging to RQ(X). 2. View quorums : For each data object X, and each view V , a view read quorum set rq(X; V ) and a view write quorum set wq(X; V ) are defined. Each view write quorum in wq(X; V ), if any, intersects each quorum in RQ(X) and each view read quorum in rq(X; V ), if any. If OE), then X is said to be writable (readable) in V . 3. Reading and writing : A logical write of X by a user transaction T executing in view V with view-id V id is required to update or initialize all copies in a view write quorum in wq(X; V ) (where each copy has a version number that contains a view-id V id), giving them the same new version number hV id; ki larger than their previous version number. A logical read of X by user transaction T is allowed in view V only if at least one copy of X accessed by the read has been initialized in V . A logical read of X by T reads a view read quorum from sites that have the same view as T and takes the value of the copy with the highest version number. For our protocol, GVP is adopted as follows: if a view V is fs can be chosen as follows: Choose a set fR 1 which is a subset of the set of rows in the diamond structure. We call R i a row in RQ(X). RQ(X) is the set of all sets of the following 22 FU, WONG AND WONG or or and if V contains a quorum Q in RQ(X), then 1. is the set of any whole row of sites in V , and 2. is the set of (i) all nodes of any one row, in V ,plus (ii) an arbitrary node for each complete row, in V , and each row in Q. ffl As rq(X; V ) is in V , must intersect any view write quorum in view V , and ffl as wq(X; V ) contains a whole row in V , it must intersect any view write quorum in V . In addition, as wq(X; V ) contains one node in each row in Q, wq(X; V ) must intersect any quorum in RQ(X). ffl All quorums in RQ(X) intersect each other. An example of a possible set of global read quorum, view read quorum and view write quorum is shown in Figure 14. The black nodes are the unreachable sites for the non-black nodes, so they may be considered as a partition. The set of non-black nodes will form a view for the functional sites. The example shows the intersection properties among the quorums. For a diamond structure with a maximum level of k, we may choose the second, k-th and (2k \Gamma 2)-th row to be the rows in RQ(X) because the second and (2k \Gamma 2)th can minimize the cost of rq(X; V ), the k-th row is the row that contains the largest number of node within one row. Choosing this row can provide a high tolerance ability of partition failure. 5. Conclusion We summarize our major findings in the table in Figure 15. In the table, N is the number of sites. The quorum size (best case) refers to the smallest read or write quorum size. The quorum size (worst case) refers to the greatest read or write quorum size. Note that although the minimal quorum size of the tree protocol is 1, the resulting protocol will have problems in read-capacity, write quorum size and also availability. The cost of one node failure refers to the effect of a single node failure on the consensus of the smallest quorum. Suppose we have decided on a smallest quorum, but one of the nodes in the quorum has failed, then we need to form another quorum, and this cost refers to the number of additional number of nodes one needs to access. Therefore, for the diamond protocol, the cost of one node failure is the number of nodes in the second smallest quorum, which is 2. DIAMOND QUORUM CONSENSUS 23 Protocol read- quorum size quorum size cost of one node capacity (best case) (worst case) failure (worst case) Alg. 1 Tree log 3 N 1 N 1 Grid - Diamond l p l p Figure 15. Properties of some quorum consensus protocols From the table and our previous discussions, we have shown that the diamond quorum consensus is a good choice for high throughput and efficiency in replicated data management. There are two main contributions in this paper. First we define a new metric of read-capacity which we show to capture the characteristics of workload-capacity, which measures how well the system can maintain high throughput and low response time under heavy workload. Secondly we propose a new protocol, the diamond protocol that can achieve high read-capacity, low quorum size, and other desirable features for replicated data management. We also show that the new protocol can easily merge with the generalized virtual partition failure protocol. One open problem that should be investigated is the quorum selection strategy. We suggest to use a random selection since it requires no overhead. We believe that it should be able to distribute the workload quite well during a busy period. By such a selection method, and with the proposed protocol, the chance of a node having no workload during a busy period would be small. Hence the throughput should be high and the average response time can be low. Another possible strategy for busy periods is for each site to pick the quorums in a round robin fashion. For example, if there are k quorums then if the previous quorum chosen at site s is Q i , the next quorum to be used will be Q (i mod k)+1 . In this way, we ensure that all quorums have similar chances of being chosen and hence the workload will also be distributed quite evenly. We can try to confirm these expectations by more detailed analysis and experiments. Acknowledgements The authors would like to thank the anonymous referees for their very thorough review and very helpful comments which enhance the paper significantly. This research was supported by the RGC (the Hong Kong Research Grants Council) grant UGC REF.CUHK 495/95E. Notes 1. Note that when the top row size is 1 and d is 0, we have the read-one-write-all protocol, which has the maximal read-capacity of N . However, this protocol has problems such as poor write availability. 2. This is a main reason why we have not set the upper and lower rows of the diamond to have size one. --R An efficient and fault-tolerant solution for distributed mutual exclusion The generalized tree quorum protocol: An efficient approach for managing replicated data. Performance characterization of the tree quorum algorithm. The grid protocol: a high performance scheme for maintaining replicated data. Maintaining availability in partitioned replicated databases. Enhancing Concurrency and Availability for Database Systems. Delay optimal quorum consensus for distributed systems. Hypercube quorum consensus for mutual exclusion and replicated data management. Hierarchical quorum consensus: A new algorithm for managing replicated data. A performance study of general grid structures for replicated data. A geometric approach for consructing coteries and k-coteries A p N algorithm for mutual exclusion in decentralized systems. The load Crumbling walls: A class of practical and efficient quorum systems. A fault-tolerant algorithm for replicated data management An analysis of the average message overhead in replica control protocols. A majority consensus approach to concurrency control for multiple copy databases. Quorum systems for distributed control protocols. Message complexity of the tree quorum algorithm. --TR Maintaining availability in partitioned replicated databases An efficient and fault-tolerant solution for distributed mutual exclusion Hierarchical Quorum Consensus Enhancing concurrency and availability for database systems The generalized tree quorum protocol Performance Characterization of the Tree Quorum Algorithm A <inline-equation> <f> <rad><rcd>N</rcd></rad></f> </inline-equation> algorithm for mutual exclusion in decentralized systems A Fault-Tolerant Algorithm for Replicated Data Management Crumbling walls An Analysis of the Average Message Overhead in Replica Control Protocols A Geometric Approach for Constructing Coteries and k-Coteries Delay-Optimal Quorum Consensus for Distributed Systems A Majority consensus approach to concurrency control for multiple copy databases Message Complexity of the Tree Quorum Algorithm The Grid Protocol Quorum-oriented Multicast Protocols for Data Replication

fault-tolerant computing;quorum consensus;mutual exclusion;distributed systems;replicated database systems

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Optimizing static calendar queues.

The calendar queue is an important implementation of a priority queue that is particularly useful in discrete event simulators. We investigate the performance of the static calendar queue that maintains N active events. The main contribution of this article is to prove that, under reasonable assumptions and with the proper parameter settings, the calendar queue data structure will have constant (independent of N) expected time per event processed. A simple formula is derived to approximate the expected time per event. The formula can be used to set the parameters of the calendar queue to achieve optimal or near optimal performance. In addition, a technique is given to calibrate a specific calendar queue implementation so that the formula can be applied in a practical setting.

INTRODUCTION The calendar queue data structure, as described by Brown [Brown 1988], is an important implementation of a priority queue that is useful as the event queue in a discrete event simulator. At any time in a discrete event simulator there are active events, where each event e has an associated event time t(e) when it is intended to occur in simulated time. The set of events is stored in the priority queue ordered by their associated event times. A basic simulation step consists of nding an event e 0 which has the smallest t(e 0 ), removing the event from the priority queue, and processing it. As a result of the processing new events may be generated. The parameter N can vary if zero or more than one new events are generated. Each new event e has an event time t(e) > t(e 0 ) and must be inserted in the priority queue accordingly. In the calendar queue the events are stored in buckets with each bucket containing events whose times are close to each other. All the events with the smallest times are in the same bucket so they can be accessed quickly and simulated. Any newly generated event can be quickly put into its bucket. When the events in one bucket are consumed, the next bucket is considered. The details of the algorithm are given later. The calendar queue has several user controllable parameters, the bucket width and number of buckets, that aect its performance. Brown [Brown 1988] provided empirical evidence that the calendar queue, with its parameters properly set, achieves expected constant time per event processed. The goal of this paper is to prove the constant time per event of the calendar queue behavior in a reasonable model where, for each new event e, the quantity t(e) t(e 0 ) is a nonnegative random variable sampled from some distribution. Generally, the number of active events may vary over time. An important case is the static case which arises when N is a constant, such as the case of simulating a parallel computer. In this case, each event corresponds to either the execution of a segment of code or an idle period by one of the processors. Thus, if there are processors, then there are exactly N active events in the priority queue. In this paper we focus on the static calendar queue. Even before Brown's paper [Brown 1988] the calendar queue was used in discrete event simulators when the number of events is large. In many of these situations the calendar queue signicantly outperforms traditional priority queue data structures [Brown 1978; Francon et al. 1978; Knuth 1973; Sleator and Tarjan 1985; Sleator and Tarjan 1986; Vuillemin 1978]. An interesting new development is the employment of a calendar queue like data structure as part of the queuing mechanism of high-speed network switches and routers [Rexford et al. 1997]. In this case the calendar queue like data structure is implemented in hardware. 1.1 Organization In Section 2 we dene the calendar queue data structure and the parameters that govern its performance. In Section 3 we present the Markov chain model of calendar queue performance. In Section 4 we present an expression that describes the performance of the calendar queue in the innite bucket case. The bucket width can be chosen to approximately minimize the expected time per event at a constant. In Section 5 we describe how to choose the number of buckets without Optimizing Static Calendar Queues 3 signicantly compromising the performance over the innite bucket calendar queue. In Section 6 we develop a technique for calibrating a calendar queue implementation and demonstrate the eectiveness of the technique. In Section 7 we present our conclusions. The Appendix contains the longer technical proofs. 2. THE CALENDAR QUEUE A calendar queue has M buckets numbered 0 to M 1, a current bucket i 0 , a bucket width -, and a current time t 0 . We have the relationship that i For each event e in the calendar queue, t(e) t 0 , and event e is located in bucket if and only if i t(e)=- mod M < (i 1). The analogy with a calendar can be stated by: there are M days in a year each of duration -, and today is i 0 which started at absolute time t 0 . Each event is found on the calendar on the day it is to occur regardless of the year. As an example choose 30. The 8 events have times 31, 54, 85, 98, 111, 128, 138, 251. { 111 128 31 { 54 { { 85 98 In this example the next event to be processed has time 31 which is in the current bucket numbered 3. Suppose it is deleted and the new event generated has time 87. Then, the new event is placed in bucket 8 next to the event with time 85. Since will not be processed until the current bucket has cycled around all the buckets once. Thus, t 0 is increased by -, and the next bucket to be examined is bucket 4 which happens to be empty. Thus, the processing of the buckets is done in cyclic order and only the events e which are in the current cycle, t 0 t(e) < are processed. A calendar queue is implemented as an array of lists. The current bucket is an index into the array, and the bucket width and current time are either integers, xed-point or oating-point numbers. Each bucket can be implemented in a number of ways most typically as an unordered linked list or as an ordered linked list. In the former case insertion into a bucket takes constant time and deletion of the minimum from a bucket takes time proportional to the number of events in the bucket. In the latter case insertion may take time proportional to the number of events in the bucket, but deletion of the minimum takes constant time. The choice of algorithm for managing the individual buckets is called the bucket discipline. In this paper we focus on the unordered list bucket discipline. 2.1 Calendar Queue Performance For the calendar queue, the performance measure we are most interested in is the expected time per event, that is, the time to delete the event with minimum time and insert the generated new event. There are two key (user controllable) parameters in the implementation of a calendar queue that eect its performance, namely, the bucket width - and the number of buckets M . The choice of the best - and M depends on the number of events N and the process by which t(e) is chosen for a newly generated event e. Assuming M is very large (innite), if - is chosen too 4 K.B. Erickson, R.E. Ladner, A. LaMarca N number of events known parameter mean of the jump known or estimated parameter time per empty bucket hidden parameter determined by calibration time per list entry hidden parameter determined by calibration d xed time per event hidden parameter determined by calibration - bucket width user controlled parameter M number of buckets user controlled parameter Table 1. Parameters of the calendar queue. large, then the current bucket will tend to have many events which is ine-cient. On the other hand if - is chosen too small, then there will be many empty buckets to traverse before reaching a non-empty bucket, which again is ine-cient. Regardless of the choice of -, if M is chosen too small, then the current bucket will again tend to have too many events in it which are not to be processed until later visits to the same bucket. In order to analyze the calendar queue we make some simplifying assumptions on the process by which t(e) is chosen for a new event e. The main assumption we make is that the quantity t(e) t(e 0 ), called the jump, is a random variable sampled from some distribution that has a mean , where e 0 is the event with minimum time t(e 0 ). We will fully delineate the simplifying assumptions later. The choice of a good - certainly depends on both and N . As grows so should -. As N grows should decrease. Determining exactly how - should change as a function of and N to achieve optimal performance is a goal of this paper. Assume that we have innitely many buckets. In addition to the two parameters and N the choice of a good - also depends on three hidden implementation parameters b, c, and d where b is the incremental time to process an empty bucket, c is the incremental time to traverse a member of a list in search of the minimum in the list, and d is the xed time to process an event. If m empty buckets are visited before reaching a bucket with n events (n 1), then the time to process an event is dened to be: KN (-) to be the stationary expected value of bm d. Then KN (-) is the expected time per event in the innite bucket calendar queue. In a real implementation of a calendar queue the number of buckets M is nite. In this case, it may happen that some events in the bucket have times that are not within - of the current time and are not processed until much later. Dene K M to be the expected time per event in the M bucket calendar queue. Generally, N (-) KN (-) because extra time may be spent traversing events in buckets that are not processed until later. Another goal of this paper is to determine how to choose M so that K M N (-) is the same or only slightly larger than KN (-). Table 1 summarizes the various parameters that aect the performance of the calendar queue. Optimizing Static Calendar Queues 551525 time per event bucket width Fig. 1. Graph of bucket width, -, vs. the expected time per event, KN (-), in the simulated innite bucket calendar queue with 100 events, exponential jump with mean 1, and time per event number of buckets Fig. 2. Graph of number of buckets, M , vs. the expected time per event, K M N (-), for - chosen optimally in the M bucket simulated calendar queue with 1,000 events, exponential jump with mean 1, and Figure 1 illustrates the existence of an optimal - for minimizing the expected time per event. Figure 2 illustrates the eect of selection of M on the expected time per event. The graphs in both Figures were generated by simulating the calendar queue with an exponential jump with mean 1 and were taken after a suitably long warm-up period and over a long enough period so that average time per event was very stable. The simulation of Figure 1 uses an innite number of buckets with 100 events. The simulation of Figure 2 uses the optimal bucket width for for the innite bucket calendar queue, then varying the number of buckets. In choosing or 3; 000 the performance curve is almost at approaching the performance with innitely many buckets. 6 K.B. Erickson, R.E. Ladner, A. LaMarca 3. MODELING THE CALENDAR QUEUE PERFORMANCE To model the calendar queue performance we begin by specifying the properties of the random variable, is the event with current minimal time, and e is the newly generated event. We assume that is a random variable with density f dened on [0; 1), the nonnegative reals. We call f the jump density and its random variable simply the jump. Successive jumps are assumed to be mutually independent and identically distributed. Let be the mean of the jump, that is: Z 1f1 F (z)g dz: (1) where F is the distribution function of the : F R xf(z) dz. We call F the jump distribution. We dene the support of the jump distribution to be The value 1 is not excluded. Technical Assumptions about f and F . In order to facilitate the proofs we make several technical assumptions about f and F that will be in force throughout, except as noted. J1. The density f(x) > 0 for all x in the interval (0; ). J2. The mean is nite. J3. There is an 0 > 0 and c 0 such that F (x) c 0 x for all x 0 . Assumption J2 is crucial; it guarantees the existence of a non-trivial \steady state". Note that J3 holds if the density f is bounded in a neighborhood of 0. 3.1 The Markov Chain We model the innite bucket calendar queue as a Markov chain b X with state space in [0; 1) N . For denote the state of the chain at time t. The state (X 1 represents the positions, relative to the beginning of the current bucket, of the N events (indexed 1 to N) in the calendar queue at step t. A step of the calendar queue consists of examining the current bucket, and either moving to the next bucket, if the current bucket is empty, or removing the event with smallest time from the current bucket and inserting a new event (with the same index) according to the jump distribution. Accordingly, the transitions of b are as follows: Let m be the index such that for all i. If X m (t) < -, then for i are independent non-negative random variables. It is assumed that these random variables t ; t 0, all have the same probability density f . The parameter - is a xed non-negative real number. We can think of X i (t) as the position of the i-th particle in an N particle system. If no particle is in the interval [0; -), then all particles move - closer to the origin. Otherwise, the particle closest to the origin, in the interval [0; -), jumps a random Optimizing Static Calendar Queues 7 distance from its current position and the other particles remain stationary. Thus, a particle in the Markov chain b X represents an event in the innite bucket calendar queue where the position of the particle corresponding to an event e is the quantity . The interval [0; -) corresponds to the currently active bucket in the innite bucket calendar queue. It is important to note that a step of the Markov chain b X does not correspond to the processing of an event in the calendar queue. The processing of an event in the calendar queue corresponds to a number of steps of the Markov chain where the interval [0; -) is empty followed by one step where the interval [0; -) is nonempty. to be the limiting probability, as t goes to innity, that the interval [0; -) has exactly i particles in it. Technically, q is a function of N and -, but we drop the N and - to simplify the notation. The quantity q 0 is the probability that the interval [0; -) is empty. It is not obvious that q i exists for 0 i N , so we prove the following Lemma in Appendix A. Lemma 3.1. If the jump density has properties J1, J2, and J3, then the limiting probabilities exist and are independent of the initial state of b X. Let us also dene EN (-) to be the limiting expected number of particles in the interval [0; -), that is, 4. EXPECTED TIME PER EVENT IN INFINITE BUCKET CASE The expected time to process an event in the innite bucket calendar queue is closely related to the function EN (-) as we see from the following Lemma. Lemma 4.1. The expected time per event in the innite bucket calendar queue is Proof. The Markov chain b X models the calendar queue. Thus q 0 is the portion of buckets visited which are empty and for j > 0, q j is the portion of buckets visited which have j events. Each empty bucket visited, which happens with probability cost b, but does not result in nding an event to process. Each bucket visited with j > 0 events, which happens with probability q j , has cost cj + d, and results in nding an event to process. Thus, the expected cost per event in the calendar queue is d) which yields equation (3) using equation (2). 8 K.B. Erickson, R.E. Ladner, A. LaMarca Let us dene the following important quantity Z -[1 F (x)] dx: (4) The second part of equation (1) implies that 0 p 1. Note also that -[1 F (-)]= p -=. In order to derive a good approximating formula for KN (-) we rst need to nd good bounds for the quantities q i for 0 i N . The following technical Lemma, proved in the Appendix, Section B, provides those bounds. Lemma 4.2. For N 2 and all - > 0 we have and for where B(j) is the tail of the binomial distribution for N trials with \success" parameter p: The simple exact formula for q 0 is interesting. It is possible to write down some very complicated integrals which give exact expressions for the other q j , but these are highly unwieldy and their proofs are not informative (cf. [Erickson 1999]). It is also interesting to note that our assumption J1 requiring the probability density f to be positive on its support can be removed, but the proofs of the theorems become even longer. Without J1, if the density f has the property that there is a constant c > 0 such that we have exact expressions for q j for all j 1. Lemma 4.2 yields the following upper and lower bounds on KN (-). Lemma 4.3. For N 2 and all - > 0 KN (-) d b Proof. From (5) and (6) we have +N- to be the standard binomial distribution with N trials and success parameter p. Since and b(i) has mean Np and Optimizing Static Calendar Queues 9 second moment (Np) 2 +Np(1 p) we sum by parts to derive which, upon substituting into equation (3) and doing a little rearranging, yields the left side of (8). Similarly, one derives the right side of (8). The range of - that gives good calendar queue performance is when In this case the bounds of (8) give us a wonderfully simple, and accurate, approximating formula for KN (-). Theorem 4.1. If then the expected time per event in the innite bucket calendar queue with bucket width - is In fact, there are numbers 1 ; 2 such that for any xed > 0 the O(N 1 ) term is bounded by ( 2 )=N uniformly for 0 < - =N . The proof of this Theorem is almost an immediate consequence of (3), (5), and but it is also postponed to the Appendix, Section C. Interestingly, the expected time depends on the mean of the jump and not on the shape of its probability density. Note that one immediate consequence of Theorem 4.1 is that if the bucket width is chosen to be =N for in a xed interval, then the innite bucket calendar queue has constant expected time per event performance. Indeed, a formula for the optimal performance of the calendar queue can be derived as seen in the following Theorem. Theorem 4.2. The expected time per event, KN (-), achieves a global minimum in the interval (0; 1) at - opt where r c KN (- opt 2bc +O N 1 The proof of this Theorem is in the Appendix, Section D. Theorem 4.2 shows that the optimal choice of - only depends on the ratio of b to c, the mean of the jump, and N . 5. CHOOSING THE NUMBER OF BUCKETS Now that we have found how to select - so as to approximately minimize the expected time per event in the innite bucket calendar queue our next goal is to select M , the number of buckets, so that the M bucket calendar queue has the same or similar performance as the innite bucket calendar queue. For the case in which the jump distribution has nite support ( < 1), there is a natural choice for M which guarantees that the calendar queue with M buckets has A. LaMarca exactly the same performance as the innite bucket calendar queue. If M =-+1, then it is guaranteed that in the long run all the events e in the current bucket will have In this case, eventually each event in the current bucket will be processed during the current visit to the bucket and not postponed until future visits to the bucket. For the case in which the support of the jump distribution is either innite or is nite but =- is too large to be practical, then it will be necessary to choose a number M which gives performance less than that of the innite bucket calendar queue. 5.1 Expected Time per Event in the nite Bucket Case The same Markov chain b X can be used to analyze this case. Let L M N (-) be the (steady state) expected number of particles in the set In terms of the M bucket calendar queue, if an event e has t(e) t 0 2 , then the event is in the current bucket but is not processed. The occurrence of such an event will cause the M bucket calendar queue to run less e-ciently than the innite bucket calendar queue. The following Lemma quanties the dierence between the performance of the nite and innite bucket calendar queues. Lemma 5.1. The expected time per event in the M bucket calendar queue with bucket width - is Proof. In the Markov chain b X, let q ij be the limiting probability that there are particles in the interval [0; -) and j particles in . The probabilities q ij can be shown to exist in the same way as we did for the probabilities q i in Lemma 3.1 by using Corollary A.1 in the Appendix, Section A. In the M bucket calendar queue the cost of visiting a bucket with i events whose times are in the interval [t and j events whose times are in the set Thus, the expected cost per event K M But (equation (5) of Lemma 4.2) and (-). By using equation (3) in the proof of Lemma 4.1 we derive the equation for K M (-). In the Appendix , Section E, we indicate how to derive the following rather horrible looking bounds for L M (-). Lemma 5.2. The function L M N (-) is bounded above by Optimizing Static Calendar Queues 11 and bounded below by +N- where p and F have the same meaning as before (see equation (4)) and Z jM- [1 F (x)] dx; [F (jM- y) F (jM- y)]: It should be noted that under the hypothesis < 1 (J2), the above series converge and can be given bounds in terms of ; -, and M . However, using the bounds as stated in the Lemma, we can derive a more useful asymptotic expression for L M (-). The Lemma is proven in the Appendix, Section F. Lemma 5.3. If are constants, then [1 F (jxr)]: 1 5.2 Degradation in Performance due to Finitely Many Buckets M to be the degradation in performance in choosing M buckets instead of innitely many buckets, that is, If we choose - optimally, then Lemma 5.3 and Theorem 4.2 yield the following asymptotic expression for M . Theorem 5.1. If M=N is constant and c [1 F (jM-)] (12) The following asymptotic bound is implied by Theorem 5.1. Theorem 5.2. If M=N is constant and c r c 12 K.B. Erickson, R.E. Ladner, A. LaMarca Proof. By Theorem 5.1 it su-ces to show that To see this let k 2, then Z 1xf(x)dx Z jD+D x f(x)dx The niteness of implies that x[1 F (x)] ! 0 as x ! 1. Therefore, if we let k !1, we get =D Equation (13) shows that for a xed > 0 (like .01) M can be chosen to be O(N) so that M . In other words, one can always choose the number of buckets M to be a multiple of N and still obtain a performance almost as good as that of the innite bucket case. For the interesting case of the exponential jump density we can calculate the series in the equation (12) exactly: e c Let us suppose optimally equal to 2=N allows us to solve equation (14) for M=N when given an acceptable M . For example, if we choose should be approximately 1:92 and if M=N should be approximately 3:02. Figure 3 illustrates that asymptotic equation provides an excellent choice of M over a wide range of N . Using our simulation of the calendar queue we plot for a wide range of N the value of M for each of 3:02. Again, measurements were taken after a suitably long warm-up period and over a long enough period so that average time per event was very stable. Both plots are relatively at near the asymptotic values .05 and respectively. Thus, equation (14) seems quite accurate. The bound of Theorem 5.2 is not necessarily tight because we are crudely approximating an integral. For example, if we choose then the formula (13) requires M=N to be at least Optimizing Static Calendar Queues 130.030.07100 1000 degredation Fig. 3. Graph of N vs degradation, M , from simulations for 6. CALIBRATION OF A CALENDAR QUEUE IMPLEMENTATION In an actual calendar queue implementation we would like to nd the best bucket width - and number of buckets M . The preceding theory tells us how to do so if we know the hidden implementation parameters b, c, and d. In this Section we give a relatively simple method of estimating these parameters simply by timing executions of the simulation for various values of - proportional to =N . The key to the method is equation (9) for the expected time per event. We can write KN (-) as a linear function of the unknowns b, c, and d. The general calibration method is as follows: rst estimate M to be large enough so that the degradation in using M buckets over innitely many is small. Second, nd K M N (-) for a number of dierent -'s by timing executions of the implementation, and third, use a linear least squares approximation to nd the b, c and d that best ts the function We illustrate this method with an example. We developed a calendar queue implementation in C++ and ran it on a DEC alphastation 250. We chose and an exponential jump with mean 10; 000. Just by examining the code we felt that b, the time to process an empty bucket, was considerably larger than c, the cost of traversing a list entry. We made an educated guess that the optimal - was certainly greater than 5. We chose or larger there is only a small chance that an event in the current bucket is not processed because its time is too large. We timed the calendar queue for 20 values of - ranging over several orders of magnitude, namely, Using this data we used linear least squares approximation to compute using equation (15). Figure 4 shows the curve of equation (15) using these parameters. The Figure also shows the time per event for 200. Thus, this method accurately predicts data points that were not used in the linear least squares approximation. It is interesting to note that using these values of b, c, and d in equations (10) and (11), we obtain - opt 61:5 and K(- opt ) 1756:38. By contrast, the best - among the 40 executions is 14 K.B. Erickson, R.E. Ladner, A. LaMarca with execution time 1754:92.1800220026003000 time per event bucket width measured predicted Fig. 4. Measured and predicted expected time per event for a calibrated implementation of a calendar queue. Care must be taken in applying this calibration method. In the method, the hidden parameters, b, c, and d, are measured indirectly by measuring the expected time per event. For xed M , N , and -, the expected time per event can vary over dierent runs because of interruptions by other processes, page faults, or other ef- fects. However, in our experimental setting we carefully controlled the environment so that our running times varied little for a xed parameter setting. In addition, measurements were taken after a suitably long warm-up period and over a long enough period so that average time per event was very stable. In a real computing environment that cannot be controlled this calibration method might not yield such good results. Ideally, using a xed N , M (large enough), and we can estimate the hidden parameters b, c, and d which then could be used for any other N , M (large enough), and , and -. However, because of the cache behavior of modern processors, the values of b, c, and d are not actually constant independent of M , N , and properties of the jump distribution other than its mean. For example, a smaller M might achieve fewer cache misses reducing the running time and thereby eectively lowering the values of these constants. It may be that in applying the calibration method, - is chosen so large that the original M chosen is far larger than necessary. In this case, it might be wise to choose a smaller M , then recalibrate the calendar queue starting with a larger -. In a real application of the calendar queue it is unlikely that the jumps are mutually independent, identically distributed random variables as described in our model. Nonetheless, the mean of the jump can be empirically estimated, the calibration done, and equation (10) for the optimal - applied to nd a potentially good -. Optimizing Static Calendar Queues 15 7. CONCLUSION We have shown that there is an expression for the expected time to process an event in the innite bucket calendar queue and that the bucket width can be chosen optimally. With the bucket width near the optimal bucket width the calendar queue has expected constant time per event. The optimal bucket width depends only on a few parameters, the incremental time to process an empty bucket (b), the incremental time to traverse a list item (c), the mean of the jump (), and the number of events (N ). We have shown that the number of buckets M can be chosen to be O(N) so as to achieve minimal or almost minimal expected time per event. Finally, we have shown that the implementation parameters can be determined by using approximation based on the method of linear least squares. Although the calendar queue runs very fast for certain applications it has the disadvantage that its performance depends on the choice of parameters - and M . An interesting problem would be to design a priority queue based on the calendar queue that automatically determines good choices for - and M . We believe that the calibration method described in this paper might give insight into the design of a dynamic calendar queue where N and/or can vary over time. Section A of the Appendix sets up the notation and concepts that are used through-out the Appendix. A. INVARIANT DISTRIBUTION, POSITIVITY, AND LIMITS Consider the Markov chain b described in Section 3. The symbol P shall denote the probability measure induced on the trajectory space of the chain b X when the initial distribution is , and P b x shall denote trajectory space probabilities when the chain starts at the point b x. (Note: P R P b x d(b x), the integration being carried out over the entire state space.) Integration (better known as expectation) with respect to P and P b x is denoted E and E b x , respectively. stand for the set of points b such that and let A . For an b x in the state space and a (measurable) subset A, the one-step transition probability (T. P.) that the chain will move from b x to a point in A is given by (b Z 11A (b x the standard i th unit coordinate vector, and 1A (b x) is the function which is 1 for b x in A and 0 otherwise. 3 3 Actually, (16) does not dene a proper transition probability at all points! Indeed, if b x is any point with two or more coordinates that are equal and strictly less than -, then b x does not lie in A 0 nor in B i for any i. Hence P (b x; subsets A. However, the jump distribution F has a density so it has no atoms (discrete points of positive probability). This and the dynamical description of the chain in Section 3 imply that two points which start at the same position A. LaMarca Let be the set [0; - It follows from (16) that (b in and A C . In other words, is an absorbing set for the chain. Moreover, the dynamical description of the chain implies that a particle which starts outside of will reach in a nite (but possibly random) number of steps. (One can show, using the method in the proof in B.3 that the number of steps required to eventually enter has a nite expectation.) Thus C is transient for the chain. (Of course, measure m is an invariant measure for the chain if m is -nite and for every measurable subset A of the state space Z (b x; A)mfdb xg; (17) A Markov chain is called a Harris recurrent chain, or simply a Harris chain, if there exists a unique, up to positive multiples, invariant measure m such that if A is any Borel subset with m(A) > 0, then P b x ( b x in the state space. (The initials i.o. stand for \innitely often".) A Harris chain with an invariant probability measure, necessarily unique, is called positive. The state space of a Harris chain can be written as a disjoint union: 6. The sets C are known as the recurrent cyclic classes. The integer d is nite and if d = 1, the chain is called aperiodic. Theorem A.1. If the jump density satises J1, J2, and J3, then the Markov chain b X with T.P. (16) is a positive, aperiodic, recurrent Harris chain. Its invariant probability is concentrated on Corollary A.1. If ' is any bounded measurable function, then for any initial distribution , we have lim Z '(b x) dm(b x); P a.s. (18) Note: The left-most term is a limit of averages of random quantities and the assertion is that the limit exists with P -probability 1 and equals the (non-random) quantities on the right. If ' is unbounded but integrable with respect to m and if the chain is Harris and positive, then the above limit relations remain valid at least in the case that is point mass at some b x or that = m. Proof. of Corollary A.1. The deterministic limit statements of Corollary A.1 are immediate consequences of Proposition 2.5 in Ch.6, x2, of [Revuz 1984]. The eventually become and remain separated w. p. 1. Indeed, this occurs as soon as one of them jumps to the right. Thus, it does no harm if we banish such points from the state space initially. With this understanding P is indeed a transition probability on its state space. Optimizing Static Calendar Queues 17 a.s. limit-of-averages assertion is a consequence of the ergodic Theorem for Harris chains. See Theorem 4.3, and its companion remark, in [Revuz 1984], Ch. 4, x4. Remark. The main signicance of aperiodicity is that it justies the existence of the limit of E b x f( b occurring in (18). The existence of limits of averages does not require aperiodicity. Proof. of Lemma 3.1. Let exactly j components of b x lie in [0; -)g: the number of particles in interval [0; -) at time t and from (18) it follows immediately that q g. The proof of Theorem A.1 will be postponed to the very last Appendix below. It is lengthy and somewhat tedious, but there are some interesting features. B. PROOF OF LEMMA 4.2 B.1 The computation of q In this Section we prove (5) of Lemma 4.2. The sets B i , dened in the last Section, are disjoint and their union is the complement (in ) of A 0 . Since m assigns 0 mass to [0; 1) N \ c , we have Let be any bounded or positive function on the state space. Equation (17) has an analogue for functions which reads: R (b x)mfdb R mfdb xg R (b y)P (b x; db y). Noting that R R (b y)P (b x; db R A0 (b x - b 1)mfdb xg, by (16), and doing a little rearranging, we get Z (b x - b 1) (b x) Z mfdb xg (b x) Z (b y)P (b x; db y) Fix i and let (b complex number with nonnegative real part. Then for b x in B i , Z (b y)P (b x; db R 1e z f(z) dz is the Laplace transform of F . Also, for b x in B j with Z (b y)P (b x; db All but the i th term on the right side of (20) vanishes and it becomes [1 ()] Z A. LaMarca Simplication of the left-hand side of (20) leads to: Z Z Divide (21) by and make ! 0. The result is -mfA 0 g= 0 (0)mfB i Equation (5) follows immediately from this and (19). B.2 The Case If we observe the successive positions of a single one of our N particles at only those times at which it actually moves, we get a 1-dimensional version of the N - dimensional chain. For From the description of the chain in terms of the independent random variables , one concludes: (i) Each sequence fX i is itself a Markov chain on the line; (ii) These N Markov chains are mutually independent. If one can nd an increasing sequence of times fS k g such that each S k is a common value of every one of the u i (that is, for each k there are numbers r i (k) not necessarily the same, such that S mutually independent components. Here is such a sequence: let S and for k > 0, let The times T are the successive (random) times at which the interval [0; -) is empty of particles (Z(T k is obtained from by adding the deterministic constant - to each of the components of b it follows that the components of b are also mutually independent. It is easy to show that the chain induced on A 0 (or trace chain), the sequence f b is also a Markov chain. See [Revuz 1984], Exercise 3.13, page 27. An important point to note is that the special structure of b X implies that for each i the chain fX i (T k ); k 0g , coincides in law with the trace chain on [-; 1) of an X. It is at least intuitively clear that the trace chain positive recurrent and has an invariant probability distribution m 0 , say, obtained by renormalizing the distribution m restricted to A 0 . (See [Revuz 1984], Ex. 3.13, p. 27, and Prop. 2.9, p. 93 for a formal proof.) Thus for subsets B, But because this trace chain also has independent components, it follows that m 0 is a \product measure" built up from the invariant distributions of each of its component chains. These component chains have identical T.P.'s, so the factors in are the same. Let us call this common factor distribution m 10 . Once computed, (concentrated on [-; 1)) may be used to compute the limit, as k !1, of the Optimizing Static Calendar Queues 19 probability of nding exactly j particles in the interval [0; -) at the times S +1. By now it should be clear that the limiting distribution of Z(S k ) is a binomial distribution corresponding to N Bernoulli trials with parameter (However, the limit distribution of Z(t) for t tending to innity without restriction is not a binomial.) The invariant distribution, let us call it m 1 rather than m, in the case our basic chain can be calculated explicitly and then m 10 obtained from the special case of (22). The measure m 1 turns out to be uniform on [0; -) and coincides with the (-translate) of the stationary distribution for the renewal process with interarrival distribution F . This stationary distribution has a density equal to the normalized tail-sum 1 F . See [Feller 1971], XI.4. One can give queuing theory arguments for the above description of m 1 , but, since equation (21) leads to this result almost immediately, we use that equation to give a quick proof. In the case equation (21), simplies to valid for any complex number ; Re() 0. If we set is an arbitrary integer, we nd that the left-hand side vanishes. The density assumption implies that () 6= 1 for any 6= 0. Hence results in the theory of Fourier series implies that we must have dm 1 some constant C. From (5) in the case -). For the Laplace transform of m 1 on [-; 1), we get Z -e x Inverting the Laplace transforms in this equation reveals that the density g 1 of m 1 on [-; 1), for x -, is given by g 1 From (22) it is then clear that m 10 has the density The upshot of the preceding is that we can now conclude that the limit distribution of Z(S k ) is: lim (w.p.1), for x Remark. It follows from the work of the last two Sections that the measure m, when restricted to A 0 , is a product measure because its restriction to A 0 coincides with However, the m-measures of subsets of A 0 have no particular interest; it is only the m-measures of the other A j (dened in the proof of Lemma 3.1) that is required and these sets are contained in the complement of A 0 . But m restricted to the complement of A 0 is not a product measure. 20 K.B. Erickson, R.E. Ladner, A. LaMarca B.3 Estimates for the q j Apart from the explicit representation of m on A 0 discussed in the last Section, a simple expression for m on all of [0; 1) N for N > 1 is not available. This means that, with one exception (the case F (-) = 0), we do not have simple explicit formulae for the values of q and must resort to approximations. It turns out, however, that the approximate formulae are quite amenable to analysis particularly in the region of interest: In this Section we nish the proof of the two inequalities of (6) which we henceforth designate (LH-6), for the left side, and (RH-6) for the right. To simplify the notation a little, the starting distribution will be omitted, if not forgotten, when it is not essential. For this proof we introduce the objects I A j ( b I A j where A . The variable #(t; 0) diers from n - (t) by at most 1 because the T k 's are the zeros of Z. Therefore, by the ergodic limit theory for b lim Hence Sn Sn and then Sn Sn a sequence of random variables fV k g (k 1) by V shall be the total number of times t in the interval that the counting variable Z(t) has the value j: I A j ( b The occurrence of the event V k > r implies that Z(S k 1 and that there are at least r jumps, counting from the rst time in [S k 1 magnitudes smaller than -. Hence Optimizing Static Calendar Queues 21 and therefore, assuming F (-) < 1, (Indeed all moments, Ef(V k ) g, are nite.) This inequality, (23), and the ergodic limit theory now yield lim lim which is (RH-6). As to (LH-6) note rst that Hence, see (25), and (23), lim n which is (LH-6). C. PROOF OF THEOREM 4.1 For the purposes of this proof let us write The conclusion of Theorem 4.1 is equivalent to the assertion that KN (-) d x c x uniformly on bounded x-intervals. We base the proof on (8) which states c x DN (-) KN (-) d b x in the new notation. Fix a number > 0 and conne x to the interval (0; =] so that 0 < - =N . Let 0 and c 0 be the numbers introduced in assumption J3 in Section 3. Keeping N > maxf2c 0 gets 22 K.B. Erickson, R.E. Ladner, A. LaMarca Also DN (-) 1(N 2 =N. Hence x c x DN (-) From this, and a little algebra, it is easily seen that to nish the proof of (28), it su-ces to nd a number C 2 , depending on , such But, because Np N- x x x The numbers C 1 (multiplied by c) and C 2 yield estimates for 1 and 2 mentioned in Theorem 4.1: D. PROOF OF THEOREM 4.2 Throughout this Section we will write a = b=c, Moreover, there is no harm in also supposing that Step 1. As a function of -, KN (-) is continuous on (0; 1). The reader is asked to turn to the formula (3). To begin with the variable q 0 is =( + N-) which is obviously continuous. The only possible discontinuous term in the formula for KN is EN (-). However, the continuity of this function is an immediate consequence of the following exact formula which will be discussed after the proof of the Theorem is complete. Lemma D.1. Step 2. The function - 7! EN (-) is nondecreasing. One can prove this by dierentiating the expression for E(-) of Lemma D.1 and checking that the result is non-negative. Optimizing Static Calendar Queues 23 Here is an outline of an alternative, but more intuitive, proof: Consider two chains b with the same N and jump density but with dierent bucket . If we follow the trajectory of an individual particle in each chain, then we nd that on average in the chain with the larger bucket size, - 2 , the particle gets back to the interval [0; - 2 ) quicker than it would get back to the interval [0; in the chain with the smaller bucket size, - 1 . Since a typical particle of the chain more often in the interval [0; - 2 ) than in [0; - 1 ) of the chain b the average number of particles in [0; - 2 ) of b 2 is at least as large as the average number of particles in [0; - 1 ) of b Step 3. We next establish lim uniformly on [- for each xed - 1 > 0. The explicit formula for q 0 yields that nondecreasing function of -. By equation (3) and step 2, we nd that the function (1 q 0 (-))KN (-) q 0 (-)b is also nondecreasing in -. Hence for b: For 1. From the above inequality we have for - 1 - 2 and N =- 1 , From this inequality, it follows that KN (-) goes to innity uniformly in the interval for each xed - 1 > 0. This follows from equation (8). Step 4. For 2a; we have min 0<-< Moreover, if - 1 is the minimizing - on this interval, then where - o;N is dened above. To prove all this, let By straightforward calculus for each N , HN (-) has a global minimum on (0; 1) at the point - o;N . Let - opt be a value of - which gives the minimum value of KN (-) on the given interval (0; =N ]. By Theorem 4.1, on this interval we can nd a constant C, depending on , such that for all N su-ciently large, uniformly for 0 < - < =N . Since is in the interval (0; =N ]. On this interval, KN (-) is thus sandwiched between the two convex functions, HN (-)C=N , both of which have a global minimum at the same point - o;N interior to the interval. For a xed N , - opt must be between the two solutions to the equation (in -) HN (-) . By a simple calculation one nds that the dierence A. LaMarca between the two solutions is O(N 3=2 ) and this yields - any - between the two solutions one nds that KN Step 5. The next step is to show that for any 0 , there is such that for all N N 1 KN Thus, the minimum exhibited in step 4, extends to the xed interval (0; - 1 ]. This fact and step 3 imply that for all N su-ciently large, min completing the proof of the Theorem. For the moment we x - 1 such that 0 < F (- 1 ) < 1. We will choose - 1 later. By the inequality (8) and the fact that p(-[1 F (-)]= we have for all - 1 KN (-) cQ a=z -. Dene LN . For each N , the horizontal line at height K cuts the graph of the convex function LN at two points, the larger of which we call z N . Thus, z N is the larger root of the equation As a sequence in N , the values z N converge to a bounded positive limit. Dene N =QN . Hence, the sequence N- also converges to a limit, . Note that tends to 2a as Q approaches 1. We choose - 1 (and hence Q) so that < Now, choose N 1 such that - Since the minimum of KN (-) is bounded above by K bounded below by the function LN (QN=(-)), then the minimum of KN (-) in the interval (0; must already lie in the interval (0; - N ], and hence in the interval (0; D.1 A discussion of the exact formula for EN (-). The proof of this result is quite long and is based on some exact, though very complicated, integral formulae for the q j 's. See [Erickson 1999] for the details. The exact formula for EN (-) leads to an exact formula for KN (-), but our work has led us to the conclusion that the excellent asymptotic formulae of Theorems 4.1 and 4.2 (and the simple inequalities of Lemma 4.2 which lead to them) are of much greater practical use and are certainly easier to prove. For this reason we have not included the long proof of the exact formula. Our main use of Lemma D.1 was to shorten the proof, slightly, of the global minimization of KN . Note that D.1 yields, immediately, the continuity of KN as a function of -. One requires continuity in order to speak sensibly of the existence of a minimizing -. Even without the continuity, however, the basic result of Theorem 4.2 is essentially correct; only the language used to express it needs to be changed. (One must use the term \greatest lower bound" in place of \minimum" and one can only assert that there are points - at which the greatest lower bound is approximately attained.) Optimizing Static Calendar Queues 25 We will write L M N (-) and Z A (t) for the number of particles in set A at time t. is a subset of [-; 1) we have are the successive times at which the interval [0; -) is empty of particles.) Hence, st Z Z w.p.1, where Suppose that at time T k 1 there are particles at positions x 1 2-). Then at there will be r particles in [0; -) at positions x where according as the i-th of these particles lands in or not when it is nally removed from the interval [0; -). (This removal must occur during Writing by the strong Markov property, where F is the eld of the random variables T k and where H t (b) is the probability that a particle starting at the origin lands in the interval it rst jumps over t. This H satises where U is the renewal measure. See [Feller 1971], page 369. 4 In general U(z) Uf[0; z]g [1 F (z)] 1 for distributions on [0; 1) so that sup [F (jM- x z) F (jM- x z)] Ufdzg (Recall the denition of 2 in the statement of Theorem 5.2.) Calling the right-hand side p and noting that conditional on the -eld F , the variables u i are 4 Feller denes H in terms of the open interval (-; 1) whereas we are using the closed interval. Because F has no atoms, this dierence in denition has no consequence. 26 K.B. Erickson, R.E. Ladner, A. LaMarca independent of k , we have [Z +- [Z +- r [1 F (-)] 1 Z +- , and Z fg means Z fg (T j k. Now at the times fT j g the particles are independent so the limiting joint distribution of Z +- is a trinomial. (Separately, they have binomial limit distributions.) Letting k !1 we get EfZ [-;2-) Z +- where p is dened at (4) and Going back to (32) with these calculations we obtain Z +- Z [-;2-) (Recall the basic property of conditional expectations EfE[ j F Replacing p with 2 =[1 F (-)] and q 0 with =( +N-) combining fractions and dropping the factor 1 F (-)] which will occur in the numerator, we nally obtain the upper bound on L M N . For the lower bound we have which evaluates to the lower bound on L M N . F. PROOF OF LEMMA 5.3 In the following we let lim, lim sup, and lim inf stand for the limits of various quantities as N !1 with the other variables constrained to vary as stated in the hypothesis. First let us note that lim F jrx. Note that t j does not vary with N . For all su-ciently large N , - will be so small that the intervals f(t j -; t j +-]g 1 are non-overlapping. -], then for all N we have JN+1 JN : Hence by continuity of F (no atoms). (The letter F stands for both the distribution function and the induced probability measure as is customary.) From this (and the limits Optimizing Static Calendar Queues 27 Next [1 F (jrx)]: As the rst sum on the left goes to it then follows that [1 F (jrx)] Using these limits in the upper bound for L M N (-) we get lim sup L M [1 F (jrx)]: Similarly, from the lower bound for L M mu+N- [1 F (jrx)]: Thus the lim sup L M so the limit of L M exists and its value is as stated. G. PROOF OF THEOREM A.1. We have seen that for each k 1 and given initial positions, the N components of are mutually independent random variables. Also, it is not hard to see that the conditional distribution of X the distribution of the residual waiting time at epoch - of a delayed renewal process starting at epoch x with interarrival distribution F . See [Feller 1971], page 369, and [Erickson 1999]. (We have already seen this distribution in the proof of Lemma 5.2, Section E, though it was was described in slightly dierent language). Letting H s fIg denote the probability that the residual waiting time at epoch s lies in I for a pure renewal process starting at 0 (H s f[0; E), it follows that for any xed x > 0 and every integer k >> x=- and any Borel I [0; ), Using the Markov property it then follows that for xed b Borel sets I i [0; ), Y 28 K.B. Erickson, R.E. Ladner, A. LaMarca If U denotes the renewal measure, then the assumption (a) implies that U has an absolutely continuous part which possesses a strictly positive density on (0; 1). But, see [Feller 1971], page 369, Consequently, the measure I 7! H k- x fIg also has an absolutely continuous part which is strictly positive on [0; ). The conclusion one may draw from the preceding is that f b X(S k )g is irreducible with respect to the measure ' N , the Lebesgue measure in R N (restricted to (0; ) N ). See [Revuz 1984], ch. 3 x2. This implies that the trace chain f b X(T k )g is also ' N irreducible on its state space A 0 . Together, these two assertions imply that the full chain b But we can also draw additional useful conclusions from (34) and (35). From Stone's decomposition Theorem, [Revuz 1984], ch 5, x5, we can write is a nite measure and U 1 is absolutely continuous with a bounded continuous density u such that lim x!1 1=. For any Borel I [0; 1) and Hence, by dominated convergence, lim Z I [1 F (x)] dx: This and the product formula (34) yield that for any Borel set A [0; 1) N lim 0 is the product measure F 0 F 0 F 0 and F 0 is the probability distribution on [0; ) with density f1 F (x)g=. Not only does (36) give us one of the limit Theorems we have used earlier, but it implies that subchains the b and b are both Harris recurrent (with invariant probabilities m translate by - b implies that, with probability 1, b for innitely many times k whatever be the initial position, but b is obtained from b adding - to each component so the previous assertion is also correct for b with respect to its invariant probability. See [Revuz 1984], ch 2, x3. Consider now the full chain b g. If ' N (A) > 0, A , then the preceding makes it clear that b will hit A with positive probability. This implies that b X is ' N -irreducible. According to [Revuz 1984], ch 2, Theorem 2.3, 2.5, and Denition 2.6, either b X is a Harris chain with a (unique up to constant multiples) invariant measure m, or else the potential kernel is proper. The potential kernel, K, is dened by K(b x; Ag. If K is proper, then can be written as an increasing sequence of subsets Dn each of which has bounded potential. But, eventually any such sets must have positive Lebesgue measure, and in that case Optimizing Static Calendar Queues 29 (34) implies that P b x f b for innitely many positions x. But K(b x; Dn ) < 1 implies that the expected total number of hits in Dn is nite which implies that the number of hits must be nite with probability 1. We thus cannot have a proper potential kernel and therefore b X must be a Harris chain. Consider next the aperiodicity issue. Seeking a contradiction, let us suppose that X is periodic. Let fC i g d be the recurrent cyclic classes in the decomposition of the state space. These subsets have positive Lebesgue measure. Without loss of generality we may suppose that mfC 1 \AN g > 0, to be the smallest S k such that b This stopping time is nite on account of . s be a doubly indexed sequence of independent random variables each with distribution F . The earliest possible epoch after at which b can arrive in A no more than one particle can move at any particular step. For any integer r 1 and any Borel rectangle A 0 we have Y On account of hypotheses J1, the distribution F and each of its convolutions F r puts positive mass on every subinterval in (0; ). Hence the right-hand side of (37) is strictly positive whenever the cylinder set A has positive Lebesgue measure. Standard measure theory implies that this is also correct for any Borel set A [-) of positive measure. probability 1, at time , b X() belongs to C 1 \ AN C 1 . So, if the chain is periodic, then, w.p.1, at all future epochs of the form nd the chain will always be found in C 1 . Also, for each d 1, at times t mod d; the chain must belong to the set C 1+k(d) which is disjoint from the other classes including C 1 . But in (37), for any N + r N the right hand side is strictly positive. By choosing A in (37) to be any set in C k \ [-) N of positive measure and letting r take on dierent ( mod d) values, we get a contradiction to the previous assertion about belonging to disjoint sets. There is no contradiction if X is aperiodic. It remains to show that the invariant measures m (unique up to constant multi- ples) of the full chain b are nite: m() < 1. The trace chain f b X(TK )g is positive recurrent with invariant probability m 0 . But m 0 is also a multiple of m restricted to A 0 . Hence, It follows from the Renewal Theorem that the mean return time to A 0 is also nite. Let 0 be a bounded measurable function on A 0 . Then is m 0 -(and hence m-) summable and lim Z (b x)dm 0 (b x) > 0: denote the number of visits to A 0 by the full chain during 0 s t. A. LaMarca ih But if the chain b were null, that is if m had innite mass, then for any bounded m-summable function , ([Revuz 1984], Theorem 2.6, page 198), 0: By Fatou's Lemma, one can readily see that this would contradict (38). ACKNOWLEDGMENTS We would like to thank the two referees and the editor for their many valuable suggestions for improving the paper. --R Implementation and analysis of binomial queue algorithms. Calendar queues: A fast o(1) priority queue implementation for the simulation event set problem. Calendar queue expectations. An Introduction to Probability Theory and Its Applications Description and analysis of an efcient The Art of Computer Programming Markov Chains. Scalable architectures for integrated tra-c shaping and link scheduling in high-speed atm switches A data structure for manipulating priority queues. --TR Self-adjusting binary search trees adjusting heaps Calendar queues: a fast 0(1) priority queue implementation for the simulation event set problem A data structure for manipulating priority queues The Art of Computer Programming, 2nd Ed. (Addison-Wesley Series in Computer Science and Information --CTR Wai Teng Tang , Rick Siow Mong Goh , Ian Li-Jin Thng, Ladder queue: An O(1) priority queue structure for large-scale discrete event simulation, ACM Transactions on Modeling and Computer Simulation (TOMACS), v.15 n.3, p.175-204, July 2005 Rick Siow Mong Goh , Ian Li-Jin Thng, Twol-amalgamated priority queues, Journal of Experimental Algorithmics (JEA), v.9 n.es, 2004 Farokh Jamalyaria , Rori Rohlfs , Russell Schwartz, Queue-based method for efficient simulation of biological self-assembly systems, Journal of Computational Physics, v.204 n.1, p.100-120, 20 March 2005

priority queue;optimization;calendar queue;discrete event simulation;data structures;algorithm analysis;markov chain

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Critical Motions for Auto-Calibration When Some Intrinsic Parameters Can Vary.

Auto-calibration is the recovery of the full camera geometry and Euclidean scene structure from several images of an unknown 3D scene, using rigidity constraints and partial knowledge of the camera intrinsic parameters. It fails for certain special classes of camera motion. This paper derives necessary and sufficient conditions for unique auto-calibration, for several practically important cases where some of the intrinsic parameters are known (e.g. skew, aspect ratio) and others can vary (e.g. focal length). We introduce a novel subgroup condition on the camera calibration matrix, which helps to systematize this sort of auto-calibration problem. We show that for subgroup constraints, criticality is independent of the exact values of the intrinsic parameters and depends only on the camera motion. We study such critical motions for arbitrary numbers of images under the following constraints: vanishing skew, known aspect ratio and full internal calibration modulo unknown focal lengths. We give explicit, geometric descriptions for most of the singular cases. For example, in the case of unknown focal lengths, the only critical motions are: (i) arbitrary rotations about the optical axis and translations, (ii) arbitrary rotations about at most two centres, (iii) forward-looking motions along an ellipse and/or a corresponding hyperbola in an orthogonal plane. Some practically important special cases are also analyzed in more detail.

Introduction One of the core problems in computer vision is the recovery of 3D scene structure and camera motion from a set of images. However, for certain con-gurations there are inherent ambiguities. This kind of problem was already studied in optics in the early 19th century, for example, by Vieth in 1818 and Muller in 1826. Pioneering work on the subject was also done by Helmholtz. See [13] for references. One well-studied ambiguity is when the visible features lie on a special surface, called a critical surface, and the cameras have a certain position relative to the surface. Critical surfaces or igef#hrlicher Ortj were studied by Krames [21] based on a monograph from 1880 on quadrics [32]. See also the book by Maybank [24] for a more recent treatment. Another well-known ambiguity is that when using projective image measurements, it is only possible to recover the scene up to an unknown projective transformation [8, 10, 35]. Additional scene, motion or calibration constraints are required for a (scaled) Euclidean reconstruction. Auto-calibration uses qualitative constraints on the camera calibration, e.g. vanishing skew or unit aspect ratio, to reduce the projective ambiguity to a sim- ilarity. Unfortunately, there are situations when the auto-calibration constraints may lead to several possible Euclidean reconstructions. In this paper, such degeneracies are studied under various auto-calibration constraints. In general it is possible to recover Euclidean scene information from images by assuming constant but unknown intrinsic parameters of a moving projective camera [26, 7]. Several practical algorithms have been developed [39, 2, 30]. Some of the intrinsic parameters may even vary, e.g. the focal length [31], or the focal length and the principal point [14]. In [29, 15] it was shown that vanishing skew suOEces for a Euclidean reconstruction. Finally in, [16] it was shown that given at least 8 images it is suOEcient if just one of the intrinsic parameters is known to be constant (but otherwise unknown). However, for certain camera motions, these auto-calibration constraints are not suOEcient [42, 1, 40]. A complete categorization of these critical motions in the case of constant intrinsic parameters was given by Sturm [36, 37]. The uniformity of the constant-intrinsic constraints makes this case relatively simple to analyze. But it is also somewhat unrealistic: It is often reasonable to assume that the skew actually vanishes whereas focal length often varies between images. While the case of constant parameters is practically solved, much less is known for other auto-calibration constraints. In [43], additional scene and calibration constraints are used to resolve ambiguous reconstructions, caused by a -xed axis rotation. The case of two cameras with unknown focal lengths is studied in [12, 28, 4, 20]. For the general unknown focal length case, Sturm [38] has independently derived results similar to those presented here and in [20, 19]. In this paper, we generalise the work of Sturm [37] by relaxing the constraint constancy on the intrinsic parameters. We show that for a large class of auto-calibration constraints, the degeneracies are independent of the speci-c values of the intrinsic parameters. Therefore, it makes sense to speak of critical motions rather than critical con-gura- tions. We then derive the critical motions for various auto-calibration constraints. The problem is formulated in terms of projective geometry and the absolute conic. We start with fully calibrated cameras, and then continue with cameras with unknown and possibly varying focal length, principal point, and -nally aspect ratio. Once the general description of the degenerate motions has been completed, some particular motions frequently occurring in practice are examined in more detail. This paper is organized as follows. In Section 2 some background on projective geometry for vision is presented. Section 3 gives a formal problem statement and reformulates the problem in terms of the absolute conic. In Section 4, our general approach to solving the problem is presented, and Section 5 derives the actual critical motions under various auto-calibration constraints. Some particular motions are analyzed in Section 6. In order to give some practical insight of critical and near- critical motions, some experiments are presented in Section 7. Finally, Section 8 concludes. 2. Background In this section, we give a brief summary of the modern projective formulation of visual geometry. Also, some basic concepts in projective and algebraic geometry are introduced. For further reading, see [6, 24, 33]. A perspective (pinhole) camera is modeled in homogeneous co-ordinates by the projection equation world point, image, P is the 3 \Theta 4 camera projection matrix and ' denotes equality up to scale. Homogeneous coordinates are used for both image and object coordinates. In a Euclidean frame, P can be factored, using a QR-decomposition, cf. [9], as Here the extrinsic parameters (R; t) denote a 3 \Theta 3 rotation matrix and a 3 \Theta 1 translation vector, which encode the pose of the camera. The columns of de-ne an orthogonal base. The standard base is de-ned by e . The intrinsic parameters in the calibration matrix K encode the camera's internal geometry: f denotes the focal length, fl the aspect ratio, s the skew and the principal point. A camera for which K is unknown is said to be uncalibrated. It is well-known that for uncalibrated cameras, it is only possible to recover the 3D scene and the camera poses up to unknown projective transformation [8, 10, 35]. This follows directly from the projection equation (1): Given one set of camera matrices and 3D points that satis-es (1), another reconstruction can be obtained from where T is a non-singular 4 \Theta 4 matrix corresponding to a projective transformation of P 3 . A quadric in P n is de-ned by the quadratic form where Q denotes a (n homogeneous point coordinates. The dual is a quadric envelope, given by where \Pi denotes homogeneous coordinates for hyper-planes of dimension are tangent to the quadric. For non-singular matrices, it can be shown that Q ' (Q ) \Gamma1 (see [33] for a proof). A quadric with a non-singular matrix is said to be proper. Quadrics with no real points are called virtual. In the plane, are called conics. We will use C for the 3 \Theta 3 matrix that de-nes the conic points x T and C for its dual that de-nes the envelope of tangent lines l T C (where C ' C \Gamma1 ). The image of a quadric in 3D-space is a conic, i.e. the silhouette of a 3D quadric is projected to a conic curve. This can be expressed in envelope forms as Projective geometry encodes only cross ratios and incidences. Properties like parallelism and angles are not invariant under dioeerent projective coordinate systems. An aOEne space, where properties like parallelism and ratios of lengths are preserved, can be embedded in a projective space by singling out a plane at in-nity \Pi 1 . The points on \Pi 1 are called points at in-nity and be interpreted as direction vectors. In P 3 , Euclidean properties, like angles and lengths, are encoded by singling out a proper, virtual conic on \Pi 1 . This absolute scalar products between direction vectors. Its dual, the dual absolute quadric\Omega products between plane normals.\Omega 1 is a 4 \Theta 4 symmetric rank 3 positive semide-nite matrix, where the coordinate system is normally chosen such is\Omega 1 's unique null vector:\Omega 1 The similarities or scaled Euclidean transformations in projective space are exactly those transformations that invariant. The transformations that leave \Pi 1 invariant are the aOEne transformations. The dioeerent forms of the absolute conic will be abbreviated to (D)AC for (Dual) Absolute C onic. Given image conics in several images, there may or may not be a 3D quadric having them as image projections. The constraints which guarantee this in two images are called the Kruppa constraints [22]. In the two-image case, these constraints have been successfully applied in order to derive the critical sets, e.g. [28]. For the more general case of multiple images, the projection equation given by (4) can be used for each image separately. 3. Problem Formulation The problem of auto-calibration is to -nd the intrinsic camera parameters denotes the number of camera positions. In general, auto-calibration algorithms proceed from a projective re-construction of the camera motion. In order to auto-calibrate, some constraints have to be enforced on the intrinsic parameters, e.g. vanishing skew and/or unit aspect ratio. Thus, we require that the calibration matrices should belong to some proper subset G of the group K of 3 \Theta 3 upper triangular matrices. Once the projective reconstruction and the intrinsic parameters are known, Euclidean structure and motion are easily computed. For a general set of scene points seen in two or more images, there is a unique projective reconstruction. However, certain special con-g- urations, known as critical surfaces, give rise to additional ambiguous solutions. For two cameras, the critical con-gurations occur only if both camera centres and all scene points lie on a ruled quadric surface [24]. Furthermore, when an alternative reconstruction exists, then there will always exist a third distinct reconstruction. For more than two cameras, the situation is less clear. In [25], it is proven that when six scene points and any number of camera centres lie on a ruled quadric, then there are three distinct reconstructions. If there are other critical surfaces is an open problem. We will avoid critical surfaces by assuming unambiguous recovery of projective scene structure and camera motion. In other words, the camera matrices and the 3D scene are considered to be known up to an unknown projective transformation. We formulate the auto-calibration problem as follows: If all that is known about the camera motions and calibrations is that each calibration matrix K i lies in some given constraint set G ae K, when is a unique auto-calibration possible? More Problem 3.1. Let G ae K. Then, given the true camera projections G, is there any projective transformation T (not a similarity) such that ~ ~ calibration matrices ~ lying in G? Without constraints on the intrinsic parameters T can be chosen arbitrarily, so auto-calibration is impossible. Also, T is only de-ned modulo a similarity, as such transformations leave K in the decomposition invariant. Based on the above problem formulation, we can de-ne precisely what is meant by a motion being critical. De-nition 3.1. Let G ae K and let (P denote two projectively related motions, with calibration matrices (K respectively. If the two motions are not related by a Euclidean transformation and are said to be critical with respect to G. A motion is critical if there exists an alternative projective motion satisfying the auto-calibration constraints. Without any additional as- sumptions, it is not possible to tell which motion is the true one. One natural additional constraint is that the reconstructed 3D structure should lie in front of all cameras. In many (but by no means all) cases this reduces the ambiguity, but it depends on which 3D points are observed. According to (4) the image of the absolute i\Omega I 0 Thus, knowing the calibration of a camera is equivalent to knowing its image Also, if there is a projectively related motion ( ~ the false image of the true absolute conic is the true image of a ifalsej absolute conic: i\Omega ~ T\Omega i\Omega T\Omega virtual quadric of rank 3. This observation allows us to eliminate the ifalsej motion ( ~ i=1 from the problem and work only with the true Euclidean motion, but with a false absolute dual quadric\Omega f . Problem 3.2. Let G ae K. Then, given the true motion G, is there any other proper, virtual conic \Omega f , dioeerent 1 , such that P i\Omega Given only a 3D projective reconstruction derived from uncalibrated images, the true\Omega 1 is not distinguished in any way from any other proper, virtual planar conic in projective space. In fact, given any such potential conic\Omega f , it is easy to -nd a 'rectifying' projective transformation that converts it to the Euclidean DAC and hence de-nes a false Euclidean structure. To recover the true struc- ture, we need constraints that single out the true\Omega 1 and \Pi 1 from all possible false ones. Thus, ambiguity arises whenever the images of some non-absolute conic satisfy the auto-calibration constraints. We call such conics potential absolute conics or false absolute conics. They are in one-to-one correspondence with possible false Euclidean structures for the scene. A natural question is whether the problem is dependent on the actual values of the intrinsic parameters. We will show that this is not the case whenever the set G is a proper subgroup of K. Fortunately, according to the following easy lemma, most of the relevant auto-calibration constraints are subgroup conditions. Lemma 3.1. The following constrained camera matrices form proper subgroups of the 3 \Theta 3 upper triangular matrices K: (i) Zero skew, i.e. (ii) Unit aspect ratio, i.e. (iii) Vanishing principal point, i.e. (iv) Unit focal length, i.e. (v) Combinations of the above conditions. Independence of the values of the intrinsic camera parameters is shown as follows: Lemma 3.2. Let G i 2 G for m, where G is a proper subgroup of K. Then, the motion (P is critical w.r.t. G if and only if the motion is critical w.r.t. G. Proof. If is critical with the alternative motion ( ~ calibrations are also critical, because G i K by the closure of G under multipli- cation. The converse also holds with G \Gamma1 , by the closure of G under inversion. Camera matrices with prescribed parameters do not in general form a subgroup of K, but it suOEces for them to be of the more general form known matrix and K belongs to a proper subgroup of K. For example, the set of all camera matrices with known focal length f has the form 2f 0 0 The invariance with respect to calibration parameters simpli-es things, especially if one chooses G m. With this in mind, we restrict our attention to proper subgroups of K and formulate the problem as follows. Problem 3.3. Let G ae K be a proper subgroup. Then, given the true motion for calibrated cameras, where any other false absolute conic\Omega f , dioeerent i]\Omega ~ where ~ 4. Approach We want to explicitly characterize the critical motions (relative camera placements) for which particular auto-calibration constraints are insuf- -cient to uniquely determine Euclidean 3D structure. We assume that projective structure is available. Alternative Euclidean structures correspond one-to-one with possible locations for a potential absolute conic in P 3 . Initially, any proper virtual projective plane conic is potentially absolute, so we look for such conics\Omega whose images also satisfy the given auto-calibration constraints. Ambiguity arises if and only if more than one such conic exists. We work with the true camera motion in a Euclidean frame where the true absolute conic\Omega 1 has its standard coordinates. Several general invariance properties help to simplify the problem: Calibration invariance: As shown in the previous section, if the auto-calibration constraints are subgroup conditions, the speci-c parameter values are irrelevant. Hence, for the purpose of deriving critical motions, we are free to assume that the cameras are in fact secretly calibrated, even though we do not assume that we know this. (All that we actually know is K which does not allow some image conics I to be excluded outright). Rotation invariance: For known-calibrated cameras, K i can be set to identity, and thus the image ! I of any false AC must be identical to the image of the true one. Since i\Omega i\Omega hold for any rotation R, the image ! i is invariant to camera rota- tions. Hence, criticality depends only on the camera centres, not on their orientations. More generally, any camera rotation that leaves the auto-calibration constraints intact is irrelevant. For example, arbitrary rotations about the optical axis and 180 ffi AEips about any axis in the optical plane are irrelevant if (a; s) is either (1; 0) or unconstrained, and Translation invariance: For true or false absolute conics on the plane at in-nity, translations are irrelevant so criticality depends only on camera orientation. In essence, Euclidean structure recovery in projective space is a matter of parameterizing all of the possible proper virtual plane conics, then using the auto-calibration constraints on their images to algebraically eliminate parameters until only the unique true absolute conic remains. More abstractly, if C parameterizes the possible conics and X the camera geometries, the constraints cut out some algebraic variety in (C; X) space. A constraint set is useful for Euclidean structure from motion recovery only if this variety generically intersects the subspaces in one (or at most a few) points (C; X 0 ), as each such intersection represents an alternative Euclidean structure for the reconstruction from that camera geometry. A set of camera poses X is critical for the constraints if it has exceptionally (e.g. in-nitely) many intersections. Potential absolute conics can be represented in several ways. The following parameterizations have all proven relatively tractable: (i) Choose a Euclidean frame in which\Omega f is diagonal, and express all camera poses relative this frame [36, 37]. This is symmetrical with respect to all the images and usually gives the simplest equations. How- ever, in order to -nd explicit inter-image critical motions, one must revert to camera-based coordinates which is sometimes delicate. The cases of a -nite false absolute conic and a false conic on the plane at in-nity must also be treated separately, e.g.\Omega with either d 3 or d 4 zero. (ii) Work in the -rst camera frame, encoding\Omega f by its -rst image ! and supporting plane (n T ; 1). Subsequent images ! are given by the inter-image homographies H is the i th camera pose. The output is in the -rst camera frame and remains well-de-ned even if the conic tends to in-nity, but the algebra required is signi-cantly heavier. Parameterize\Omega f implicitly by two images ! 2 subject to the Kruppa constraints. In the two-image case this approach is both relatively simple and rigorous - two proper virtual dual image conics satisfy the Kruppa constraints if and only if they de-ne a (pair of) corresponding 3D potential absolute conics - but it does not extend so easily to multiple images. The derivations below are mainly based on method (i) . 5. Critical Motions In this section, the varieties of critical motions are derived. In most situations, the problem is solved in two separate cases. One is when there are potential absolute conics on the plane at in-nity, \Pi 1 , and the other one is conics outside \Pi 1 . If the potential conics are all on \Pi 1 , it is still possible to recover \Pi 1 and thereby obtain an aOEne reconstruction. Otherwise, the recovery of aOEne structure is ambiguous, and we say that the motion is critical with respect to aOEne reconstruction. The following constraints on the camera calibration are considered: (i) known intrinsic parameters, (ii) unknown focal lengths, but the other intrinsic parameters known, (iii) known skew and aspect ratio. These constraints form a natural hierarchy and they are perhaps the most interesting ones from a practical point of view. In Section 3, it was shown in that it is suOEcient to study the normalized versions of the auto-calibration constraints, since critical motions are independent of the speci-c values of the intrinsic parameters. That is, when some of the intrinsic parameters are known, e.g. the principal point is (10; 20), we may equivalently analyze the case of principal point set to (0; 0). The corresponding camera matrices give rise to subgroup conditions according to Lemma 3.1. 5.1. Known intrinsic parameters We start with fully calibrated perspective cameras. The results may not come as a surprise, but it is important to know that there are no other possible degenerate con-gurations. Proposition 5.1. Given projective structure and calibrated perspective cameras at m - 3 distinct -nite camera centres, Euclidean structure can always be recovered uniquely. With distinct camera centres, there is always exactly a twofold ambiguity. Proof. Assuming that the cameras have does not change the critical motions. The camera orientations are irrelevant because any false absolute conic must have the same (rotation invariant) images as the true one. Calibrated cameras never admit false absolute conics on as the (known) visual cone of each image conic can intersect \Pi 1 in only one conic, which is the true absolute conic. Therefore, consider a -nite absolute conic\Omega f , with supporting plane outside \Pi 1 . As all potential absolute conics are proper, virtual and positive semi-de-nite [34, 37], a Euclidean coordinate system can be chosen such f has supporting plane z = 0, and matrix coordinates \Omega Since the cameras are calibrated, their orientations are the conic projection (4) in each camera becomes \Gammat]\Omega y x z z x y x y x z z y1 optical centre Figure 1. A twisted pair of reconstructions. As the conic should be proper, both d 4 6= 0 and t 3 6= 0, which gives Thus the only solutions are t and\Omega implies that there are at most two camera centres, and the false conic is a circle of imaginary radius i z, centred in the plane bisecting the two camera centres In the two-image case, the improper self-inverse projective transfor- mation interchanges the true\Omega 1 and the f , according to T\Omega '\Omega and takes the two projection matrices P to While the -rst camera remains -xed, the other has rotated 180 ffi about the axis joining the two centres. This twofold ambiguity corresponds exactly to the well-known twisted pair duality [23, 18, 27]. The geometry of the duality is illustrated in Figure 1. The 'twist' T represents a very strong projective deformation that cuts the scene in half, moving the plane between the cameras to in-nity, see Figure 2. By considering twisted vs. non-twisted optical ray intersec- tions, one can also show that it reverses the relative signs of the depths, so for one of the solutions the structure will appear to be behind one visual optical centre supporting planes potential conics cone Figure 2. Intersecting the visual cones of two image conics satisfying the Kruppa constraints generates a pair of 3D conics, corresponding to the two solutions of the twisted pair duality. camera, cf. [17]. To conclude, Proposition 5.1 states that any two-view geometry has a 'twisted pair' projective involution symmetry and any camera con-guration with three or more camera centres has a unique projective-to-Euclidean upgrade. 5.2. Unknown focal lengths In the case of two images and internally calibrated cameras modulo unknown focal lengths, it is in general possible to recover Euclidean structure. Since we know that the solutions always occur in twisted pairs (which can be disambiguated using the positive depth constraint), it is more relevant to characterize the motions for which there are solutions other than the twisted pair duality. Therefore, the two-camera case will be dealt with separately, after having derived the critical motions for arbitrary many images. 5.2.1. Many images If all intrinsic parameters are known except for the focal lengths, the camera matrix can be assumed to be which in turn implies that the image of a potential absolute conic satis-es We start with potential absolute conics on \Pi 1 . Potential absolute conics on \Pi 1 Let C f denote a 3 \Theta 3 matrix corresponding to a false absolute conic (in locus form) on the plane at in-nity. Since C f is not the true one, I. The image of C f is according to (4) Notice that criticality is independent of translation of the camera. Two cameras are said to have the same viewing direction if their optical axes are parallel or anti-parallel. Proposition 5.2. Given \Pi 1 and known skew, aspect ratio and principal point, then a motion is critical if and only if there is only one viewing direction. Proof. Choose coordinates in which camera 1 has orientation R Suppose a motion is critical. According to (6) and (7), this implies that 3 for some - ? \Gamma1. For camera 2, let apply (7), I This implies that r in turn, R which is equivalent to a -xed viewing direction of the camera. Conversely, suppose the viewing direction is -xed, which means that R Then, it is not possible to disambiguate between any of the potential absolute conics in the pencil C f (-) ' I 3 , since R i C f R T Potential absolute conics outside \Pi 1 Assume we have a critical motion (R with the false dual absolute conic\Omega f . If the supporting plane f is \Pi 1 , the critical motion is described by Proposition 5.2, so assume f is outside \Pi 1 . As in the proof of Proposition 5.1, one can assume without loss generality that a Euclidean coordinate system has been chosen such f has supporting plane z = 0, and matrix coordinates \Omega The image of\Omega f is according to (4), i\Omega d 2i optical centres critical ellipse critical hyperbola Figure 3. Two orthogonal planes, where one plane contains an ellipse and the other contains a hyperbola. A necessary condition for degeneracy is that R i should diagonalize C f to the form (6), i.e. the matrix C f must have two equal eigenvalues. As it is always possible to -nd an orthogonal matrix that diagonalizes a real, symmetric matrix [5], all we need to do is to -nd out precisely when C f has two equal eigenvalues. Lemma A.1 in the Appendix characterizes matrices of this form. Applying the lemma to C f in (8), with oe results in the following cases: (i) If d 1 6= d 2 , then a. b. These equations describe a motion on two planar conics for which the supporting planes are orthogonal. On the -rst plane, the conic is an ellipse, while on the other the conic is a hyperbola (depending on whether d 1 ? d 2 or vice versa), see Figure 3. (ii) If d arbitrary. Notice that the second alternative in case (ii) of Lemma A.1 does not occur, since it implies t T e making C f rank-de-cient. Also, case (iii) is impossible, since oe It remains to -nd the rotations that diagonalize C f . Since rotations around the optical axis are irrelevant, only the direction of the optical axis is signi-cant. Suppose the optical axis is parameterized by the camera centre t and a direction d, i.e. Rg. Any point on the axis projects to the principal point, The direction d should equal the third row of R, which corresponds to the eigenvector of the single eigenvalue of C f . Regarding the proof of Lemma A.1, it is not hard to see that the eigenvectors are v ' in the two sub-cases in (i) above. Geometrically, this means that the optical axis must be tangent to the conic at each position, as illustrated in Figure 4(b). Similarly in (ii), it is easy to derive that which means that the optical axis should be tangent to the translation direction, cf. Figure 4(c). An exceptional case is when C f has a triple eigenvalue, because then any rotation is possible. However, according to Proposition 5.1, it occurs only for twisted pairs. To summarize, we have proven the following. Proposition 5.3. Given known intrinsic parameters except for focal lengths, a motion is critical w.r.t. aOEne reconstruction if and only if the motion consists of (i) rotations with at most two distinct centres (twisted pair ambiguity), or (ii) motion on two conics 1 (one ellipse and one hyperbola) whose supporting planes are orthogonal and where the optical axis is tangent to the conic at each position, or (iii) translation along the optical axis, with arbitrary rotations around the optical axis. The motions are illustrated in Figure 4. In case (i) and (ii), the ambiguity of the reconstruction is twofold, as there is only one false absolute conic, whereas in case (iii) there is a one-parameter family of potential planes at in-nity (all planes z=constant). Case (iii) can be seen as a special case of the critical motion in Proposition 5.2, which also has a single viewing direction, but arbitrary translations. 5.2.2. Two images For two cameras, projective geometry is encapsulated in the 7 degrees of freedom in the fundamental matrix, and Euclidean geometry in the 5 degrees of freedom in the essential matrix. Hence, from two projective images we might hope to estimate Euclidean structure plus two additional calibration parameters. Hartley [10, 11] gave a method for the case where the only unknown calibration parameters are the focal lengths of the two cameras. This was later elaborated by Newsam et. al. 1 The actual critical motion is the conics minus the two points where the ellipse intersects the plane z = 0, since the image ! is non-proper at these points. potential conic absolute optical centre (b) conic potential absolute optical centres critical ellipse critical hyperbola (c) conic potential optical centres absolute Figure 4. Critical motions for unknown focal lengths: (a) A motion with two -xed centres. (b) A planar motion on an ellipse and a hyperbola. (c) Translation along the optical axis. See Proposition 5.3. y x x z z y (b) x x y Figure 5. Critical con-gurations for two cameras: (a) Intersecting optical axes. (b) Orthogonal optical axis planes. See Proposition 5.4. [28], Zeller and Faugeras [41] and Bougnoux [4]. All of these methods are Kruppa-based. We will derive the critical motions for this case based on the results of the previous sections. Proposition 5.4. Given zero skew, unit aspect ratio, principal point at the origin, but unknown focal lengths for two cameras, then a motion (in addition to twisted pair) is critical if and only if (i) the optical axes of the two cameras intersect or (ii) the plane containing the optical axis of camera 1 and camera centre 2, is orthogonal to the plane containing optical axis of camera 2 and camera centre 1. Proof. Cf. [28]. Suppose a motion is critical. Regarding Proposition 5.2, we see that if there is only one viewing direction, the optical axes are parallel and intersect at in-nity, leading to (i) above. Examining the three possibilities in Proposition 5.3, we see that the -rst one is the twisted pair solution. The second one, either both cameras lie on the same conic (and hence their axes are coplanar and intersect) or one lies on the hyperbola, the other on the ellipse (in which case their optical axes lie in orthogonal planes) leading to (ii). Conversely, given any two cameras with intersecting or orthogonal-plane optical axes, it is possible to -t (a one-parameter family of) conics through the camera centres, tangential to the optical axes. The two critical camera con-gurations are shown in Figure 5. 5.3. Known skew and aspect ratio Consider the image ! of\Omega 1 . Inserting the parameterization of K in (2) into its dual !, it turns out that ! . Since f and fl never vanish, requiring that the skew vanishes is equivalent to ! The constraint can also be expressed in envelope form using ! dually ! If in addition to zero skew, unit aspect ratio is required in K, it is equivalent to ! 22 . This follows from the fact that ! and . The constraint can also be transfered to ! , dually ! Analyzing the above constraints on ! in locus form, results in the following proposition when the plane at in-nity \Pi 1 is known. Proposition 5.5. Given \Pi 1 , a motion is critical with respect to zero skew and unit aspect ratio if and only if there are at most two viewing directions. Proof. For each image we have the two auto-calibration constraints (9), given by (7). Choose 3D coordinates in which the -rst camera has orientation R I. The image 1 constraints become simply so we can parameterize C f with C 11 , C 12 and C 13 . Given a subsequent image 2, represent its orientation R 2 by a quaternion its two auto-calibration constraints, and eliminate C 11 between them to give: One of the 3 factors must vanish. If the -rst vanishes the motion is an optical axis rotation, q 2 If the second vanishes it is a 180 ffi AEip about an axis orthogonal to the optical one, q 2 In both cases the viewing direction remains unchanged and no additional constraint is enforced on C f . Finally, if the third factor vanishes, solving for C f in terms of q gives a linear family of solutions of the form (the third row of R 2 (q)) are the two viewing directions and (ff; fi) are arbitrary parameters. Conversely, given any potential AC C f 6' I, there is always exactly one pair of real viewing directions that make C f critical under (11). The linear family ff contains three rank 2 members, one for each eigenvalue - of C f (with fi 0 =ff calculation shows that each member can be decomposed uniquely (up to sign) into a pair of viewing direction vectors supporting (11), but only the pair corresponding to the middle eigenvalue is real. (Coincident eigenvalues correspond to coincident viewing directions and can be ignored). Hence, no potential AC C f can be critical for three or more real directions simultaneously. Table I. Summary of critical motions in auto-calibration. Auto-calibration Critical motions Reconstruction constraint ambiguity Known calibration twisted pair duality projective Unknown focal length (i) optical axis rotation aOEne but otherwise known (ii) motion on two planar conics projective calibration (iii) optical axis translation projective Unknown focal length (i) intersecting optical axes projective (two images only) (ii) orthogonal optical axis planes projective Zero skew and (i) two viewing directions aOEne unit aspect ratio (ii) complicated algebraic variety projective For potential absolute conics outside \Pi 1 things are more compli- cated. For each image, there are two auto-calibration constraints. So in order to single out the true absolute conic (which has 8 degrees of freedom), at least 4 images are necessary. For a f the polynomial constraints in (9) and (10) determine a variety in the space of rigid motions. We currently know of no easy geometrical interpretation of this manifold. It is easy to see that given a critical camera motion, the ambiguity is not resolved by rotation around the camera's optical axis. 5.4. Summary A summary of the critical motions for auto-calibration under the auto-calibration constraints studied is given in Table I. The reconstruction ambiguity is classi-ed as projective if the plane at in-nity cannot be uniquely recovered, and aOEne if it is possible. As mentioned earlier, the twisted pair duality is not a true critical motion, since the positive-depth constraint can always resolve the ambiguity. 6. Particular Motions Some critical motions occur frequently in practice. In this section, a selection of them is analyzed in more detail. 6.1. Pure rotation In the case of a stationary camera performing arbitrary rotations, no 3D reconstruction is possible. There always exist many potential absolute conics outside \Pi 1 . However, it is still possible to recover the internal camera calibration, provided there are no potential absolute conics on \Pi 1 , cf. [37]. Proposition 5.2 and Proposition 5.5, regarding critical motions and potential ACs on \Pi 1 tells us when such auto-calibration is possible for a purely rotating camera. 6.2. Pure translation If a sequence of movements only consists of arbitrary translations and no rotations, all proper, virtual conics on \Pi 1 are potential absolute conics. Still, one could hope to recover the plane at in-nity correctly, and thus get an aOEne reconstruction. Proposition 6.1. Let (t be a general sequence of translations, where m is suOEciently large. Then, the motion is (i) always critical w.r.t. aOEne reconstruction under the constraints zero skew and unit aspect ratio. (ii) not critical w.r.t. aOEne reconstruction under the constraints zero skew, unit aspect ratio and vanishing principal point. Proof. (i) We need to show that there exists a potential DAC\Omega f outside which is valid for all . Choose a coordinate system such that \Theta I . Then for instance\Omega is a potential DAC (multiply P i\Omega i to get ! and check that it ful-lls (9) and (10)). (ii) follows directly from Proposition 5.3. Note that translating only along the optical axis in case (ii) above results in a critical motion. 6.3. Parallel axis rotations Sequences of rotations around parallel axes with arbitrary translations are interesting in several aspects. They occur frequently in practice and are one of the major degeneracies for auto-calibration with constant intrinsic parameters [36, 43]. See Figure 6. It follows directly from Proposition 5.5 that given zero skew, unit aspect ratio and general rotation angles, the -xed-axis motion is not critical unless it is around the optical axis. If we further add the vanishing principal point constraint, the optical axis remains critical according Figure 6. Rotations around the vertical axes with arbitrary translations. to Proposition 5.2. If we know only that the skew vanishes, we have the following proposition. Proposition 6.2. Let (R be a general motion whose rotations are all about parallel axes, where R suOEciently large. Given \Pi 1 , the motion is critical w.r.t. zero skew if and only if the rotation is around one of the following axes: or (1; \Gamma1; 0), where each denotes an arbitrary real number. Proof. Let C f denote a false AC on \Pi 1 . The zero skew constraint in using the parameterization in (7) gives C arbitrary rotation around a -xed axis (q can be parameterized by R. Inserting this into the zero skew constraint in yields a polynomial in R[-]. Since - can be arbitrary all coeOEcients of the polynomial must vanish. The solutions to the system of vanishing coeOEcients are the ones given above. Some of these critical axes may be resolved by requiring that the camera calibration should be constant. In [37], it is shown that parallel axis rotations under constant intrinsic parameters are always critical and give rise to the following pencil of potential absolute conics: Combining constant intrinsic parameters, and some a priori known values of the intrinsic parameters, some of the critical axes are still critical. Corollary 6.1. Let (R be a general motion with parallel axis rotations, where R suOEciently large. Given \Pi 1 , and constant intrinsic parameters, the following axes are the only ones still critical: (ii) (0; 0; 1) w.r.t. zero skew and unit aspect ratio, w.r.t. an internally calibrated camera except for focal length. Proof. (i) Using the potential ACs in (12) in the proof of Proposition 6.2, one -nds that the only critical axes remaining under the zero skew constraint are (0; ; ) and ( ; 0; ). (ii) and (iii) are proved analogously. 7. Experiments In practice, a motion is never exactly degenerate due to measurement noise and modeling discrepancies. However, if the motion is close to a critical manifold it is likely that the reconstructed parameters will be inaccurately estimated. To illustrate the typical eoeects of critical motions, we have included some simple synthetic experiments for case of two cameras with unknown focal lengths but other intrinsic parameters known. We focus on the question of how far from critical the two cameras must be to give reasonable estimates of focal length and 3D Euclidean structure [20]. The experimental setup is as follows: two unit focal length perspective cameras view 25 points distributed uniformly within the unit sphere. The camera centres are placed at (\Gamma2; \Gamma2; and (2; \Gamma2; 0) and their optical axes intersect at the origin, similar to the setup in Figure 5(a). Independent Gaussian noise of 1 pixel standard deviation is added to each image point in the 512 \Theta 512 images. In the experiment, the elevation angles are varied, upwards for the left camera and downwards for the right one, so that their optical axes are skewed and no longer meet. For each pose, the projective structure Unknown focal bundle: - Focal Length Elevation Angle (degrees) Unknown focal bundle: - Calibrated bundle: -x- Point Elevation Angle (degrees) Figure 7. Relative errors vs. camera elevation for two cameras. and the fundamental matrix are estimated by a projective bundle adjustment that minimises the image distance between the measured and reprojected points [3]. Then, the focal lengths are computed analytically with Bougnoux' method [4]. For comparison, a calibrated bundle adjustment with known focal lengths is also applied to the same data. The resulting 3D error is calculated by Euclidean alignment of the true and reconstructed point sets. Figure 7 shows the resulting root mean square errors over 100 trials as a function of elevation angle. At zero elevation, the two optical axes intersect at the origin. This is a critical con-guration according to Proposition 5.4. A second critical con-guration occurs when the epipolar planes of the optical axes become orthogonal at around 35 ffi elevation. Both of these criticalities are clearly visible in both graphs. For geometries more than about 5-10 ffi from criticality, the focal lengths can be recovered quite accurately and the resulting Euclidean 3D structure is very similar to the optimal 3D structure obtained with known calibration. 8. Conclusion In this paper, the critical motions in auto-calibration under several auto-calibration constraints have been derived. The various constraints on the intrinsic parameters have been expressed as subgroup conditions on the 3 \Theta 3 upper triangular camera matrices. With this type of condition, we showed that the critical motions are independent of the speci-c values of the intrinsic parameters. It is important to be aware of the critical motions when trying to auto-calibrate a camera. Additional scene or motion constraints may help to resolve the ambiguity, but clearly the best way to avoid degeneracies is to use motions that are ifarj from critical. Some synthetic experiments have been performed that give some practical insight to the numerical conditioning of near-critical and critical stereo con-gurations. Acknowledgments This work was supported by the European Union under Esprit project LTR-21914 Cumuli. We would like to thank Sven Spanne for constructing the proof of Lemma A.1. Appendix Lemma A.1. Let A be a real, symmetric 3 \Theta 3 matrix of the form real 3 vector and ae a non-zero real scalar. Let oe 1 ,oe 2 and oe 3 be given real scalars. Then, necessary and suOEcient conditions on (t,ae) for A to have two equal eigenvalues can be divided into three cases: for at least one i (where or 3). Furthermore, ae can take the values: for any i for which t T e (ii) If oe a. t T e b. t T e (iii) If oe arbitrary. Proof. It follows from the Spectral Theorem [5] that if A is real and symmetric with two equal eigenvalues -, then there is a third eigenvector v and a scalar - such that corresponding to v is -). This gives Multiplying this matrix equation with e 1 , e 2 and e 3 , results in three vector equations, To prove (i), assume oe 1 6= oe 2 6= oe 3 . The orthogonal bases e 1 , e 2 and e 3 are linearly independent and cannot all be linear combinations of t and v, so one of oe must vanish, and thereby exactly one. Suppose If one of the coeOEcients is non-zero, then t and v would be linearly dependent. However, this is impossible because e 2 and e 3 are linearly independent and oe Analogously, ae 6= 0 because otherwise e 2 and e 3 would be linearly dependent according to (13). Therefore t T e If t is orthogonal to e 1 and calculations yield that ae must be chosen as which is also suOEcient. When two or three of oe i are equal, similar arguments can be used to deduce (ii) and (iii). --R Close Range Photogrammetry and Machine Vision. Matrices and Linear Transformations. Matrix Computation. Handbuch der Physiologischen Optik. Theory of Reconstruction from Image Motion. Algebraic Projective Geometry. Analytical Quadrics. 'Vision 3D Non Calibr#e: Contributions # la Reconstruction Projective et #tude des Mouvements Critiques pour l'Auto-Calibrage' --TR Multiple Interpretations of the Shape and Motion of Objects from Two Perspective Images A theory of self-calibration of a moving camera Three-dimensional computer vision Reconstruction from Calibrated CamerasMYAMPERSANDmdash;A New Proof of the Kruppa-Demazure Theorem Estimation of Relative Camera Positions for Uncalibrated Cameras Camera Self-Calibration What can be seen in three dimensions with an uncalibrated stereo rig Self-Calibration from Image Triplets Euclidean 3D Reconstruction from Image Sequences with Variable Focal Lenghts Minimal Conditions on Intrinsic Parameters for Euclidean Reconstruction Autocalibration and the absolute quadric Euclidean Reconstruction from Image Sequences with Varying and Unknown Focal Length and Principal Point Critical Motion Sequences for Monocular Self-Calibration and Uncalibrated Euclidean Reconstruction Metric calibration of a stereo rig The Modulus Constraint Euclidean Reconstruction from Constant Intrinsic Parameters Ambiguity in Reconstruction From Images of Six Points From Projective to Euclidean Space Under any Practical Situation, a Criticism of Self-Calibration Self-Calibration and Metric Reconstruction in spite of Varying and Unknown Internal Camera Parameters --CTR Antonio Valds , Jos Ignacio Ronda, Camera Autocalibration and the Calibration Pencil, Journal of Mathematical Imaging and Vision, v.23 n.2, p.167-174, September 2005 Gang Qian , Rama Chellappa, Bayesian self-calibration of a moving camera, Computer Vision and Image Understanding, v.95 n.3, p.287-316, September 2004 P. Sturm , Z. L. Cheng , P. C. Y. Chen , A. N. Poo, Focal length calibration from two views: method and analysis of singular cases, Computer Vision and Image Understanding, v.99 n.1, p.58-95, July 2005 Pr Hammarstedt , Fredrik Kahl , Anders Heyden, Affine Reconstruction from Translational Motion under Various Autocalibration Constraints, Journal of Mathematical Imaging and Vision, v.24 n.2, p.245-257, March 2006 Toms Svoboda , Daniel Martinec , Toms Pajdla, A convenient multicamera self-calibration for virtual environments, Presence: Teleoperators and Virtual Environments, v.14 n.4, p.407-422, August 2005 Kalle strm , Fredrik Kahl, Ambiguous Configurations for the 1D Structure and Motion Problem, Journal of Mathematical Imaging and Vision, v.18 n.2, p.191-203, March Lourdes Agapito , E. Hayman , I. Reid, Self-Calibration of Rotating and Zooming Cameras, International Journal of Computer Vision, v.45 n.2, p.107-127, November 2001 Loong-Fah Cheong , Chin-Hwee Peh, Depth distortion under calibration uncertainty, Computer Vision and Image Understanding, v.93 n.3, p.221-244, March 2004 Xiaochun Cao , Jiangjian Xiao , Hassan Foroosh , Mubarak Shah, Self-calibration from turn-table sequences in presence of zoom and focus, Computer Vision and Image Understanding, v.102 n.3, p.227-237, June 2006 Han , Takeo Kanade, Multiple Motion Scene Reconstruction with Uncalibrated Cameras, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.25 n.7, p.884-894, July Marta Wilczkowiak , Peter Sturm , Edmond Boyer, Using Geometric Constraints through Parallelepipeds for Calibration and 3D Modeling, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.27 n.2, p.194-207, February 2005

critical motions;absolute conic;calibration;structure and motion;projective geometry;auto-calibration;3D reconstruction

361263

Tolerance to Multiple Transient Faults for Aperiodic Tasks in Hard Real-Time Systems.

AbstractReal-time systems are being increasingly used in several applications which are time-critical in nature. Fault tolerance is an essential requirement of such systems, due to the catastrophic consequences of not tolerating faults. In this paper, we study a scheme that guarantees the timely recovery from multiple faults within hard real-time constraints in uniprocessor systems. Assuming earliest-deadline-first scheduling (EDF) for aperiodic preemptive tasks, we develop a necessary and sufficient feasibility-check algorithm for fault-tolerant scheduling with complexity $O(n^2 \cdot k)$, where $n$ is the number of tasks to be scheduled and $k$ is the maximum number of faults to be tolerated.

Introduction The interest in embedded systems has been growing steadily in the recent past, specially those ples include autopilot systems, satellite and launch vehicle control, as well as robots, whether in collaborating teams or not. For some of these systems, termed hard real-time systems (HRTSs), the consequences of missing a deadline may be catastrophic. The ability to tolerate faults in HRTSs is crucial, since a task can potentially miss a deadline when faults occur. In case of a fault, a deadline can be missed if the time taken for recovery from faults is not taken into account during the phase that tasks are submitted/accepted to the system. Clearly, accounting for recovery from faults is an essential requirement of HRTSs. When dealing with such HRTSs, permanent faults can be tolerated by using hot-standby spares [KS86], or they can be masked by modular redundancy techniques [Pra86]. In addition to permanent faults, tolerance to transient faults is very important, since it has been shown to occur much more frequently than permanent faults [IR86, IRH86, CMS82]. In a study, an orbiting satellite containing a microelectronics test system was used to measure error rates in various semiconductor devices including microprocessor systems [CMR92]. The number of errors, caused by protons and cosmic ray ions, mostly ranged between 1 and 15 in 15-minute intervals, and was measured to be as high as 35 in such intervals. More examples of such safety critical applications can be found in [LH94]. Transient faults can be dealt with through temporal redundancy, that is, allowing extra time (slack) in the schedule to re-execute the task or to execute a recovery block [HLMSR74]. The problem solved in this paper is as follows. Given a set of n aperiodic tasks, we seek to determine if each task in the set T is able to complete execution before its deadline under EDF scheduling even if the system has to recover from (at most) k faults. We consider a uniprocessor system and assume that each task may be subjected to multiple transient faults. A simple solution would be to check the feasibility of each of the schedules generated by the possible combination of faults using the approach described in [LLMM99] for each schedule. The high complexity of this scheme provides the impetus for searching for a more efficient solution. The solution presented in this paper develops an optimal (necessary and sufficient) feasibility check that runs in O(n 2 \Delta time in the worst case. Although we consider aperiodic tasks, we note that the technique presented in this paper can be used to verify the fault-tolerance capabilities of a set of periodic tasks by considering each instance of a periodic task as an aperiodic task within the Least Common Multiple of the periods of all the periodic tasks. Moreover, scheduling aperiodic tasks is the basis for scheduling periodic tasks in frame-based systems, where a set of tasks (usually having precedence constraints) is invoked at regular time intervals. This type of systems is commonly used in practice because of its simplicity. For example, in tracking/collision avoidance applications, motion detection, recognition/verification, trajectory estimation and computation of time to contact are usually component sub-tasks within a given frame (period) [CBM93]. Similarly, a real-time image magnification task might go through the steps of non-linear image interpolation, contrast enhancement, noise suppression and image extrapolation during each period [NC96]. Even though these are periodic tasks, the system period is unique and therefore the scheduling of the each instance (corresponding in our nomenclature to an "aperiodic" task) can be done within a specific time interval. The rest of this paper is organized as follows. In Section 2 we present the model and notation for the aperiodic, fault-tolerant scheduling problem. In Section 3 we introduce an auxiliary function that will aid in the presentation of our solution. In Section 4 we describe the feasibility tests for a set of tasks under a specific fault pattern and generalize it in Section 5 for any fault pattern, examining the worst case behavior with respect to k faults. In Section 6, we survey some related work and, in Section 7, we finalize the paper with concluding remarks and directions for future work. Model and Notation We consider a uniprocessor system, to which we submit a set T of n tasks: g. A task - i is modeled by a tuple - is the ready time (earliest start time of the task), D i is the deadline, and C i is the maximum computation time (also called worst case execution time). The set of tasks that become ready at a given time t is denoted by RS(T ; t). That is, RS(T We assume EDF schedules with ties in deadlines broken arbitrarily. The schedule of T is described by the function " if EDF does not schedule any task between t and t time. We will use EDF (T ) to refer to the EDF schedule of T . We define e i to be the time at which task - i completes execution in EDF (T ), and we define the function to be the number of free slots between is, the number of slots for which EDF (T ; (excluding the slot that starts at t 2 ). EDF (T ) is said to be feasible if e i - D i for all It is assumed that faults can be detected at the end of the execution of each task. The time required by the fault detection mechanism can be added to the worst case computation time C i of the task and does not hinder the timeliness of the system. Many mechanisms have been proposed for fault detection at the user level, the operating system level, and the hardware level. At the user level, a common technique is to use consistency or sanity checks, which are procedures supplied by the user, to verify the correctness of the results [HA84, YF92]. For example, using checksums, checking the range of the results or substituting a result back into the original equations can be used to detect a transient error. Many mechanisms that exist in operating systems and computer hardware may be used for error detection and for triggering recovery. Examples are the detection of illegal opcode (caused by bus error or memory corruption), memory range violation, arithmetic exceptions and various time-out mechanisms. Hardware duplication of resources can also be used for detecting faults through comparison of results. It should be noted, however, that while each of the mechanisms described above is designed for detecting specific types of faults, it has been long recognized that it is not possible for a fault detection mechanism to accomplish a perfect coverage over arbitrary types of faults. When a fault is detected, the system enters a recovery mode where some recovery action must be performed before the task's deadline. We assume that a task - i recovers from a fault by executing a recovery block [HLMSR74, LC88], - i;1 at the same priority of - i . A fault that occurs during the execution of - i;1 is detected at the end of - i;1 and is recovered from by invoking a second recovery block, - i;2 , and so on. It is assumed that the maximum time for a recovery block of - i to execute is . The recovery blocks for each task may have a different execution time from the task itself; in other words, the recovery is not restricted to re-execution of the task. Recovery blocks can be used for avoiding common design bugs in code, for providing a less accurate result in view of the limited time available for recovery, or for loading a "safe" state onto memory from some stable source (across the network or from shielded memory). We shall denote a pattern of faults over T as a set g, such that f i is the number of times the task - i 2 T or its recovery blocks will fail before successful completion. We use to denote the EDF schedule of T under the fault pattern F , that is, when - i is forced to execute f i recovery blocks. EDF F (T ) is said to be feasible if for all its recovery blocks complete by D i . Note that EDF F (T ) cannot be feasible if for any i. Given a task set T and a specific fault pattern F , we define two functions. The first function, defines the amount of work (execution time) that remains to be completed at time t in EDF (T ). This work is generated by the tasks that became ready at or before time t, that is by the tasks in f- Specifically, where the - \Gamma operator is defined as a - At a time t, any positive amount of work in W decreases by one during the period between t, while when a task becomes ready at t, the work increases by the computation time of this task. The second function, W F (T ; t) is defined in a similar way, except that we include time for the recovery of failed tasks at the point they would have completed in the fault-free schedule. Specifically, The two functions defined above will be used to reason about the extra work needed to recover from faults. Note that although task - i may complete at a time different than e i in EDF F (T ), the function W F has the important property that it is equal to zero only at the beginning of an idle time slot in EDF F (T ). This, and other properties of the two functions defined above are given next. only if there is no work to be done at time t in EDF (T ), which means that any task with R i - t finishes at or before time t in the fault-free case. Property 2: W F only if there is no work to be done at time t in EDF F (T ), which means that any task with R i - t finishes at or before time t when the tasks are subject to the fault pattern F . Property 3: W F That is, the amount of work incurred when faults are present is never smaller than the amount of work in the fault-free case. That is, the slot before the end of a task is never idle. The above four properties follow directly from the definition of W 3 The ffi-Function In order to avoid explicitly deriving the EDF schedule in the presence of faults, we define a function, ffi, which loosely corresponds to the "extra" work induced by a certain fault pattern, F . Intuitively, is the amount of unfinished "extra" work that has been induced by the fault pattern F at time t. In other words, it is the work needed above and beyond what is required in the fault-free schedule for T . The idle time in the fault-free EDF schedule is used to do this extra work. The ffi -function will play an important role in the process of checking if each task meets its deadline in EDF F (T ). Following is a method for computing ffi directly from the fault-free EDF schedule of T and the fault pattern F . (2) In order to show that the above form for ffi(T ; t; F) is equivalent to W F consider the four different cases above. case 1: At no task can end, and thus t 0 6= e j for any j. From the definitions of W and W F , this implies that W (T ; case 2: When implies that W (T which by Property 3 implies that also W F case 3: When EDF (T ; states that W (T ; states that j. In this case, ffi (T ; t; case 4: When t 6= e i and EDF implies that W (T which by Property 3 implies that also W F Hence, the - operations in the definitions of W and W F reduce to the usual subtraction, and it is straightforward to show that For illustration, Figure 1 shows an example of a task set and the corresponding values of the function for a specific F . In this example, we consider the case in which only - 1 and - 3 may be subject to a fault. Note that the value of ffi decreases when EDF (T ) is idle and increases at the end of each task that is indicated as faulty in F . R C D V223000 Fault-free EDF Schedule Figure 1: Task Set, EDF schedule and ffi values for f As we have mentioned above, the ffi function is an abstraction that represents the extra work to be performed for recovery. This extra work reduces to zero when all ready tasks complete execution and recovery, as demonstrated by the following theorem. Theorem in both EDF (T ) and EDF F (T ), any task with R i - t finishes at or before time t. Proof: If Equation (2), this decrease in the value of the ffi-function is only possible if EDF (T ; ", which from Property 1 leads to W (T ; Equation (1) gives W F and the proof follows from Properties 1 and 2 ffl 4 Feasibility Test for a Task Set Under a Specific Fault Pattern Given a task set, T , and a fault pattern, F , we now present a method for checking whether the lowest priority task, denoted by - ' 2 T , completes by its deadline in EDF F (T ). Theorem Given a task set, T , and a fault pattern, F , the lowest priority task, - ' , in T completes by D ' in EDF F (T ), if and only if Proof: To prove the if part, assume that t 0 is the smallest value such that e ' - t 0 - D ' and are identical from which implies that - ' completes by e ' - D ' in both schedules. If, however, be the latest time before t 0 such that ffi(T ; - t; F) ? 0. Note that - is the first value after e ' at which the definition of - t). Hence, by Theorem 1, all tasks that are ready before - t finish execution by - t in both EDF (T ) and EDF F (T ). Moreover, ffi(T ; t; which means that thus EDF (T ) is identical to EDF F (T ) in that period. But - ' completes in EDF (T ) at e ' , which means that it also completes at e ' in EDF F (T ). We prove the only if part by contradiction: assume that finishes in EDF F (T ) at - t for some e ' - t - D ' . The fact that the lowest priority task, - ' , executes between time - means that no other task is available for execution at - t \Gamma 1, and thus 1. Given the assumption that implies that which by Property 1 implies that EDF leads to which is a contradiction ffl The next corollaries provide conditions for the feasibility of EDF F (T ) for the entire task set, T . Corollary 1: A necessary and sufficient condition for the feasibility of EDF F (T ) for a given T and a given F can be obtained by applying Theorem 2 to the n task sets T j , contains the j highest priority tasks in T . Proof: The proof is by induction. The base case is trivial, when since there is only a single task. For the induction step, assume that EDF F (T j ) is feasible and consider T where - ' has a lower priority than any task in T j . In EDF F (T j+1 ), all tasks in T j will finish at exactly the same time as in EDF F (T j ), since - ' has the lowest priority. Hence, the necessary and sufficient condition for the feasibility of EDF F (T j+1 ) is equivalent to the necessary and sufficient condition for the completion of - ' by D ' ffl Corollary 2: A sufficient (but not necessary) condition for the feasibility of EDF F (T ) for a given T and a given F is Proof: Note that the proof of the only if part in Theorem 2 relies on the property that - ' is the lowest priority task, which, in EDF, means the task with the latest deadline. The if part of the theorem, however, is true even if - ' is not the lowest priority task. Hence, any - i 2 T , completes by D i in EDF F (T ), if which proves the corollary ffl Figure 2: The fault-tolerant schedule for the task set in Figure 1 We clarify the conditions of the above corollaries by examples. First, we show that the condition given in Corollary 2 is not necessary for the feasibility of EDF F (T ). That is, we show that, for any given - i , it is not necessary that in order for - i to finish by D i in EDF F (T ). This can be seen from the example task set and fault pattern shown in Figure 1. The value of ffi(T ; t; F) is not zero between e 7. Yet, as shown in Figure 2, - 1 and - 2 will finish by their deadlines in EDF F (T ). In other words, the condition that stated in Theorem 2, is necessary and sufficient for the feasibility of only the lowest priority task in EDF F (T ) (task - 4 in the above example). Next, we show that, as stated in Corollary 1, we have to repeatedly apply Theorem 2 to all task sets T j , to obtain a sufficient condition for the feasibility of the entire task set. In other words, it is not sufficient to apply Theorem 2 only to T . This can be demonstrated by modifying the example of Figure 1 such that D 7. Clearly, this change in D 3 may still result in the same EDF schedule for T and thus will not change the calculation of Although the application of Theorem 2 guarantees that - 4 will finish by its deadline in EDF F (T ), the recovery of - 3 will not finish by D as seen in Figure 2. Assume, without loss of generality, that the tasks in a given task set, T , are numbered such that to be the extra work that still needs to be done due to a fault pattern F , at time Noting that ffi (T ; t; F) increases only at Equation (2) can be rewritten using the slack() function defined in Section 2 as follows: where The application of Theorem 2 for a given T and F requires the simulation of EDF (T ) and the computation of e i , as well as slack(e The values of ffi i computed from Equations (3) and (4) can then be used to check the condition of the theorem. Each step in the above procedure takes O(n) time, except for the simulation of the EDF schedule. Such simulation may be efficiently performed by using a heap which keeps the tasks sorted by deadlines. Each task is inserted into the heap when it is ready and removed from the heap when it completes execution. Since each insertion into and deletion from the heap takes O(logn) time, the total simulation of EDF takes O(nlogn) time. Thus, the time complexity of the entire procedure is O(nlogn). Hence, given a task set, T , and a specific fault pattern, F , a sufficient and necessary condition for the feasibility of EDF F (T ) can be computed using Corollary 1 in O(n 2 logn) time steps. This is less efficient than simulating EDF F (T ) directly, which can be done in O(nlogn) steps. However, as will be described in the next section, simulating EDF (T ) only is extremely advantageous when we consider arbitrary fault patterns rather than a specific fault pattern. 5 Feasibility Test for a Task Set Under Any Fault Pattern We now turn our attention to determining the feasibility of a given task set for any fault pattern with k or less faults. We use F w to denote a fault pattern with exactly w faults. That is, We also define the function ffi w which represents the maximum extra work at time t induced by exactly w faults that occurred at or before time t. In other words, it is the extra work induced by the worst-case fault pattern of w faults: Note that, although the use of F w in the above definition does not specify that all w faults will occur at or before time t, the value of reach its maximum when all possible w faults occur by time t. Theorem 3 : For a given task set, T , a given number of faults w, and any fault pattern, F w , the lowest priority task, - ' , in T completes by D ' in EDF F w t, e ' - t - D ' . Proof: This theorem is an extension of Theorem 2 and can be proved in a similar manner ffl In order to compute ffi w efficiently, we define the values and use them to compute which is directly derived from Equation (3). The value of each ffi w defined as the maximum extra work at induced by any fault pattern with w faults. This maximum value can be obtained by considering the worst scenario in each of the following two cases: ffl all w faults have already occurred in - . Hence, the maximum extra work at e i is the maximum extra work at e i\Gamma1 decremented by the slack available between e i\Gamma1 and e i . have already occurred in - additional fault occurs in - i . In this case, the maximum extra work at e i is increased by V i , the recovery time of - i . Hence, noting that e and the function slack() are derived from EDF (T ) and do not depend on any particular fault pattern, the values of ffi w can be computed for using the following recursive formula: \Gammaslack(e The computations in Equation (6) can be graphically represented using a graph, G, with n columns and k rows, where each row corresponds to a particular number of faults, w, and each column corresponds to a particular e i (see Figure 3b). The node corresponding to row w and column e i will be denoted by N w . A vertical edge between N w and N w+1 represents the execution of one recovery block of task - i , and thus is labeled by V i . A horizontal edge between N w and N w means that no faults occur in task - i , and thus is labeled by - \Gammaslack(e to indicate that the extra work that remained at e i\Gamma1 is decremented by the slack available between e i\Gamma1 and e i . Then, each path starting at N 0 1 in G represents a particular fault pattern (see Figure 4). The value of ffi w corresponding to the worst case pattern of w faults at is computed from Equation 6, which corresponds to a dynamic programming algorithm to compute the longest path from N 0 1 to N w R e (a) The task set d slack (b) the fault free schedule and the computation of00 04 d Figure 3: The calculation of 4.3 e e d d d00 Figure 4: Two fault patterns for the task set of Figure 3 and the corresponding paths in G. Figure 3 depicts an example of the computation of ffi w i for a specific task set and 2. The value of ffi w i is written inside node N w . We can see that, for this example, from Equation (5), which satisfies the condition of Theorem 3, and thus, the lowest priority task, - 3 , will finish before D in the presence of up to any two faults. Similar to Corollary 1 discussed in the last section, a necessary and sufficient condition for the feasibility of EDF F (T ) requires the repeated application of Theorem 3. Corollary 3: A necessary and sufficient condition for the feasibility of EDF F (T ) for a given T R C D V1 122 6 29 1 e 14e e 3(b) the fault free schedule and the computation of d (a) The task set d 120+2 Figure 5: An example with three tasks. and any fault pattern F with k or less faults can be obtained by applying Theorem 3 to the n task sets contains the j highest priority tasks in T . Figure 5 shows the computation of ffi for an example with three tasks. Note that although the application of Theorem 3 to this example shows that the lowest priority task, - 3 will finish by its deadline in the presence of any two faults, the set of three tasks is not feasible in the presence of two faults in - 1 since in this case, either - 1 or - 2 will miss the deadline. This is detected when Theorem 3 is applied to the task set T g. To summarize, given a task set and the maximum number of faults, k, the following algorithm can be used to optimally check if EDF F (T ) is feasible for any fault pattern of at most k faults. Algorithm "Exact" is the highest priority task in T /* the one with earliest deadline */, is the lowest priority task (the only task) in T 1 */ ffl For do 1. Simulate EDF (T j ) and compute e well as slack(), 2. Renumber the tasks in T j such that e 1 - e j , 3. Compute Equation (6), 4. Let e this is just for computational convenience */ 5. If ffi w 6. If (j = n) then EDF F (T ) is feasible ; EXIT. 7. Let - ' be the highest priority task in 8. note that - ' is the lowest priority task in T j+1 */ Hence, in order to determine if the lowest priority task in a task set can finish by its deadline in the presence of at most k faults, steps 1-5 (which apply Theorem steps to both generate EDF (T j ) and apply Equation (6). In order to determine the feasibility of repeat the for loop n times for Note, however, that with some care, EDF (T j+1 ) can be derived from EDF (T j ) in at most O(n) steps, thus resulting in a total of O(n 2 for the feasibility test. Compared with the O(n k+1 logn) complexity required to simulate EDF under the possible O(n k ) fault patterns, our algorithm has a smaller time complexity, even for As indicated in Corollary 2, a sufficient but not necessary feasibility test may be obtained by computing ffi from a simulation of EDF (T ), and then making sure that, for each task - i , ffi is equal to zero between e i and D i . This can be completed in O(nlog(n)+nk) time as shown in the following algorithm. Algorithm "Sufficient" 1. Simulate EDF (T ) and compute e as well as slack(), 2. Renumber the tasks in T j such that e 1 - e n , 3. Compute Equation (6), 4. Let e this is just for computational convenience */ 5. For do d d tee d 200 0021 (a) The task set (b) the computation of d Figure An example in which f - can tolerate any two faults The example shown in Figure 6 shows that a task - i , even if the value of ffi computed from the simulation of EDF (T ) does not equal zero between e i and . In this example, Yet, it is easy to see that the shown EDF schedule can tolerate any two faults (two faults in - 1 , two faults in - 2 or one fault in each of - 1 and - 2 ). To intuitively explain this result, we note that, although 2 represents the maximum recovery work that needs to be done at no information is kept about the priority at which this recovery work will execute in EDF F (T ). Specifically, in the given example, some of the work in ffi 2 will execute in EDF F (T ) at the priority of - 1 , which is lower than the priority of Thus, it is not necessary that to finish before its deadline. This, in general, may happen only because it is possible for a lower priority task to finish before a higher priority task. That is, if for some i and j, e Finally, we note that, from the observation given in the last paragraph, algorithm "Sufficient" will provide a sufficient and necessary feasibility test in the special case where tasks complete execution in EDF (T ) in the order of their priorities (deadlines). That is, if computed from EDF (T ) satisfy e i - e i+1 and D i - D i+1 . In this case, the recovery work in any ffi w would have to execute in EDF F (T ) at a priority higher than or equal to that of - i , and thus it is necessary for this work to be completed by D i if - i is to complete by its deadline. 6 Related Work Earlier work dealing with tolerance to transient faults for aperiodic tasks was carried out from the perspective of a single fault in the system [LC88, KS86]. More recently, the fault models were enhanced to encompass a single fault occurring every interval of time, for both uniprocessors and multiprocessor systems [BJPG89, GMM94, GMM97]. Further, tolerance to transient faults for periodic tasks has also been addressed for uniprocessors [RT93, RTS94, OS94, PM98, GMM98] and multiprocessor systems [BMR99, OS95, LMM98]. In [KS86], processor failures are handled by maintaining contingency or backup schedules. These schedules are used in the event of a processor failure. To generate the backup schedule, it is assumed that an optimal schedule exists and the schedule is enhanced with the addition of "ghost" tasks, which function primarily as standby tasks. Since not all schedules will permit such additions, the scheme is optimistic. More details can be found in [KS97]. Duplication of resources have been used for fault-tolerance in real-time systems [OS92]. How- ever, the algorithm presented is restricted to the case where all tasks have the same period. More- over, adding duplication for error recovery doubles the amount of resources necessary for scheduling. In [BJPG89], a best effort approach to provide fault tolerance has been discussed in hard real-time distributed systems. A primary/backup scheme is used in which both the primary and the backup start execution simultaneously and if a fault affects the primary, the results of the backup are used. The scheme also tries to balance the workload on each processor. More recently, work has been done on the problem of dynamic dispatching algorithms of frame-based computations with dynamic priorities, when one considers a single fault. In [LLMM99], it was shown that simply generating n EDF schedules, one for each possible task failure, is sufficient to determine if a task set can be scheduled with their deadlines. Also, the work in [Kop97] describes the approach taken by the Mars system in frame-based fault tolerance. Mars was a pioneer system in the timeline dispatching of tasks through the development of time-triggered protocols. It takes into account the scheduling overhead, as well as the need for explicit fault tolerance in embedded real-time systems. However, MARS requires special hardware to perform fault-tolerance related tasks such as voting and, thus, it cannot be used in a broad range of real-time systems. 7 Conclusion We have addressed the problem of guaranteeing the timely recovery from multiple faults for aperiodic tasks. In our work, we assumed earliest-deadline-first scheduling for aperiodic preemptive tasks, and we developed a necessary and sufficient feasibility-test for fault-tolerant admission con- trol. Our test uses a dynamic programming technique to explore all possible fault patterns in the system, but has a complexity of O(n 2 \Delta k), where n is the number of tasks to be scheduled and k is the maximum number of faults to be tolerated. EDF is an optimal scheduling policy for any task set T in the sense that, if any task misses its deadline in EDF (T ), there is no schedule for T in which no deadlines are missed. EDF is also an optimal fault-tolerant scheduling policy. Specifically, EDF F (T ) for a fault pattern F is equivalent to EDF (T 0 ) where T 0 is obtained from T by replacing the computation time, C i , of each task - i in Hence, the work presented in this paper answers the following question optimally: Given a task set, T , is there a feasible schedule for T that will allow for the timely recovery from any combination of k faults? Acknowledgments The authors would like to thank Sanjoy Baruah for proposing the problem of tolerating k faults in EDF schedules and for valuable discussions and feedback during the course of this work. The authors would also like to acknowledge the support of DARPA through contract DABT63-96-C- 0044 to the University of Pittsburgh. --R Workload Redistribution for Fault Tolerance in a Hard Real-Time Distributed Computing System Layered control of a Binocular Camera Head. Single Event Upset Rates in Space. Derivation and Caliberation of a Transient Error Reliability Model. Implementation and Analysis of a Fault-Tolerant Scheduling Algorithm A Program Structure for Error Detection and Recovery. A Measurement-Based Model for Workload Dependence of CPU Errors Measurement and Modeling of Computer Reliability as Affected by System Activity. On Scheduling Tasks with a Quick Recovery from Failure. A Fault-tolerant Scheduling Problem Architectural Principles for Safety-Critical Real-Time Applications Global Fault Tolerant Real-Time Scheduling on Multiprocessors An Efficient RMS Admission Control and its Application To Multiprocessor Scheduling. An Imprecise Real-Time Image Magnification Algorithm An Algorithm for Real-Time Fault-Tolerant Scheduling in a Multiprocessor System Enhancing Fault-Tolerance in Rate-Monotonic Scheduling Allocating Fixed-Priority Periodic Tasks on Multiprocessor Sys- tems Minimum Achievable Utilization for Fault-tolerant Processing of Periodic Tasks Fault Tolerant Computing: Theory and Techniques. Enhancing Fault Tolerance of Real-Time Systems through Time Redundancy Scheduling Fault Recovery Operations for Time-Critical Applications Algorithm Based Fault Tolerance for Matrix Inversion With Maximum Pivoting. --TR --CTR Alireza Ejlali , Marcus T. Schmitz , Bashir M. Al-Hashimi , Seyed Ghassem Miremadi , Paul Rosinger, Energy efficient SEU-tolerance in DVS-enabled real-time systems through information redundancy, Proceedings of the 2005 international symposium on Low power electronics and design, August 08-10, 2005, San Diego, CA, USA Alireza Ejlali , Bashir M. Al-Hashimi , Marcus T. Schmitz , Paul Rosinger , Seyed Ghassem Miremadi, Combined time and information redundancy for SEU-tolerance in energy-efficient real-time systems, IEEE Transactions on Very Large Scale Integration (VLSI) Systems, v.14 n.4, p.323-335, April 2006 Xiao Qin , Hong Jiang, A novel fault-tolerant scheduling algorithm for precedence constrained tasks in real-time heterogeneous systems, Parallel Computing, v.32 n.5, p.331-356, June 2006

real-time scheduling;fault-tolerant schedules;fault recovery;earliest-deadline first

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On Load Balancing in Multicomputer/Distributed Systems Equipped with Circuit or Cut-Through Switching Capability.

AbstractFor multicomputer or distributed systems that use circuit switching, wormhole routing, or virtual cut-through (the last two are collectively called the cut-through switching), the communication overhead and the message delivery time depend largely upon link contention rather than upon the distance between the source and the destination. That is, a larger communication overhead or a longer delivery delay occurs to a message when it traverses a route with heavier traffic than the one with a longer distance and lesser traffic. This characteristic greatly affects the selection of routes for interprocessor communication and/or load balancing. We consider the load-balancing problem in these types of systems. Our objective is to find the maximum load imbalance that can be eliminated without violating the (traffic) capacity constraint and the route to eliminate the imbalance while keeping the maximum link traffic as low as possible. We investigate the load-balancing problem under various conditions. First, we consider the case in which the excess load on each overloaded node is divisible. We devise a network flow algorithm to solve this type of load balancing problem optimally in polynomial time. Next, we impose the realistic assumption that the system uses a specific routing scheme so that the excess load transferred from an overloaded node to an underloaded node must use the route found by the routing scheme. For this case, we use a graph transformation technique to transform the system graph to another graph to which the same network flow algorithm can be applied to solve the load balancing problem optimally. Finally, we consider the case in which the excess load on each overloaded node is indivisible, i.e., the excess load must be transferred as an entity. We show that the load-balancing problem of this type becomes NP-complete and propose a heuristic algorithm as a solution.

Introduction In multicomputer or distributed systems, dynamic creation/deletion of data and/or files may temporarily overload some nodes'/sites' storage space while leaving some others' underloaded. Since storage resources at a node/site are usually limited, uneven data/file distribution may result in inefficient use of storage space and affect future data/file creation. For example, some nodes/sites may not have sufficient space to store new data/files even if the overall system has sufficient space for all the data/files. Load balancing in this respect is thus to transfer the excess (data) load on overloaded nodes to underloaded nodes to balance the (data) load among all the nodes in the system. For multicomputer or distributed systems that use circuit switching, wormhole routing [9], or virtual cut-through [8], the communication overhead and the message delivery time depend largely upon the link contention rather than upon the distance between the source and the destination. That is, a larger communication overhead or a longer delivery delay results when a message traverses a route with heavier traffic than one with a longer distance and lesser traffic. This characteristic greatly affects the selection of routes for interprocessor communication (IPC) and/or load balancing. The objective of selecting a route for IPC or load balancing is thus to minimize the traffic volume on each link so that the communication overhead/delay due to link contention can be minimized. (Note that this objective also reduces the probability of blocking future messages.) While transferring load from overloaded nodes to underloaded ones balances the storage load among all nodes, minimizing the maximum link contention among all links balances the communication load among all links. The major difference between IPC and load balancing is that in the former case we must select a route or routes for each pair of communicating processors, while in the latter case we can select a route or routes from an overloaded node to one or more underloaded nodes. (Note that the excess load on an overloaded node can be transferred to any underloaded node or nodes, instead of a particular one.) Because of this difference, most, if not all, of the variations of the IPC routing problem are NP-hard, while optimal algorithms of polynomial-time complexity exist for several variations of the load balancing problem. Kandlur and Shin [7] studied the route selection problem for interprocessor communication in multicomputer networks equipped with virtual cut-through switching capability. In this paper, we study instead the route selection problem for load balancing in multicomputer or distributed systems that use circuit switching, wormhole routing, or virtual cut-through. Our main concern is to find the maximum load imbalance that can be eliminated (and the routes to eliminate the imbalance) without violating the (traffic) capacity constraint 3 on each link, while keeping the maximum link contention as low as possible. Our work is a significant extension to Bokhari's work [3]. He solved the load balancing problem under several restricted assumptions: 1) there is only unit load imbalance on each overloaded or underloaded node, i.e., each node has either one unit of excess load or one unit of deficit load, or is neutral, and 2) contention is not allowed on any link, i.e., no more than one unit of excess load can be transferred via any link. Moreover, his solution approach does not take into account the contention between the excess load transferred among processors and the other IPC traffic. In this paper, we relax these assumptions: the load imbalance on each node can be any arbitrary value instead of one unit only, and more than one unit of excess load can be transferred via a link as long as the link's (traffic) capacity constraint is not violated. Two cases are studied. First, we consider the case in which the excess load on each overloaded node is divisible, i.e., can be arbitrarily divided and transferred to one or more underloaded nodes. Second, we consider the case in which there may be one or more entities of excess load on each node, and each of them is indivisible and must be transferred to an underloaded node as an entity. We also take into account the effect of existing IPC traffic on route selection for transferring excess load. As a result, the load balancing problem considered in this paper is much more general and practical, and more difficult to solve, than the one treated in [3]. In [3], Bokhari considered multicomputer systems that use some specific routing schemes. In particular, he considered mesh and hypercube interconnection networks that use row-column (column-row) and e-cube routing schemes, respectively. He used a graph transformation technique and a network flow algorithm to solve the load balancing problem in these systems. The graph transformation schemes used for meshes and hypercubes are different, and their correctness is not trivial to prove, especially for the case of hypercube interconnection networks. In contrast, we propose a simple, unified graph transformation scheme and a network flow algorithm to solve the load balancing problem in multicomputer/distributed systems with and without specific routing schemes. The proposed graph transformation scheme, together with the network flow algorithm, works for a larger class of routing schemes, including both the row-column (column-row) and e- cube schemes. The proposed scheme also has an intuitive appeal, and its correctness is very easy to prove. With the proposed graph transformation scheme and the network flow algorithm, we show that for the case of divisible excess load, the load balancing problem with or without specific routing schemes can be solved optimally in polynomial time, i.e., we can find the maximum load 3 to be defined later. imbalance that can be eliminated without violating the traffic capacity constraint on each link while minimizing the maximum contention among all links. For the case of indivisible excess load, we first prove that the load balancing problem is NP-complete, and then propose a heuristic algorithm for it. The rest of the paper is organized as follows. In Section 2, we formally define the load balancing problem considered in this paper and briefly review a network flow problem whose solution algorithms will be used to solve our load balancing problem. In Section 3, we discuss how to apply the network flow algorithm described in Section 2 to optimally solve the load balancing problem under the assumptions that excess load is divisible and there is no specific routing scheme in the system under consideration. In Section 4, we show how to transform the representing graph of a system with a specific routing scheme to another graph, so that the technique described in Section 3 can be used to find an optimal solution for the load balancing in the system. In Section 5, we give an NP-complete proof and a heuristic algorithm for the load balancing problem with indivisible excess load. The paper concludes with Section 6. Problem formulation and a network flow algorithm 2.1 Problem formulation The system under consideration is either a distributed point-to-point network or a multicomputer with an interconnection structure, such as a mesh or a hypercube. We will use a directed graph E) to represent the system, where the vertex/node set V represents the set of nodes/processors in the system, and the edge set E represents the set of communication links. Also, a traffic capacity (or simply, capacity) function C is defined on the edge set E, i.e., each edge associated with a (traffic) capacity C(v which is the maximum communication volume (measured in data units, such as bits, bytes, or packets) that can take place from node v i to node v j . If there is no such constraint on a link (u; v), C(u; v) is defined to be 1. Note that the traffic capacity defined in this paper is not the link bandwidth which is the maximum data transmission rate of a link. We can think of the traffic capacity of a link as the maximum contention (to be defined later) allowed on the link. Also note that using the notation (v to denote an edge allows at most one edge from a vertex to another vertex. This, however, does not impose any unnecessary constraints as multiple edges from a vertex to another vertex can be transformed into single edges by introducing a new vertex for each of them and properly redefining the capacity function. We assume that when the system needs to perform load balancing, each node is either over- loaded, underloaded, or neutral. The (total) excess load on an overloaded node s by e i , and the deficit load on an underloaded node t j 2 V is denoted by d j . The excess and deficit loads can be any arbitrary values, as opposed to only one unit as assumed in [3]. As in [3], we assume that the global state of the system and the degree of load imbalance on each node are known to the central load balancing controller. (The determination of the degree of load imbalance on each node is beyond the scope of this paper.) We require at most e i units of load to be transferred from an overloaded node s i to other underloaded nodes, and at most d j units of load to be transferred to an underloaded node t j from other overloaded nodes. We call this requirement the load transfer constraint. We also assume, without loss of generality, that transferring one unit of load over a link incurs one unit of communication volume on the link. The link capacity constraint requires that the total communication volume on a link (v not exceed the the traffic capacity C(v of the link. The system may or may not use a specific routing scheme. If a specific routing scheme is used, a path P from v i to v j is said to be feasible (under the underlying routing scheme) if the routing scheme will find P as one possible path from v i to v j . We assume that there is at least one feasible path from a vertex to any other vertex (so jEj - jV j). Note that there may be one or more feasible paths from a vertex to another vertex. Since we assume there is at most one edge from a vertex to another vertex, we can use a sequence of vertices , to denote a path. In this paper, we consider only the routing schemes that satisfy the following properties: P1. Each edge in E is a feasible path, i.e., if (v is a feasible path from v i to v j . P2. Any sub-path of a feasible path is also feasible, i.e., if path v i 1 is feasible then path is also feasible, for all 1 P3. If the last edge of a feasible path overlaps the first edge of another feasible path, then the path formed by combining these two feasible paths is also feasible, i.e., if v i 1 and are two feasible paths, then v i 1 is also feasible, for Fig. 1). The routing schemes commonly used in meshes and hypercubes are the row-column (column- row) and e-cube routing algorithms, respectively. A mesh interconnection network can be considered as a two dimensional array, in which each processor, denoted by hx; yi, is connected to its four neighboring processors hx \Sigma 1; yi and hx; y \Sigma 1i (the boundaries of the array may be wrapped around). On the other hand, an n-dimensional hypercube (n-cube) has 2 n processors labeled from each processor is labeled by the binary representation/address, hb feasible feasible Figure 1: Property P3 of routing schemes considered in this paper. Two processors are connected to each other if and only if their binary addresses differ in exactly one bit. The row-column algorithm on meshes routes a message/packet first horizontally from its source node to the node that is at the same column as its destination node, and then vertically to its destination node. For example, a message with source node h3; 4i and destination node h5; 2i will be routed via the path h3; 4i, h4; 4i, h5; 4i, h5; 3i, h5; 2i. The e-cube algorithm on hypercubes always routes a message to the node that more closely matches the address of the destination node with the comparison beginning from the least significant bit of the addresses. For example, a message with source node h01110i and destination node h10101i will be routed via the path h01110i, h01111i, h01101i, h00101i, h10101i. It is easy to check that the row-column (column-row) routing algorithm on meshes satisfies properties P1-P3, and the e-cube routing algorithm on hypercubes satisfies properties P1- P2. To check that the e-cube routing algorithm also satisfies P3, let v and be two feasible paths in an n-cube under the e-cube routing algorithm, and let the addresses of v i j and v i j+1 be hb respectively (recall that since and v i j+1 are two adjacent nodes, their binary addresses differ only in one bit). According to the e-cube algorithm, the addresses of all the nodes v i j+1 must have the form it is easy to see that the path v i 1 formed by combining the two feasible paths is also a feasible path. We want to find the maximum load imbalance that can be eliminated (and the routes to eliminate the imbalance) without violating the link capacity and load transfer constraints while minimizing the maximum link contention. 2.2 The minimax flow algorithm Our solution approach to the load balancing problem considered in this paper is based on the flow problem and its solution algorithm described in [6]. 4 This minimax flow algorithm finds a maximum flow for a network with a 0/1 weight function that also minimizes the maximum edge cost (the cost of an edge is defined to be the weight times the flow of the edge). Applying this algorithm to the load balancing problem, one can view the maximum flow as the maximum load imbalance that can be eliminated, and the edge cost as link contention. Before describing our solution approach for the load balancing problem, we first give a brief review on the minimax flow problem/algorithm. Details on the network flow problem and the flow problem/algorithm can be found in [10] and [6], respectively. t; C) be a network with vertex/node set V , edge set E, source s, sink t, and capacity function E) is the underlying directed graph 5 with is the set of positive real numbers. Each edge (u; v) 2 E is also associated with a nonnegative real-valued weight w(u; v). For ease of discussion, we define both C and w are defined on V \Theta V ). If we say that the network has a 0/1 weight function w. A flow in a network N is a function that satisfies the following properties: 1. Capacity constraint 2. Conservation condition: For each edge (u; v) 2 E, f(u; v) is called the flow in (u; v). For each (u; v) is called the net flow from u to v. The capacity constraint states that the flow in (u; v) is bounded by the capacity C(u; v), and the conservation condition states that the net flow going into a node, except the source and the sink, is equal to the net flow going out of the node. The value of a flow f , denoted as jf j, is the net flow going out of the source, i.e., For the load balancing problem with divisible excess load, transferring f(u; v) units of load from u to v and f(v; u) units of load from v to u (assuming that f(u; v) - f(v; u)) can be replaced 4 It has been brought to our attention that Ahuja [2] has designed a similar algorithm to solve the minimax transportation problem, and his algorithm can also be adapted to solve the minimax flow problem. 5 Without loss of generality, we assume that G is a simple graph, i.e., it has no loop (an edge from a vertex v to itself) and no multiple edges (edges from a vertex u to another vertex v). Therefore, each edge can be represented by the two end vertices of the edge. Note, however, that this assumption can be easily enforced by introducing "dummy" vertices and properly redefining the capacity and the weight functions if graph G does not originally satisfy the assumption. by transferring f(u; v) \Gamma f(v; u) units of net load from u to v without changing the value of the flow (load) and without increasing the contention (flow) in any edge of the network. Therefore, f is a flow with f(u; v) - f(v; u) ? 0, the flow in (u; v) can be simply replaced by f(u; v) \Gamma f(v; u) and the flow in (v; u) by 0 (note that f(u; v) - f(v; u) ? 0 implies that both (u; v) and (v; u) belong to E). Note, however, that the above replacement operation is not valid if excess load is indivisible since one indivisible excess load of f(u; v) units transferred from u to v and another indivisible excess load of f(v; u) units transferred from v to u cannot cancel each other and be replaced by a single load transferred from one vertex to the other. If (u; v) 2 E and f(u; we say that flow f saturates edge (u; v) and call (u; v) an f-saturated edge in N . The (edge) cost (with respect to flow f) of each edge (u; v) 2 E is defined to be w(u; v) \Delta f(u; v), and the (total) cost of a flow f is defined to be v). The flow problem [6] is to find a maximum flow f which minimizes the maximum edge cost, i.e., minimizes max (u;v)2E w(u; v) \Delta f(u; v). We will show that our load balancing problem can be transformed to the minimax flow problem with a 0/1 weight function. We henceforth concentrate on networks with 0/1 weight functions. Definition. Given a network t; C) with a 0/1 weight function w, define to be a new network with for each edge (u; v) 2 E. An edge (u; v) 2 E fi is called a critical edge if w(u; Let f be a maximum flow in N and f fi a maximum flow in N(fi). Since C fi (u; v) - C(u; v) for all (u; v) 2 E, we have jf v)g. It is easy to see that fi. Therefore, jf fi. The capacity fi of the critical edges in N(fi) is the maximum edge cost (note that the weight of a critical edge is 1) allowed for the network N(fi), and hence, the minimum value of the maximum edge cost for a maximum flow in N is fi , where fi is the minimum value of fi such that jf fi is a maximum flow in N(fi). 2 We propose in [6] a minimax flow algorithm, MMC01, as a solution to the minimax flow problem with a 0/1 weight function. MMC01 simply finds fi and constructs a maximum flow for the network N(fi ). For completeness, we list Algorithm MMC01 in Fig. 2 and summarize it below. However, for the sake of conciseness, we omit the proofs of the correctness and time complexity of the algorithm. The interested reader is referred to [6] for details. The idea behind Algorithm MMC01 is that in each iteration, variable fi of the constructed network N(fi) is set to the maximum edge cost allowed in that iteration. With this maximum edge Algorithm MMC01 Step 1. Find a maximum flow f and its value jf j for the network t; C). Step 2. Let ' be the number of edges with nonzero weights in N (w.l.o.g. assume ' - 1). Step 3. Construct network Find a maximum flow f fi and its value jf fi j for N(fi). If jf to Step 5. Step 4. Let \Delta := jf Let R be the set of f fi -saturated critical edges in N(fi), i.e., Go to Step 3. Step 5. A maximum flow, f fi , that minimizes the maximum edge cost is found, and the maximum edge cost with respect to flow f fi is fi. Figure 2: Algorithm for minimax flow problem with a 0/1 weight function. cost, the capacity of an edge (u; v) with w(u; set to min(C(u; v); fi), i.e., the flow allowed to go through edge (u; v) is restricted to min(C(u; v); fi), and hence, the cost of edge (u; v) is bounded by min(C(u; v); fi) \Delta w(u; v) - fi. The algorithm repeatedly constructs maximum flows for networks N(fi) with increasing values of fi (Steps 3-4). Initially, fi := 0 (Step 2). If jf there is a maximum flow with zero cost. Otherwise, if jf fi, the optimal value of fi (i.e., the minimum value of the maximum edge cost, fi ) is found. In Step 3, if jf the optimal value of fi has not been found. For each (u; v) if w(u; is an f fi -saturated critical edge in N(fi). Therefore, to get a larger flow, we need to increase the capacities of critical edges. Let \Delta and - be defined as in the algorithm (Step 4). It has been shown in [6] that fi + \Delta=- fi . Hence, we set \Delta=- and repeat the process (Step 4). This assignment guarantees that the value of fi is always less than or equal to the optimal value fi , and upon termination jf . It has also been shown in [6] that Algorithm MMC01 terminates in at most ' iterations, and hence has a time complexity of O(' \Delta M(n;m)), where ' is the number of edges with nonzero weight and M(n;m) is the time complexity of the algorithm used in Algorithm MMC01 to find a maximum flow in a network with jV vertices and edges. 3 Systems without specific routing schemes In this section, we discuss the load balancing problem for systems without being constrained by any specific routing scheme, i.e., the excess load to be transferred from an overloaded node s i to an underloaded node t j can use any route (path) from s i to t j . We first consider the case in which the excess load on each overloaded node can be arbitrarily divided and transferred to one or more underloaded nodes. For example, if node v i has excess load e i , and nodes v j and v k have deficit load d j and d k , respectively, we can transfer e ij and e ik units of load from v i to v j and v k , respectively, where of e are real numbers. In the case where excess load is indivisible, i.e., each overloaded node may have one or more entities of excess load each of which can only be transferred to an underloaded node as an entity, the load balancing problem becomes more difficult. We defer the discussion of this case until Section 5. For the case that excess load is arbitrarily divisible, Algorithm MMC01 described in Section 2.2 can be easily applied to find the maximum amount of load imbalance that can be eliminated while minimizing the maximum link contention. Given the graph representation E) of a multicomputer or distributed system, and its capacity function C, let ae V be the set of overloaded nodes with node s i having excess load e i , and be the set of underloaded nodes with node t j having deficit load d j , where e i 's and d j 's are all real numbers (note that Recall that e i is the (maximum) amount of load on s i to be transferred to other underloaded nodes, and d j is the maximum amount of load can receive from other overloaded nodes. (As mentioned in Section 1, we assume that e i 's and d j 's are given and their determination is beyond the scope of this paper.) We construct a new graph G 0 by adding to G a new source node s, a new sink node t, and t)g. Define a new capacity function C 0 for (v is the current communication volume on link (v due to the interprocessor communication traffic, i.e., C 0 (v is the maximum amount of load that can be transferred on link (edge) (v violating its traffic capacity constraint. Finally, the weight function w is defined as w(u; Recall that in the load balancing problem considered in this paper, we want to find the maximum amount of load imbalance that can be eliminated while minimizing the maximum link contention. This is equivalent to find a minimax flow in the network with the 0/1 weight function w. Let f(v be the amount of load that will be transferred on link (v when the load balancing procedure is activated. There are two possible ways to define the link contention, depending on the type of communication traffic to be minimized on a link: C1) if we are concerned with minimizing the amount of total communication traffic on link (v the contention to be F (v if we are concerned with minimizing the amount of load to be transferred on link (v we define the contention to be f(v For C2, we simply use Algorithm MMC01 to find a minimax flow f for the network (with the weight function w). The value jf j and the maximum edge cost of the flow f found by MMC01 are the maximum load imbalance that can be eliminated and the maximum edge contention under that flow, respectively. For C1 where the link contention is defined as F (v the network N(fi) should be redefined as follows. F be defined as above. Define to be a new network with for each edge (u; v) 2 E. An edge (u; v) 2 E fi is called a critical edge if w(u; Moreover, the initial value of fi in Step 2 of MMC01 should be changed to max (u;v)2E F (u; v). In this case, the value of fi in each iteration of MMC01 is the maximum contention, F (u; v) allowed for that iteration. Since both cases C1 and C2 can be solved similarly except that the graph representations need to be appropriately defined, unless otherwise stated, we assume in the following discussion that the contention of an edge (u; v) is defined to be f(u; v), i.e., the total amount of excess load that is to be transferred on that edge. Suppose excess load is not arbitrarily divisible. Without loss of generality, we assume that the smallest indivisible load entity is one unit. In this case, the flow f and the capacity C should be redefined as functions from V \Theta V to Z is the set of positive integers. Algorithm MMC01 can still be applied to find a (integral) minimax flow for the network N , except that the statement fi := fi + \Delta=- in Step 4 should now be changed to fi := fi G G' x xy xz jl ik Figure 3: Illustration of graph transformation. 4 Systems with specific routing schemes In this section, we discuss the load balancing problem for systems with special routing schemes (that satisfy properties P1-P3). As discussed in Section 2, both the row-column (column-row) routing scheme for meshes and the e-cube routing scheme for hypercubes satisfy properties P1-P3. In [3], these two routing schemes were handled differently in solving the load balancing problem. In fact, using the approach described in [3], different graph transformation methods need to be designed for different routing schemes. In contrast, we propose a unified graph transformation scheme which can be applied to different routing schemes as long as they satisfy properties P1-P3. Given a system graph E) and a specific routing scheme that satisfies properties P1- P3, we first transform G into another graph G according to the following rules (see Fig. R1. For each vertex v x 2 V , there are d(v x x;d are the in-degree, out-degree, and total degree, respectively, of vertex v x in G. Each vertex v i called an in-vertex/node (of v x ), corresponds to an incoming edge (v and each vertex v xz , called an out-vertex/node (of v x ), corresponds to an outgoing edge (v x v x in G. R2. Let v ik and v i xy correspond to the edge (v G. There is a corresponding edge (v ik xy in G 0 with the same capacity as (v and with the weight of 1. R3. If v is a feasible path from v i to v j in G, add an edge (v i xz ) in graph G 0 with a capacity of 1 and a weight of 0, where (v xy jl are the edges in G 0 corresponding to the edges (v R4. For each overloaded node v i , add to G 0 a node s i and d ik Figure 4: The transformed graph of a 3 \Theta 3 mesh that uses the row-column routing scheme. each of which has a capacity of e i and a weight of 1, where e i is the (total) excess load on v i . For each underloaded node v j , add to G 0 a node t j and d i (v j ) edges (v i jl each of which has a capacity of d j and a weight of 1, where d j is the deficit load on v j . R5. Add to G 0 a source s, a sink t, an edge (s; s i ) with a capacity of e i and a weight of 0 for each overloaded node v i (s i ), and an edge with a capacity of d j and a weight of 0 for each underloaded node v j (t j ). After the system graph G is transformed into G 0 , we can treat the system represented by G 0 as one without any specific routing scheme, and solve the load balancing problem by finding a minimax flow for the network with the weight function w as described in Section 3, are the capacity and weight functions defined in rules R1-R5. The transformed graphs (obtained by applying only rules R1-R3) of a 3 \Theta 3 mesh that uses the row-column routing scheme and a 3-cube that uses the e-cube routing scheme are shown in Fig. 4 and Fig. 5, respectively. Note that we assume each link between two adjacent nodes u and v in a mesh or a hypercube is a bi-directional communication link, and thus, there are two directed edges (u; v) and (v; u) corresponding to this link in the graph representation of the mesh or the hypercube. From Fig. 4, it is easy to see that whenever a routing path uses a horizontal edge, it will no longer be able to use a vertical edge, and hence, each directed path from an out-vertex to an Figure 5: The transformed graph of a 3-cube that uses the e-cube routing scheme. in-vertex in the transformed graph corresponds to a feasible path found by the row-column routing scheme in the mesh and vice versa. The transformed graph also satisfies the requirement of the e-cube routing scheme for hy- percubes. For example, consider node h011i in Fig. 5. From the definition of the e-cube routing scheme, a path going into node h011i from node h010i can go to either node h001i or node h111i, a path going into node h011i from node h001i can only go to node h111i, and a path going into node h011i from node h111i can go nowhere. To formally prove the correctness of the transformation, it suffices for us to prove the following theorem. Theorem 1: Suppose the routing scheme used in the system under consideration satisfies properties P1-P3. Every feasible routing path from a vertex v x to another vertex v y in G corresponds to a unique directed path from an out-vertex v xz of v x to an in-vertex v i yw of v y in the transformed graph G 0 and vice versa (assume G 0 is derived by applying rules R1-R3 to G). y is a feasible path from v x to v y in G. From property P2, we know that each (sub-)path v j l feasible path in G, and from transformation rules R1-R3, it is easy to see that there exists a directed path v yw in G 0 . On the other hand, suppose zw is a directed path in G 0 . From the transformation rules, it is easy to see that each edge must satisfies that either u 0 is an in-vertex and v 0 is an out-vertex or vice versa. Moreover, if u 0 is an in-vertex and v 0 is an out-vertex, then u 0 and v 0 correspond to the same vertex in G. Therefore, k must be even and all are out-vertices, and all v 0 are in-vertices. Moreover, each pair of vertices 2l and v 0 correspond to the same vertex, say v 2l is an in-vertex of v j l and v 0 is an out-vertex of v j l , for some i l and . Now, from the transformation rules R1 and R2, we know that v x y is a (directed) path from v x to v y in G. For notational simplicity, let v and . We next prove by induction that path is feasible. Specifically, we will show that all the paths v j 0 are feasible. Since (v E, from property P1, we have that path v is feasible in G. Suppose path v j 0 feasible in G. Since (v 0 ) is an edge in G 0 , from the transformation rule R3 we know that v j is a feasible path in G. Then, by property P3, we conclude that path v 5 Systems with indivisible excess load In this section, we discuss the case in which excess load is indivisible. We assume that there is no specific routing scheme in the system. For systems that use certain specific routing schemes, one can first apply the graph transformation rules described in Section 4 to the representing graph and then treat the transformed graph as a system with no specific routing scheme. As discussed in Section 3, in a system with indivisible excess load, each overloaded node s i has one or more entities of excess load e i1 , for some k i - 1, each of which can only be transferred to an underloaded node as an entity. Without loss of generality, we assume that each overloaded node s i has exactly one entity of indivisible excess load e i since if s i has of excess load, we can add a new overloaded node s ij with one entity of indivisible excess load e ij and a new edge each entity of excess load e ij neutral node. Note that we use e to refer to either the entity of excess load or its amount. We first show that the load balancing problem with indivisible excess load is NP-hard in the strong sense [5] (in fact, we show that the problem of finding the maximum load imbalance that can be eliminated without considering the link contention is already NP-hard in the strong sense if Figure Instance construction in the NP-completeness proof. the excess load is indivisible). We then propose a heuristic algorithm as a solution to the NP-hard case of the load balancing problem. The decision version of the load balancing problem of finding the maximum load imbalance that can be eliminated is to ask, given a number B, whether or not it is possible to eliminate at least B units of load imbalance (without violating the link capacity and load transfer constraints). Theorem 2: The decision version of the load balancing problem of finding the maximum load imbalance that can be eliminated is NP-complete in the strong sense if the excess load is indivisible. Proof: It is easy to see that the decision version of the load balancing problem is in NP. To complete the proof, we reduce to it the multiprocessor scheduling problem [5]: Given a set A of n tasks, a length l(a i ) for each 1 - i - n, a number p of processors, and a deadline D, is there a partition of A such that max 1-i-p ( Given an instance of the multiprocessor scheduling problem, we construct an instance of the load balancing problem (shown in Fig. 6) in which (1) each s i an overloaded node with indivisible excess load of l(a i ) units, and t is an underloaded node with deficit load of units; (2) there are p node-disjoint paths from u to v, all the edges on these paths have a capacity D, all the other edges have an infinite capacity, and Note that the construction can be done in polynomial time. It is easy to see that at least B units of load imbalance can be eliminated without violating the link capacity and load transfer constraints if and only if there exists a solution for the multiprocessor scheduling problem. Since the multiprocessor scheduling problem is NP-compete in the strong sense, the decision version of the load balancing problem with indivisible excess load is also NP-complete in the strong sense. 2 Since it is unlikely to find a polynomial time optimal algorithm for the load balancing problem with indivisible excess load, we propose below a heuristic algorithm for the problem. Let E) be the graph representation of the multicomputer or distributed system under consideration, and C(u; v) be the capacity (for load transferring purpose) of edge (u; v), for all (u; v) 2 E. Let s i , be the overloaded nodes and t i q, be the underloaded nodes. Each overloaded node s i has indivisible excess load e i which must be routed to an underloaded node as an entity, and each underloaded node t i has deficit load d i which is the maximum amount of load it can receive from overloaded nodes. Without loss of generality, we assume that e i 's are sorted in non-increasing The heuristic algorithm (see Fig. 7) consists of two phases. In Phase I, we treat the excess load as if it were divisible and use the network flow technique described in Section 3 to find a minimax flow f . If the excess load was indeed divisible, f would be an optimal solution in which the value jf j is the maximum load imbalance that can be eliminated with the maximum link contention (cost) minimized. In Phase II, we use the minimax flow f found in Phase I as a "template" and route the entities of excess load one by one in such a way that the resulting flow on each link will be as close to the corresponding minimax flow as possible, i.e., the value f(u; v) found in Phase I serves as the target flow (load) for edge (u; v) to be achieved in Phase II. Since in general larger amounts of excess load are more difficult to route than smaller amounts, we will route the excess load in non-increasing order of load amount. During the execution of Phase II, f 0 (u; v) is the total load currently routed through edge v). If the excess load currently being routed is e i , we say that an edge (u; v) is feasible if a path from s i to t is feasible 6 if all edges on the path are feasible. We will route excess load e i from the overloaded node s i to an underloaded node t j (actually, to node t) only via a feasible path, i.e., excess load can only be routed via a path in which each edge has enough (remaining) capacity. Note that in Phase II, vertex s and edges (s; s i are, in fact, not used, i.e., the underlying graph is G qg. The excess load e i is routed using a greedy type algorithm, called ordered depth-first search (O-DFS). Note that f(u; v) \Gamma f 0 (u; v) is the difference between the target flow f(u; v) and the total load f 0 (u; v) currently routed through edge (u; v). A large f(u; implies that the current load routed through edge (u; v) is still far from the target value (note that f(u; may be negative). Therefore, at each vertex u, we always choose to traverse next the edge (u; v) that has the largest f(u; \Delta) \Gamma f 0 (u; \Delta) value among all feasible outgoing edges at u, and hence, reduce 6 Note that we overload the word "feasible" here. The term "feasible path" defined here is different from that defined in Section 2.1. Phase I. Step 1. Construct a network and Let w(u; Step 2. Treat each excess load e i as a divisible load, and use Algorithm MMC01 to find a minimax flow f for N . Phase II. Step 1. Set f 0 (u; v) := 0, for all (u; v) Step 2. For i / 1 to p do the following: Step 2.1. Use the ordered depth-first-search (O-DFS) algorithm to find a feasible path from s i to t, where the ordered DFS algorithm is similar to the DFS graph traversal algorithm [4], except that when branching out from a vertex we always choose to traverse next the edge that has the largest f(\Delta; \Delta) \Gamma f 0 (\Delta; \Delta) value among all untraversed feasible edges at that vertex. Step 2.2. If there does not exist any feasible path from s i to t, it means that the excess load e i will not be eliminated when the next round of load balancing is performed. If there exists a feasible path from s i to t, let P be the (first) path found by the O-DFS algorithm (P will be used as the route to eliminate the excess load e i when the next round of load balancing is performed). Reset f 0 (u; v) := f 0 (u; v) each edge (u; v) on P . Figure 7: A heuristic algorithm for the case that excess load is indivisible. the maximum f(u; \Delta) \Gamma f 0 (u; \Delta) value at u. Note that the heuristic algorithm is not an optimal algorithm. It may not find the maximum load imbalance that can be eliminated, and in cases in which it does find the maximum load imbalance, it may not minimize the maximum link contention. We use the following example to further illustrate the heuristic algorithm. Example 1: Suppose the constructed network of a system graph E) and the minimax flow f found at the end of Phase I are shown in Fig. 8(a). The maximum edge 7/7 7/4 9/4 9/3 7/5 s (a) 9/3 (b) 9/3 (c) 9/3 (d) 9/3 2/2 Figure 8: An example that shows how the heuristic algorithm works. cost (link contention) shown in the figure is 4 (note that only edges in E are considered). In Phase II, we initially set f 0 (u; We first route excess load e 1 . Starting from vertex s 1 , since f(s 1 the O-DFS algorithm will first visit vertex v 2 . Since (v is the only feasible outgoing edge (i.e., the next vertex visited is t 2 . Then, vertex t is visited and a feasible path from s 1 to t is found for e i , i.e., the path s t. For each edge (u; v) on that path, we reset f 0 (u; . The current values of C 0 (u; all are shown in Fig. 8(b). We next route excess load e 2 . Using the O-DFS algorithm, we find the feasible path s for e 2 . Note that at vertex s 2 , although f(s still choose edge 5). For each edge (u; v) on the path found for e 2 , we reset f 0 (u; . The current values of are shown in Fig. 8(c). The next excess load to be routed is e 3 , and the path found for e 3 is s t. The values of routed and f 0 is updated are shown in Fig. 8(d). Finally, we route excess load e 4 . Starting from s 4 , O-DFS first traverses edge vertex t 3 , since there is no feasible outgoing edge, O-DFS backtracks to vertex s 4 , and then traverses edge since t 3 has been visited, O-DFS next traverses (v both 2, the next edge traversed is and the path found for e 4 is s t. The values of C 0 (u; and routed and f 0 is updated are shown in Fig. 8(e). The amount of load imbalance that can be eliminated in this example is 7 and the maximum link contention is 7. 2 The time complexity of the heuristic algorithm is shown in the following theorem. Theorem 3: Phase I of the heuristic algorithm has a worst-case time complexity of O(m \Delta M(n;m)), where is the time complexity of finding a maximum flow in a network of O(x) vertices and O(y) edges. Phase II of the heuristic algorithm has a worst-case time complexity of O(p is the number of excess load entities. Proof: The complexity of Phase I is discussed in Section 2.2. (note that there are m edges with nonzero weight in N ). As mentioned earlier, the underlying graph in finding paths from overloaded nodes to underloaded nodes in Phase II is G where q is the number of underloaded nodes. The well-known DFS algorithm can be done in x is the number of vertices and y is the number of edges of the graph traversed. For our O-DFS algorithm, each time when we first visit or backtrack to a vertex, we always traverse an untraversed edge with the maximum f(\Delta; \Delta) \Gamma f 0 (\Delta; \Delta) value. Therefore, traversing all outgoing edges at a vertex u takes at most O(d is the out-degree of u and the logarithm is to the base 2. Thus, the total time to route excess load entities is at most [1, 4] Since we need to route p excess load entities, e i the worst-case time complexity of Phase II is O(p m). (Note that since we assume there is only one entity of indivisible excess load on each overloaded node and there is at least one directed path from a vertex to any other vertex in G, we have m 6 Conclusion In this paper, we consider the load balancing problem in multicomputer or distributed systems that use circuit switching, wormhole routing, or virtual cut-through, with the objective of finding the maximum load imbalance that can be eliminated without violating the (traffic) capacity constraint on any link while minimizing the maximum link contention among all links. We solve the problem under various conditions. We give an O(m \Delta M(n;m)) optimal algorithm for the load balancing problem with divisible excess load, where n is the number of nodes/processors and m is the number of links in the system, and M(x; y) is the time complexity of finding a maximum flow in a network of x vertices and y edges. We propose a graph transformation technique for systems with specific routing schemes that satisfy properties P1-P3 described in Section 2.1. This graph transformation technique transforms the representing graph of a system to another graph with which the load balancing problem can be solved optimally in the same manner and with the 7 It is easy to see that if each overloaded node has more than one entity of indivisible excess load, the time complexity of Phase II should be changed to O(p since we will add p vertices and p edges to the system graph (however, we still have m - n ? q). same time complexity as the load balancing problem for systems without specific routing schemes. We also consider the load balancing problem for the case in which excess load is indivisible. We prove that the problem is NP-hard and propose, based on the O(m \Delta M(n;m)) optimal algorithm for the problem with divisible excess load, an O(m \Delta M(n;m)+ m) heuristic algorithm as a solution to the problem, where p is the number of excess load entities. The result obtained in this paper is a significant extension to Bokhari's work reported in [3]. We generalize his work in several directions: 1) we relax the assumption of unit load imbalance; 2) we relax the assumption of unit link contention; 3) we consider the effect of existing IPC traffic on the selection of routes for load balancing; 4) our graph transformation technique and network flow model can be applied to a larger class of routing schemes. Moreover, our solution approach and algorithms are more intuitive and simpler than those proposed in [3]. --R The Design and Analysis of Computer Algorithms. Algorithms for the minimax transportation problem. A network flow model for load balancing in circuit-switched multicomput- ers Introduction to Algorithms. Computers and Intractability: A Guide to the Theory of NP-Completeness A fast algorithm for the minimax flow problem with 0/1 weights. Traffic routing for multicomputer networks with virtual cut-through capability Virtual new computer communication switching technique A survey of wormhole routing techniques in direct networks. Data Structures and Network Algorithms. --TR --CTR Michael E. Houle , Antonios Symvonis , David R. Wood, Dimension-exchange algorithms for token distribution on tree-connected architectures, Journal of Parallel and Distributed Computing, v.64 n.5, p.591-605, May 2004 Patrick P. C. Lee , Vishal Misra , Dan Rubenstein, Distributed algorithms for secure multipath routing in attack-resistant networks, IEEE/ACM Transactions on Networking (TON), v.15 n.6, p.1490-1501, December 2007

minimax flow problem;overloaded/underloaded nodes;load balancing;excess/deficit load;link traffic

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An Implementation of Constructive Synchronous Programs in POLIS.

Design tools for embedded reactive systems commonly use a model of computation that employs both synchronous and asynchronous communication styles. We form a junction between these two with an implementation of synchronous languages and circuits (Esterel) on asynchronous networks (POLIS). We implement fact propagation, the key concept of synchronous constructive semantics, on an asynchronous non-deterministic network: POLIS nodes (CFSMs) save state locally to deduce facts, and the network globally propagates facts between them. The result is a correct implementation of the synchronous input/output behavior of the program. Our model is compositional, and thus permits implementations at various levels of granularity from one CFSM per circuit gate to one CFSM per circuit. This allows one to explore various tradeoffs between synchronous and asynchronous implementations.

Introduction Our purpose is to reduce the gap between two distinct models of concurrency that are fundamental in the embedded systems framework, the synchronous and asynchronous models, with application to systems written in the Esterel synchronous programming language and implemented in the POLIS system developed at UC Berkeley and Cadence. The synchronous or zero-delay model is used in circuit design and in synchronous programming languages such as Esterel [6], Lustre [12], Signal [10], and SyncCharts [2] (a synchronous version of Statecharts [13]), see [11] for a global overview. In this model, all bookkeeping actions such as control transmission and signal broadcasting are conceptually performed in zero- time, only explicit delays taking time. Thus, a conceptual global clock controls precisely when statements simultaneously compute and exchange messages. The model makes it possible to base design on deterministic concurrency, which is much easier to deal with than classical non-deterministic concurrency. Compiling, optimizing, and verifying programs is done using powerful Boolean computation techniques, see [5]. The synchronous model is well-suited for direct speci-cation and implementation of comparatively compact programs such as protocols, controllers, human-machine interface drivers, and glue logic. In this case, one can build a global clock slow enough to react to each possible environmental input. In an asynchronous model, processes exchange information through messages with non-zero travel time. Asynchronous models are well-suited for network-based distributed systems speci- -cation and for hardware/software codesign, where the relative speed of components may vary This work was begun while the -rst author was visiting Cadence Berkeley Laboratories, August 1998. widely. There are many asynchronous formalisms with varied communication policies. For ex- ample, CSP processes [14] communicate by rendezvous, while data-AEow processes [15] exchange data through queues or buoeers. The POLIS [3] mixed synchronous/asynchronous model has been developed at UC Berkeley and Cadence, with primary focus on codesign. It is a Globally Asynchronous Locally Synchronous (GALS) model, in which synchronous nodes called CFSMs (Codesign Finite State Machines) are arranged in an asynchronous network and communicate using non-blocking 1- place buoeers, and through a synthesized real-time operating system (RTOS) for the software part. The CFSMs can be programmed in a concurrent synchronous language such as Esterel, thus taking maximal advantage of the synchronous model at the node level. The model can be eOEciently simulated and implemented in hardware and/or software; notice that 1-place buoeers are much simpler to implement than FIFOs, especially at the hardware/software boundaries. However, POLIS networks have much less intrinsic semantic safety than FIFO-based dataAEow Kahn networks [15], which are behaviorally deterministic, and their behavior must be carefully controlled. In particular, buoeer overwriting in POLIS can lead to non-deterministic behaviors that can be hard to analyze and prove correct. Here, we show that the behavior of a synchronous circuit or program can be nicely implemented in a POLIS network. Of course, one can implement a synchronous program in a single CFSM node in a straightforward way. Here, we are interested in distributed implementations where the synchronous behavior is split between asynchronously communicating units, without a global clock. In practice, this is useful when the application behavior is naturally synchronous but the execution architecture is distributed and possibly heterogeneous, with physical inputs and outputs linked to dioeerent computing units. We retain the synchronous philosophy when specifying an application and we bene-t from the AEexibility and eOEciency of CFSM networks in the implementation. We propose a solution in which the CFSM granularity can be chosen at will: any part of the synchronous program can be implemented in a single synchronous CFSM, which makes it possible to partition the program according to the architecture constraints and the best synchrony/asynchrony compromise. Other authors have proposed such distributed implementations of synchronous programs on asynchronous networks, see for example [9, 8], and we draw much from their work. However, our implementation takes maximal advantage of the semantics of the objects we deal with and it is presented dioeerently, with a trivial correctness proof. Technically speaking, we present a POLIS implementation of constructive synchronous circuits [5, 18], which is a class of well-behaved cyclic circuits that generalizes the usual class of acyclic circuits. Since Esterel programs are translated into constructive circuits [4], this implementation handles Esterel as well. The key of any implementation of synchronous programs is the realization of a conceptual zero-delay reaction to an input assignment. In a distributed asynchronous network, this must be done by a series of message exchanges. In our implementation, the messages are CFSM- events that carry proven facts about synchronous circuit wire or expression values. Such facts are exactly the logical information quanta on which the constructive semantics are based. The CFSM nodes generate output facts from input facts according to the semantic deduction rules. This is done over a series of computations since conceptually simultaneous facts now arrive at dioeerent times. For a single reaction of a program, the number of events is uniformly bounded. No buoeer overwrite can occur in the network. Although the internal behavior is non-deterministic, the overall behavior respects the synchronous semantics of the original program and thus is de- terministic. This is true independently of the schedule employed by the RTOS. In addition, execution of successive synchronous reactions can be pipelined. Finally, the implementation takes full advantage of the mathematical properties of the constructive semantics. In particular, the compositionality property makes it possible to arbitrarily group elementary circuit gates into CFSM nodes: this allows any level of granularity, from one single CFSM for the program at one extreme to one CFSM per individual gate at the other. Clearly, there are many applications for which using only the synchronous formalism at spec- i-cation level makes no sense, in which case our results are not directly applicable. Nevertheless, we think that they show that the apparent distance between synchrony and (controlled) asyn- IJYXFigure 1: Circuit C 1 chrony can be reduced, and we hope that the technology we present can serve as a basis for future mixed-mode language developments. We start in Section 2, by presenting the logical, semantical, and electrical views of constructive circuits. In Section 3, we brieAEy present the POLIS CFSM network model of computation. Our implementation of constructive circuits in this model is presented in Section 4, We discuss possible applications and synchrony/asynchrony tradeooes in Section 5, and we conclude in Section 6. Constructive Circuits Constructive circuits are iwell-behavedj possibly cyclic circuits that generalize the class of acyclic circuits. Acyclic circuits can be viewed in two dioeerent ways: ffl as Boolean equation systems, then de-ning a Boolean function that associates an output value assignment with each input value assignment. ffl as electrical devices made of wires and gates that propagate voltages and have certain delays: if the inputs are kept electrically stable long enough to one of two binary voltages (say 0V and 3V), the outputs stabilize to one of the binary voltages. Relating the Boolean and electrical approaches is easy for acyclic circuits: when the outputs electrically stabilize, they take the voltages corresponding to the results of the Boolean input/output function. Constructive circuits have exactly the same characteristics even in the presence of cycles 2.1 The Behavior of Cyclic Circuits A circuit has input, output, and internal wires; the latter we also call local variables. In our examples, we use the letters I ; J for the inputs and X;Y for the outputs and locals, making it precise which are the outputs where necessary. Each output or local variable is de-ned by an equation is an expression built using variables and the operators : (negation), - (conjunction), an - (disjunction). For simplicity, we assume that an expression E is either a variable, the negation of a variable, or a single n-ary operator - or - applied to variables or the negation of variables. Any circuit can be put into this form by adding enough auxiliary variables. A circuit can also be considered as a network of gates, as pictured in Figure 1. Each wire has a single source and multiple targets. The gates correspond to the operators. As a running example, we shall consider the following circuit C 1 , with outputs X and Y : ae Notice that C 1 is cyclic: Y appears in the equation of X and conversely. 2.1.1 Circuits as Boolean Equations In the Boolean view, we try to solve the circuit equations using Boolean values 0 and 1. An input assignment i associates 0 or 1 with some input variables. An input assignment is complete if it associates a value with any input variable. For a complete assignment i, A Boolean solution of the circuit is an assignment of values 0 or 1 to the other variables that satis-es the equations. An acyclic circuit has exactly one Boolean solution for each complete input assignment. A cyclic circuit may have zero, one or several solutions for a given complete input assignment. For example, consider the case where there is no input and one output X . For there is no solution. For there is a unique solution there are two solutions For , there is a unique solution if The solution is the equations reduce to and there are two solutions, 2.1.2 Circuits as Electrical Devices In the electrical view, one preferably uses the graphical presentation and vocabulary. Wires associated with variables carry two dioeerent voltages, also called 0 and 1 for simplicity, and logic gates implement the Boolean operators. Wires and gates can have propagation delays. We shall not be very accurate here about delays; technically, the delay model we refer to is the up-bounded inertial delay model described in [7, 17]. A complete input assignment is realized by keeping the input wires stable over time at the appropriate voltages. Voltages propagate in the circuit wires according to the laws of electricity, and the property we are interested in is wire voltage stabilization after a bounded time. The non-input wires are assumed to be initially unstable. Outputs of acyclic circuits always stabilize. Outputs of cyclic circuits may or may not stabi- lize. For example, the output of stabilizes, while that of 0 and 1. The output of remains unstable. When wires stabilize, their values always satisfy the equations. Stabilization may depend on delays. For example, in the Hamlet circuit 1 de-ned by , the output X stabilizes to 1 for some delays and does not stabilize for others, see [5]. Stabilization may also depend on the input assignment: for C 1 , outputs stabilize to the right Boolean values unless I in which the behavior is delay-dependent, with no stabilization for some delays. 2.2 Constructive Boolean Logic Notice that the perfect match between Boolean and electrical solution is lost for cyclic circuits: for Hamlet, the Boolean output function is well-de-ned and yields electrical stabilization may not occur. Hamlet has a unique Boolean solution because 1 happens to be a solution while 0 is not. Finding the solution involves propagating non-causal information and this cannot be done by non-soothsaying electrons in wires. Fortunately, Boolean logic can be weakened into constructive Boolean logic, in which the solution to Hamlet is rejected, thereby rendering the Boolean and electrical results the same: no solution exists. Constructive Boolean Logic precisely models electrical behavior. 2.2.1 Facts and Proofs Constructive Boolean logic deals with facts and proofs. A fact has the form where E is a Boolean expression. An input fact is I = 0 or I = 1 for an input variable I . An input assignment i is a set of input facts. Facts are deduced from other facts by deduction rules. There are deduction rules for each type of gate and one rule to handle equations. Here are the rules for the - conjunction operator: 1 Think of X as to be. (l-and) (r-and) (b-and) The facts above the horizontal bar are the premises and the fact below the bar is the conclusion. Rule (b-and) reads as follows: from the facts 1. The rules for - (or-gate) are dual. The rules for negation are: Notice that X - Y behaves as :(:X - :Y ), just as in classical Boolean logic. The rules for a circuit equation are where b can be either 0 or 1. A proof is a sequence of facts that starts by the facts of an input assignment and such that any other fact can be deduced from the previous facts using a rule. The following consistency lemma shows the soundness of the proof system. It is easily shown by induction on the length of the proof. Lemma 1 If there exists a proof of a fact 1), then there is no proof of 2.2.2 Proof Examples We give some proof examples for C 1 . We present them in annotated proof form, writing at each step the deduced fact, the premises, and the applied deduction rule. Here is an annotated proof for with complete input assignment I = 0; from (1) by (l-and) from (2) and (5) by (b-and) Here is the dual proof for from (1) by (l-and) from (2) and (5) by (b-and) Notice that the deduction ordering is X -rst, Y next in P 01 , while it is the reverse ordering Y -rst, X next in P 10 . This is the main dioeerence between acyclic and constructive circuits: in acyclic circuits, one can -nd a data-independent variable ordering valid for all input assignments. In constructive circuits, such an ordering exists for each input assignment, but it may be data-dependent 2.2.3 Example of Non-Provable Circuits The circuits are both rejected as having no output proof, and for the very same reason: there is no way to start a proof. Notice that the existence or non-existence of a Boolean solution is not relevant. The circuit for example, has two Boolean solutions: 1. However, to verify either solution one would have to -rst make an assumption about the solution, and then verify the validity of the assumption. Constructive proofs must only propagate facts; they are not allowed to make assumptions Constructive Boolean logic rejects the Hamlet circuit which no output fact can be proven. As above, there is no way to start a proof without making an assumption. The law of excluded middle X - does not hold in constructive logic, unless X has already been proved to be 0 or 1. 2.2.4 Output Proofs and Complete Proofs An output proof is a proof that proves a fact for each output variable. A complete proof is a proof that proves a fact for each variable. A circuit is output constructive w.r.t. a complete input assignment i if there is an output proof starting with the facts in i. The circuit is completely constructive w.r.t. i if there is a complete proof starting with the facts in i. The dioeerence is that no fact is needed for an intermediate variable in an output proof if this variable is not needed to prove the output facts. It is even allowed that no fact about this variable can be proved. Consider for example only X is an output. If no fact for Y can be proved. The circuit is output constructive but not completely constructive for this input assignment. Although output constructiveness seems more general, we shall deal with complete construc- tiveness in the sequel since it is much easier to handle. Complete constructiveness is also required by the semantics of Esterel [4]. 2.2.5 Constructive Logic Matches Delay Independence Constructive Boolean logic exactly represents delay independence: given a complete input as- signment, a circuit electrically stabilizes its output wires (resp. all its wires) for any gate and wire delays if and only if it is output constructive (resp. completely constructive). This fundamental result is shown in [18, 17] using techniques originally developed for asynchronous circuit analysis [7]. Notice that a given fact can have several proofs. Delay assignments actually select proofs. Consider is the output. For there are two proofs of 0: the -rst one deduces the second one deduces the same fact from deduced from Electrically speaking, the -rst proof occurs when propagates through X's and gate before while the second proof occurs if there is a long delay on the I input wire, long enough for to propagate through X's and gate before I = 0. 2.3 Scott's Fixpoint Semantics The classical model of Boolean logic is binary, variables taking values in 1g. Constructive Boolean logic has a natural ternary semantic model. 2.3.1 The Ternary Model The ternary domain is 1g. The unde-ned value ? (read bottom) represents absence or non-provability of information. The domain is partially ordered by Scott's information ordering the total values 0 and 1 being incomparable 2 . Tuples are partially ordered componentwise: x - y ioe x k - y k for all k. Functions are required to be monotonic must have f(x) - f(y) in B n ? . A composition of monotonic functions is monotonic. Functions are partially ordered by f - g if 2.3.2 The Fixpoint Theorem The key result in Scott's semantics is the -xpoint theorem, which we state here in a simple case. ? be monotonic, and let a -xpoint of f be an element x of B n ? such that x. The theorem states that f has a least -xpoint lfp(f), which is the (-nite) limit of the increasing sequence The function lfp that associates the least -xpoint lfp(f) with f is itself monotonic. 2.3.3 The Basic Ternary Operators The Boolean operators are extended as follows to the ternary logic. There is no choice for negation, which must be monotonically de-ned by -, we choose the parallel extension, which is the least monotone function such that closely corresponds to electrical gate behavior and to our proof rules. The extension of disjunction - is dual. Other possible extensions of - are the strict extension such that 0 - the left sequential extension such that and the symmetrical right sequential extension. They are de-nable from the parallel extension in constructive logic (hint: the expression X-:X has value 1 if and only if X is de-ned).See [16, 1] for a complete discussion of these extensions. It is interesting to note that the parallel extension cannot be de-ned in sequential languages such as C and requires a parallel interpretation mechanism, hence its name. 2.3.4 Circuits as Fixpoint Operators A circuit with input vector ? and other variables in vector x ? de-nes an equation of the form where the k-th component of f is given by the right-hand-side of the equation for x k . Given an input assignment i, let us write f i from B n ? to itself. We call a solution of the circuit w.r.t. i the least -xpoint lfp(f i ) of f i . For example, in circuit C 1 , the least -xpoint for input I = 0; while the least -xpoint for The next theorem shows that the constructively deducible facts exactly correspond to the -xpoint solution. Theorem 1 Given a circuit C de-ning a function f and an input assignment i, a fact constructively provable if and only if the X-component of the least -xpoint of f i has value b. Unfortunately, some authors use f0; to mean the same thing! The proof is standard and left to the reader (use inductions on term size and proof length). Notice that the theorem does not require the input assignment to be complete. It is also valid when some inputs are ?. Then, no fact for these inputs can be used in deductions. This concludes the theory of constructive circuits: electrically stabilizing in a delay-independent way is the same as being provable in constructive Boolean logic or as having a non-? value in the least -xpoint. 2.4 Algorithms for Circuit Constructiveness There are algorithms to detect whether a circuit is constructive for a given input assignment or for all complete input assignments. Here, we present a linear-time algorithm that works for one complete input assignment. It is used in the Esterel v5 compiler, for interpretation mode (option -I). Algorithms checking constructiveness for all inputs or for some input classes are much more complex. The BDD-based algorithm used in the Esterel v5 compiler (option -causal) is presented in [18, 17, 19]. It will not be considered here. 2.4.1 An Interpretation Algorithm The running data structure of the algorithm is composed of two sets of facts called and TODO and of an array PRED of integer values indexed by non-input variable names. The TODO set initially contains the input facts, and the DONE set is initially empty. The array entry PRED[X ] is initialized to the number of predecessors of X , which is the number of variable occurrences in the de-nition equation of X , also called the fanin number in the electrical presentation. The algorithm successively takes a fact from TODO, puts it in DONE, and propagates its constructive consequences, which may add new facts to TODO and decrement the predecessor counts. Propagating the consequences of a fact works as follows: ffl All variables that refer to V in their de-nition decrement their predecessor count according to the number of occurrences of V in their de-nition. immediately determines that then that fact is added to TODO. This occurs de-ned by a conjunction where V appears positively, in which case or by a disjunction where V appears negatively, in which case (symmetrically if 1). This fact propagation rule corresponds to deduction rules such as (l-and) and (r-and ), possibly combined with (not-0 ) and (not-1 ). ffl If the predecessor count of a variable W falls to 0 and the value of W is not yet determined, a new fact is added to TODO, where c is the identity of the de-nition operator of W , i.e. 1 for - and 0 for -. This corresponds to rules such as (b-and ). 2.4.2 Execution Example For with inputs I = 0; we start in the following state: We remove I = 0 from TODO and put it in DONE. We decrement the predecessor count of X . immediately implies we add that fact to TODO: We now process 1. The only consequence is that the number of predecessors of Y is decremented, since does not determine Y by itself: We now process This fact does not directly determine the value of Y , but it exhausts its predecessor list: We can now deduce that the value of Y is 1 since Y is an empty conjunction. We add this fact to TODO: We have computed all the facts we need. However, it is useful to perform the last step, which will bring us back to a nice clean state. Processing puts this fact in DONE and decrements X's predecessor count: Since we build proofs, the result of the algorithm does not depend on the order in which we pick facts in TODO. For the input I = 0; here is a run where the output values are computed faster but cleanup is longer: For the non-constructive input I = rapidly reach a deadlock: There are no remaining facts in TODO, and yet no fact has been established for X or Y and their predecessor counts are positive. The following result shows that our algorithm is correct and complete: Theorem 2 Let C be a circuit with n variables and i be a complete input assignment. The circuit is output constructive w.r.t. i if and only if the algorithm starts with i and computes a fact for each output variable. The circuit is completely constructive w.r.t. I if and only if the algorithm terminates with all predecessor counts 0. For a completely constructive circuit, the algorithm always takes the same number of steps, which is the sum of all the fanin counts. 3 POLIS and the CFSM model Recall our goal is to implement synchronous circuits within the POLIS system. POLIS [3] is a software tool developed at UC Berkeley for the synthesis of control-dominated reactive systems that are targeted for mixed hardware/software implementations. The primary feature of POLIS is its underlying CFSM model of computation; it is within this model that we implement synchronous circuits. 3.1 Overview The model of computation consists of a network of communicating Codesign Finite State Machines (CFSMs). The communication style is called GALS: globally asynchronous locally syn- chronous. At the node level, each CFSM has synchronous semantics: when run, a CFSM reads inputs, computes, and writes outputs instantaneously. At the network level, the CFSMs communicate asynchronously: communication is done via data transmission through buoeers, and no assumptions are made about the relative delays of the computations performed by each CFSM or about the delays of the data transmission. 3.2 CFSM Communication Each CFSM has a set of inputs and outputs, and CFSMs are connected with nets. A net associates an output of one CFSM to some inputs of other CFSMs. The information transmitted between CFSMs is composed of a status and a value which are stored in 1-place communication buoeers. For each net, there is one associated value buoeer and multiple status buoeers, one for each attached CFSM input. Thus, each CFSM has a local copy of the status of each of its inputs, while the value is stored in a shared buoeer. A CFSM input buoeer is composed of the local status buoeer and the shared value buoeer. 3 The status buoeer stores either 1 or 0, representing presence or absence of valid data in the value buoeer. A CFSM input assignment is the set of values stored in the input buoeers for a CFSM. It is equivalent to the circuit input assignment given in Section 2.1.1. A CFSM input assignment may be complete or partial. A captured input assignment corresponds to the statuses and values that are actually read from the buoeers when a CFSM in run. 3.3 CFSM Computation A CFSM computation is called a CFSM execution or CFSM run. When a CFSM executes, it reads its inputs, makes its computation, writes its outputs, and resets (consumes) its inputs. Input reading: A CFSM atomically reads and resets the status buoeers: it simultaneously reads all status buoeers and sets them to 0, ready for the arrival of new inputs. 4 It subsequently reads the values of the present inputs. This determines the captured input assignment. Computation: The CFSM uses the captured input assignment to make its computation: it computes its outputs and next states based on the values given in its state transition table. The computation is done synchronously, which means that the CFSM reacts precisely to the captured input assignment, regardless of whether the inputs change while the CFSM is computing. Output writing: For each output, a CFSM writes the value buoeer and subsequently atomically sets the status buoeers for each associated CFSM input. 5 A CFSM-event consists of an output emitting its data and the corresponding input status buoeers being set to 1. 3.4 CFSM Network Computation A network computation is called a network execution or network run and corresponds to several CFSM executions. Each CFSM network has an associated scheduler(s). The scheduler continuously reads the current input assignments, determines which CFSMs are runnable, and chooses the order in 3 Note that in [3], the word event is used both for the status alone and for the status/value pair. In POLIS, a CFSM may have an empty execution, which means that it does not react to its current inputs. In this case, the current inputs are saved, and any inputs that are received while the CFSM is determining its empty reaction are added to the input assignment, which is restored and thus read at the next run. We do not use this feature here. 5 Atomic reads and writes are more expensive, since they require an implementation that guarantees that these actions can happen simultaneously. The decision was made in POLIS to make status buoeer reading and writing atomic, and not value-buoeer reading and writing, because atomically reading and writing of short bit strings can be implemented eOEciently, and because this guarantees certain desirable behavioral properties in the system. which to run them. 6 A CFSM is runnable if it has at least one input status buoeer set to 1. A CFSM is run by the scheduler sometime after it is runnable. Typically, an input assignment is given to the network, and the scheduler runs the CFSMs according to its schedule until there are no further changes in the communication buoeers. This is called a complete network execution. Time eoeectively passes when control is returned to the scheduler, and thus instantaneous communication between CFSM modules is not possible. Implementing Constructive Circuits in CFSM Networks In this section, we explain our realization of the synchronous behavior of a circuit on a CFSM network. To facilitate the exposition, we restrict ourselves to the extreme case of one CFSM per gate. More realistic levels of granularity will be handled in Section 5. In Section 2.4, we presented an algorithm to compute the behavior of a circuit for a given circuit input assignment. The essential ingredients were a set TODO of facts to propagate, a set DONE of established facts, and a predecessor counter for each variable. The basic idea of the CFSM network implementation presented here is to distribute a similar algorithm over a network of CFSMs, associating a CFSM with each circuit gate (equation). We start by studying the reaction to a single input assignment and then present various ways of chaining reactions to handle circuit input assignment sequences, to obtain the cyclic behavior characteristic of synchronous systems. 4.1 Fact propagation in a CFSM network We implement each gate as a CFSM that reads and write facts, which are encoded in POLIS CFSM-events sent by one gate to its fanouts. The arrival of a fact at a gate makes the gate runnable, and, when run, if there is a provable output fact from the facts received so far, the gate CFSM outputs it. Fact propagation between gates is directly performed by the underlying POLIS scheduling and CFSM-event broadcasting mechanisms. A POLIS execution schedule is thus precisely a proof (fact propagation) ordering. Facts arrive sequentially at a gate CFSM. Therefore, a combinational circuit gate must be implemented by a sequential CFSM that remembers which facts it has received so far. The sequential state of a gate CFSM encodes the number of predecessors of the interpretation algorithm of Section 2.4. 4.2 The Basic Gate CFSM For ease of exposition, we write the gate CFSMs in Esterel. This makes the gate speci-cation very AEexible, which will be useful in the next sections. No preliminary knowledge of Esterel is required. To handle our running example , it suOEces to describe the AndNot gate Other gates are similar. The Esterel program for AndNot has the following interface: module AndNot : Here, A, B, and C are Esterel signals of type boolean, the values of which are called true and false. Esterel signals are just like POLIS buoeers, with some additional notation. An event of a boolean-valued signal such as A has two components: a binary presence status component, also written A, which can take values present and absent , and a value component of type boolean, written ?A. We choose to encode the fact present with value true (resp. 6 In POLIS, scheduler is automatically synthesized with parameters, such as the type of scheduling algorithm, given by the user. false) 7 . Notice that we use two pieces of information, the status and the value, to represent a fact, i.e. the stable value of a wire. A present status component indicates stability, i.e. that a fact has been propagated to this point, and the value component represents the Boolean value of the fact. Like a POLIS captured input assignment, An Esterel input assignment de-nes the presence status of each input signal and the value of each present signal. For instance, for AndNot, A(true).B(false) is an Esterel input assignment in which A is present with value true and B is present with value false, encoding the facts is an input assignment where A is present with value false and B is absent, encoding the fact Like a CFSM, an Esterel program repeatedly reacts to an externally provided input assignment by generating an output assignment. The processing of an input assignment is also called a reaction or an instant. In POLIS, a run of an Esterel CFSM triggers exactly one reaction of the Esterel program, with the same input assignment. Unlike in POLIS, communication in Esterel is instantaneous: a signal emitted by a statement is instantaneously received by all the statements that listen to it. Similarly, control propagation is instantaneous; for example, in a sequence 'p; q', q immediately starts when p terminates. The only statements that break the AEow of control are explicit delays such as iawait Sj that waits for the next occurrence of a signal S. Finally, in Esterel, signal presence status is not memorized from reaction to reaction, but value is : the value of the Esterel expression ?A of A in a reaction where A is absent is the one it had in the previous reaction. Notice that the value of a signal may change only when the signal is present. Our -rst attempt to write the Esterel body of AndNot is: await A; if not ?A then emit C(false) end if if ?B then emit C(false) end if if (?A and not ?B) then emit C(true) end if The program reads as follows. First, we start two parallel threads. The -rst thread waits for the presence of A, and the second threads waits for the presence of B. The -rst input assignment can have A present, B present, or both (an empty assignment with neither A nor B present would leave the program in the same state; such an assignment is permitted in Esterel but will never be generated by the POLIS scheduler). If A is absent, the -rst thread continues waiting. If A is present, the -rst thread immediately checks A's value ?A and immediately outputs C(false) if ?A is false, thus mimicking the (l-and) deduction rule; the thread terminates immediately in either case. The second thread behaves symmetrically but checks for the truth of ?B to emit C(false). If both A and B are present, the threads evolve simultaneously. The Esterel parallel construct '-' terminates immediately when both branches have termi- nated. Therefore, the above parallel statement terminates exactly when both A and B have been received, either simultaneously or in successive input assignments. In that instant, C(true) is emitted if the possibly memorized values ?A and ?B are respectively true and false, mimicking the (b-and) deduction rule with negated second argument. 4.2.1 Avoiding Double Output Our gate CFSM almost works, but not quite, since C(false) can be emitted twice (possibly at dioeerent instants) if ?A is false and ?B is true. The gate should output C only once. To correct this problem, we use an auxiliary Boolean signal Caux: 7 Other equivalent encodings can be considered. One can for example use a pair of pure signals for each variable, one for presence and one for value. The encoding we use makes a clear dioeerence between availability and value. Sd Figure 2: Partial state transition graph for module AndNot signal Caux : combine boolean with and in await A; if not ?A then emit Caux(false) end if if ?B then emit Caux(false) end if if (?A and not ?B) then emit Caux(true) end if await Caux; emit C(?Caux) signal The -rst branch of the outermost parallel behaves as before but emits Caux instead of C. The second branch waits for Caux to emit C with the same value, and immediately terminates. If Caux is emitted twice in succession by the -rst branch, the second emission is simply unused since the iawait Cauxj statement has already terminated. The icombine boolean with andj declaration smoothly handles simultaneous double emission, also called collision. For this example, collision occurs if A(false) and B(true) occur simultaneously, in which case both iemit C(false)j statements are simultaneously executed. The combine declaration speci-es that the result value ?Caux is the conjunction of the separately emitted values. Here, we could as well use disjunction, for only false values will be combined. 4.2.2 The Gate CFSM State Graph The gate CFSM state transition graph (STG) is partially shown in Figure 2. The transitions are shown for the cases in which A is received before B, the other cases (B arriving -rst or A and B arriving simultaneously) are similar and not pictured. This partial STG is shown to help visualize the sequential state traversal in a familiar syntax, but is not a practical input mechanism for reactive modules compared to the Esterel language. For example, a module that waits for n signals concurrently will have 2 n states, while the Esterel description has size n. Note also that the C aux signal is shown in the output list for visualization purposes; it is an internal signal that is not seen by any other module. 4.2.3 Gate CFSM Execution Example To become familiar with the Esterel semantics, let us run the AndNot program on two dioeerent input assignment sequences. We start in state S 0 where we are waiting for the inputs A and B and internally for Caux, pictured by underlining the active await statements: signal Caux : combine boolean with and in await A; if not ?A then emit Caux(false) end if if ?B then emit Caux(false) end if if (not ?A and ?B) then emit Caux(true) end if await Caux emit C(?Caux) signal Assume the -rst gate input assignment is A(true) and B absent. Then, iawait Aj terminates, and we execute the test for inot ?Aj; since the test fails, the -rst parallel branch terminates without emitting Caux. We then reach state S 1 , in which we continue waiting for B and Caux: signal Caux : combine boolean with and in await A; if not ?A then emit Caux(false) end if if ?B then emit Caux(false) end if if (?A and not ?B) then emit Caux(true) end if await Caux; emit C(?Caux) signal If we now input B(false), we execute the ?B test, which also fails. Since the second parallel branch terminates, the parallel statement terminates immediately; we execute the i?A and not ?Bj test, which succeeds. We emit Caux(true), which makes the iawait Cauxj statement instantaneously terminate; the output C(true) is emitted, since We reach the dead state S d where no signal is awaited. Assume now that the -rst gate input assignment is A(false) and B absent. Then, starting from S 0 , we execute the -rst test, which succeeds and emits Caux(false). The iawait Cauxj statement immediately terminates and C(false) is emitted. We continue waiting for B, in the following signal Caux : combine boolean with and in await A; if not ?A then emit Caux(false) end if if ?B then emit Caux(false) end if if (?A and not ?B) then emit Caux(true) end if await Caux; emit C(?Caux) signal Then, when B occurs in a later input assignment, the iawait Bj statement terminates and the program reaches the dead state S d . If ?B is true, the emission of Caux(false) is performed but unused. This last step of waiting for B mimics the last cleanup step of the propagation algorithm of Section 2.4. It will be essential to chain cycles in Section 4.4. If A and B occur together in the -rst input assignment, then AndNot immediately emits C with the appropriate value and transitions directly from state S 0 to dead state S d . Notice that the number of predecessor waited for in the algorithm of Section 2.4 is exactly the number of underlined statements among iawait Aj and await Bj. Scheduler A A Y I Figure 3: CFSM network for circuit C 1 4.3 Performing a Single Reaction on a Network of Gates Given a circuit C, the CFSM network for C is obtained by creating an input buoeer for each input signal in C, an output buoeer for each output signal, and a gate CFSM for each equation in C. Gate CFSM outputs are broadcast to the gate CFSMs that use them, as speci-ed by the circuit equations. To run the network for a given circuit input assignment i, it suOEces to put the input values de-ned by i in each of the network input buoeers. Then, the gate CFSMs directly connected to inputs become runnable. As soon as a gate has computed its result, it puts it in its output buoeer, the result's value is automatically transferred to all fanout CFSM input buoeers by the network, and these CFSMs become runnable. 4.3.1 An Execution Example Consider the network for C 1 , pictured in Figure 3, where the CFSMs for X and Y are called CX and CY. The rectangular buoeers are the 1-place buoeers used to communicate CFSM-events between modules. Note that there are two information storage mechanisms at work during the execution of this circuit: 1. The CFSM-gates as implemented by the Esterel modules internally store which signals they have received and thus which they are still waiting for using their implicit states. 2. The CFSM-network as implemented in POLIS stores a copy of each CFSM-event, one for each fanout of that event, using the 1-place buoeers. Consider the input assignment I = 0 and -rst put false in I's buoeer and true in J's buoeer. The CFSMs CX and CY become runnable. Assume CX is run -rst. Then it captures the partial input assignment A(false) and B absent, which encodes I = 0. The CX CFSM outputs C(false), which is the encoding for goes to state S 1 . The false event is made visible at CY's B input buoeer after some time. ffl Assume -rst that CY is run before the arrival of CX's output. Then CY captures the partial input assignment A(true) and B absent, which encodes the fact 1. The CY CFSM emits no output and continues waiting for its B input, in state S 2 . When X's false value is written in CY's B input buoeer, CY is made runnable and runs with captured input assignment A absent and B(false); it emits C(true), which encodes Y = 1, and goes to the dead state. ffl Assume instead that CX's false output is written in CY's input buoeer B before CY is run. Then, when CY is later run, it captures the complete input assignment A(true):B(false), which encodes the facts It emits C(true) and goes directly to the dead state. Once CY has emitted its output C(true), the true value is written in CX's input buoeer B, and CX is made runnable again. Then, CX is run with input assignment B(true) and A absent, which encodes goes to the dead state. 4.3.2 Correctness of the CFSM Implementation The CFSM network computes a proof in the same way as the interpretation algorithm of Section 2.4, but with dynamic and concurrent scheduling of fact propagation. Building a new fact is equivalent to generating a CFSM-event. Propagating a fact is equivalent to broadcasting the CFSM-event to the fanouts and running the fanout CFSMs, which is exactly what the network automatically provides. The following theorem summarizes the results: Theorem 3 Let C be a circuit. Let n be the number of output or local variables (fanouts), and let f be the number of variable occurrences in the right-hand-sides of C's equations (fanins). Let i be a circuit input assignment. For any run of the network associated with C initialized with i, the following holds: 1. The number of created CFSM-events is bounded by n, and the number of CFSM runs is bounded by f . No buoeer overwrite can occur. 2. If, in some complete network execution sequence, exactly n CFSM-events have been created, then the implemented circuit is completely constructive w.r.t. i, and the output gate CFSM generated events are the encodings of the output values of C w.r.t. i. All complete execution sequences give the same result independent of the schedule, and all gate CFSMs terminate in the dead state once all CFSM-events have been processed. 3. If, for some complete run, less than n CFSM-events have been created, then this is true for all runs and C is not completely constructive w.r.t. i. Output constructive circuits can be handled by a slight modi-cation of the result, but loosing the nice fact that all gate CFSMs terminate in the dead state, which if useful when chaining reactions, which we demonstrate in the next section. 4.4 Chaining Reactions A synchronous circuit or program is meant to be used sequentially, the user or RTOS providing a sequence of input assignments and reading a sequence of output assignments. In our POLIS implementation, the user alternates writing circuit input assignments in the network input buoeers and reading the computed circuit output assignments in the network output buoeers. Since POLIS uses 1-place buoeers for communication, we must make sure that no buoeer overwrite occurs in the network. In particular, we cannot let the user overwrite an input buoeer until its value has been completely processed by the gates connected to it. Here are four possible user-level protocols: ffl Wait for a given amount of time. This is the technique used for single-clocked electrical circuits. Since the number of operations to be performed is uniformly bounded, if the underlying machinery (CPUs, network, etc.) has predictable performance, we are Figure 4: Circuit C 1 guaranteed that the reaction is complete after a maximal (predictable) time and that no buoeer overwriting occurs. This solution is often used in cycled-based control systems implemented in software and in Programmable Logic Controllers (PLCs). This protocol can be realized in our implementation with the addition of performance estimation, in order to compute the frequency with which new inputs can be fed to the synchronous circuit. ffl Compute and return a termination signal. If the circuit is completely constructive w.r.t. the input, we know that the computation has -nished when all the gate CFSMs have read all their inputs, i.e. when the network has processed a given number of CFSM-events. We can either modify the scheduler to have it report completion to the user or build an explicit termination signal by having each gate output a separate CFSM-event when it has processed all inputs. These CFSM-events are gathered by an auxiliary gate that generates a termination event for the user when all its input have arrived. These centralized solutions are not in the spirit of distributed systems. ffl Implement a local AEow control protocol at each gate CFSM. This is a much more natural solution in a distributed setting and it makes it possible to pipeline the execution: for each input, the user may enter a new value as soon as the AEow-control protocol says so, without waiting for the reaction to be complete. The protocol must ensure that an input for a conceptual synchronous cycle never interferes with values for other cycles. ffl Queue input events: this solution is used in [9, 8]. It implies that the user can always write new inputs and is never blocked. In our implementation, the same AEow control problem is simply pushed inside the network, since CFSMs do not communicate using queues. We now present a AEow-control protocol that supports pipelining. The reactions remain globally well-ordered as required by the synchronous model: the n-th value of input I is processed in the same conceptual synchronous cycle as the n-th value of input J ; however, because of pipelining, internal network CFSM scheduling and CFSM-event generation can occur in intricate orderings. To make the gate reusable, is suOEces to embed their bodies into an Esterel iloop.endj in-nite loop. Then, instead of going to the dead state, a gate CFSM returns to its initial state. This is why it is much easier to handle complete proofs. To deal with more general output proofs, we should add complicated gate reset mechanism, while reset is automatically performed by complete proofs. Thanks to the AEexibility of Esterel code, the protocol only requires a slight modi-cation of our basic gate code, and the addition of a new module. The corresponding CFSM network is shown in Figure 4. Consider an output X of a CFSM M, read for example by two other CFSMs N and P. With X and N (resp. P) we associate a signal X-Free-N (resp. X-Free-P) that is written by N (resp. P). With X and M we associate a signal X-Free-M read by M and written by an auxiliary module X-CFSM which consumes X-Free-N and X-Free-P and writes X-Free-M when both X-Free-N and X-Free-P have received a value. The buoeers in Figure 4 for each signal are those used by POLIS; the actual information determining when the signal X is free to be written by M is contained in the implicit states of X-CFSM. The new module is written as follows: module X-CFSM: input X-Free-N, X-Free-P; output X-Free-M; loop await X-Free-N await X-Free-P emit X-Free-M loop module Similarly, for a network input I broadcast to N and P, we generate a network output buoeer I-Free -lled by the auxiliary CFSM reading I-Free-N and I-Free-P, and for any network output O a network input buoeer O-Free -lled by the user when it is ready to accept a new value of O. We require M to write its X output only when X-Free-M holds 0, then consuming that value. We require N (resp. P) to write 0 in X-Free-N (resp. X-Free-P) when it reads its local copy of the input X. The AndNot CFSM is modi-ed as follows: module AndNot : output A-Free, B-Free; input C-Free; loop signal Caux : combine boolean with and in await A; emit A-Free; if not ?A then emit Caux(false) end if emit B-Free; if ?B then emit Caux(false) end if if (?A and not ?B) then emit Caux(true) end if await Caux; await C-Free emit C(?Caux) signal loop The output C is emitted only when the last of Caux and C-Free has been received. When the gate CFSM is instantiated at a node M, the A-Free, B-Free, and C-Free buoeers must be appropriately renamed A-Free-M, B-Free-M, and C-Free-M, to avoid name clashes. The AEow-control mechanism acts in two ways. First, it prevents buoeer overwriting. Second, it makes pipelining possible. Given a circuit input assignment i n at cycle n, the new value of a circuit input I for cycle can be written in I's network input buoeer as soon as I-Free is full. Therefore, it is not necessary to wait for the global end of a cycle to locally start a new one. We have a last technical problem to solve. Assume that an AndNot gate CFSM starts circuit cycle n. Assume that the gate CFSM receives an A input event, say A(false) with B absent. The gate sends back A-Free. From then on, the gate can receive two inputs: ffl The B input event that holds B's value in cycle n. This input should be processed normally since the gate CFSM is currently processing cycle n. ffl The out-of-order A input event that holds A's value for cycle n + 1. Processing this input should be deferred until B has been processed. In the current POLIS network model, a CFSM is made runnable as soon as it receives an input event. Therefore, the gate can be made runnable with input A for cycle while it is still processing cycle n. At this point, the gate should either internally memorize A's value or rewrite it in the A buoeer, leaving in both cases the A-Free AEow control buoeer empty until it has -nished cycle n. Both solutions are expensive and somewhat ugly. We suggest a slight modi-cation to the POLIS scheduling policy. A CFSM should tell the scheduler which input buoeers it is currently interested in, and the scheduler should not make the CFSM runnable if none of these buoeers holds an event. When the CFSM is run, its captured input assignment should only contain the events in the buoeers the CFSM is explicitly waiting for, leaving the rest in their input buoeers. In the above example, the gate CFSM tells the scheduler it is only waiting for B. If the new value of A comes in, the CFSM is not made runnable. When B occurs, the gate is made runnable, and it will run with input B only. Once the gate has processed B, it tells the scheduler that it is now waiting for both A and B. Since A is already there, the gate can be immediately made runnable again. The -nal version of the gate CFSM involves the auxiliary Wait signals sent to the scheduler to implement this mechanism: module AndNot : output A-Free, B-Free; output A-Wait, B-Wait; input C-Free; output C-Free-Wait; loop signal Caux : combine boolean with and in abort sustain A-Wait when A; emit A-Free; if not ?A then emit Caux(false) end if abort sustain B-Wait when B; emit B-Free; if ?B then emit Caux(false) end if if (?A and not ?B) then emit Caux(true) end if await Caux; abort sustain C-Free-Wait when C-Free emit C(?Caux) signal loop The iAwait Aj statement has become iabort sustain A-Wait when Aj. The isustain A-Waitj statement emits A-Wait in each clock cycle. the iabort p when Aj aborts its body p right away when A occurs, not executing p at abortion time. Therefore, A-Wait is emitted until A is received, that instant excluded. 5 Mixed Synchronous/Asynchronous Implementation We now have two very dioeerent levels of granularity for implementing an Esterel program in POLIS: compiling the program into a single CFSM node or building a separate CFSM for each gate of the program circuit. The -rst does not support distribution, while the second is clearly too ineOEcient: the associated overhead is unacceptable for large programs since it involves scheduling each individual gate CFSM multiple times. We now brieAEy explain how we can deal with many other implementation choices with dioeer- ent levels of granularity, using the compositional and incremental character of the constructive semantics. When doing so, we retain the full synchronous semantics of the program, but we trade ooe synchrony and asynchrony in the implementation. The idea as one moves to a larger granularity implementation is to partition the set of gates into gate clusters G 1 Each cluster G k groups its gates into a single CFSM, the clusters being connected by the POLIS network as before. The partition can be arbitrary, and chosen to match any locality or performance constraints. Facts are processed both synchronously and asynchronously, but again their proofs are derived from the synchronous constructive semantics. In particular, synchronous fact processing is done within a cluster using the algorithm of Section 2.4, in a single CFSM and in one computation of that CFSM; asynchronous fact processing is done across the network and thus between CFSMs. Some facts will be both synchronously and asynchronously processed, e.g. an output from gate g 1 that is an input to another gate g 0in the same cluster G 1 and to g 2 in another cluster G 2 What makes this possible is the ability of our centralized and distributed algorithms to deal with partial deduction: given a partial input assignment i, both algorithms generate all the facts that can be deduced from i. If a new fact is added to i, the algorithms incrementally deduce its consequences. Therefore, it does not matter whether facts are handled synchronously in a gate cluster or asynchronously in the POLIS cluster network. Consider for example the following circuit C 2 obtained by adding an output Z to C 1 Consider -rst the clusters G fZg. Assume that we receive the fact deduces outputs that fact to G 2 , which can make a local transition to reach the state S 1 where it waits only for Y also internally remembers in its local state that Y has lost a predecessor. Thus, synchronously propagated to Y in the same cluster, and asynchronously propagated to Z in the other cluster through another call to a CFSM,. If we now receive sends that fact to G 2 , which can now output With the same input sequence, consider the clusters G g. When receiving instantaneously generates the facts determined synchronously. The fact asynchronously propagated to G 2 by the network, and G 2 's CFSM transitions to a state where it waits only for J . When occurs, the CFSM outputs that fact is propagated to G 1 's CFSM, which goes back to its initial state. Optimal solutions to the problem of determining a set of clusters is beyond the scope of this paper. A number of clustering algorithms exist in the literature, and the design may be entered in a partitioned fashion that leads to a natural clustering as well. In our case, clustering according to the source code module structure is an obvious candidate for a clustering heuristic, as well as clustering according to the frequency of use of signals (like clocks in Lus- tre). Here, we simply point out that our algorithms and the semantics behind them permit any level of granularity: from individual gates implemented as separate CFSMs, to an entire synchronous program implemented as a single CFSM. Thus, the tradeooe between synchronous and asynchronous implementation of a synchronous program can be fully explored. 6 Conclusions and Future Work We have described a method for implementing synchronous Esterel programs or circuits on globally asynchronous locally synchronous (GALS) POLIS networks. The method is based on fact propagation algorithms that directly implement the constructive semantics of synchronous programs. We have developed AEow-control techniques that automatically ensure that no POLIS buoeer can be overwritten and that make pipelining possible. Initially, we have associated a POLIS CFSM with each circuit gate, which is unrealistic in practice. However, our method is fully compositional, and fact propagation can be performed either synchronously in a node or asynchronously between nodes. This makes it possible to cluster gates into bigger synchronous nodes and to explore the tradeooe between synchronous and asynchronous implementation. For simplicity, we have only dealt with the pure fragment of Esterel where signals carry no value. Extension to full value-passing Esterel constructs raises no particular diOEculty. A complete implementation is currently being developed. --R Domains and Lambda-Calculi The Constructive Semantics of Esterel. The Foundations of Esterel. The Esterel Synchronous Programming Language: Design asynchronous Circuits. Distributing automata for asynchronous networks of processors. Distributing reactive systems. Programming Real-Time Applications with Signal Synchronous Programming of Reactive Systems. The Synchronous DataAEow Programming Language Lustre. A Visual Approach to Complex Systems. Communicating Sequential Processes. The Semantics of a Simple Language for Parallel Programming. LCF as a programming language. Formal Analysis of Cyclic Circuits. Constructive Analysis of Cyclic circuits. Analyse Constructive et Optimisation S --TR Communicating sequential processes Statecharts: A visual formalism for complex systems The ESTEREL synchronous programming language Formal verification of embedded systems based on CFSM networks Hardware-software co-design of embedded systems Domains and lambda-calculi The foundations of Esterel Synchronous Programming of Reactive Systems Constructive Analysis of Cyclic Circuits Formal analysis of synchronous circuits --CTR Gerald Lttgen , Michael Mendler, The intuitionism behind Statecharts steps, ACM Transactions on Computational Logic (TOCL), v.3 n.1, p.1-41, January 2002 Mohammad Reza Mousavi , Paul Le Guernic , Jean-Pierre Talpin , Sandeep Kumar Shukla , Twan Basten, Modeling and Validating Globally Asynchronous Design in Synchronous Frameworks, Proceedings of the conference on Design, automation and test in Europe, p.10384, February 16-20, 2004 Stephen A. Edwards , Olivier Tardieu, SHIM: a deterministic model for heterogeneous embedded systems, Proceedings of the 5th ACM international conference on Embedded software, September 18-22, 2005, Jersey City, NJ, USA Stephen A. Edwards , Edward A. Lee, The semantics and execution of a synchronous block-diagram language, Science of Computer Programming, v.48 n.1, p.21-42, July

embedded systems;finite state machines;synchronous programming;asynchronous networks

363459

Wavelet and Fourier Methods for Solving the Sideways Heat Equation.

We consider an inverse heat conduction problem, the sideways heat equation, which is a model of a problem, where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. Mathematically it is formulated as a Cauchy problem for the heat equation in a quarter plane, with data given along the line x=1, where the solution is wanted for $0 \leq x < 1$.The problem is ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. We consider stabilizations based on replacing the time derivative in the heat equation by wavelet-based approximations or a Fourier-based approximation. The resulting problem is an initial value problem for an ordinary differential equation, which can be solved by standard numerical methods, e.g., a Runge--Kutta method.We discuss the numerical implementation of Fourier and wavelet methods for solving the sideways heat equation. Theory predicts that the Fourier method and a method based on Meyer wavelets will give equally good results. Our numerical experiments indicate that also a method based on Daubechies wavelets gives comparable accuracy. As test problems we take model equations with constant and variable coefficients. We also solve a problem from an industrial application with actual measured data.

Introduction In many industrial applications one wishes to determine the temperature on the surface of a body, where the surface itself is inaccessible for measurements [1]. It may also be the case that locating a measurement device (e.g. a thermocouple) on the surface would disturb the measurements so that an incorrect temperature is recorded. In such cases one is restricted to internal measurements, and from these one wants to compute the surface temperature. In a one-dimensional setting, assuming that the body is large, this situation can be modeled as the following ill-posed problem for the heat equation in the quarter plane: Determine the temperature u(x; t) for temperature measurements g(\Delta) := u(1; \Delta), when u(x; t) satisfies Of course, since g is assumed to be measured, there will be measurement errors, and we would actually have as data some function g m 2 L 2 (R), for which where the constant ffl ? 0 represents a bound on the measurement error. For can solve a well-posed quarter plane problem using g m as data. For we have the sideways heat equation 1 Note that, although we seek to recover u only for 0 - x ! 1, the problem specification includes the heat equation for x ? 1 together with the boundedness at infinity. Since we can obtain u for x ? 1, also u x (1; \Delta) is de- termined. Thus we can consider (1.3) as a Cauchy problem with appropriate Cauchy data [u; u x ] given on the line Although in this paper we mostly discuss the heat equation in its simplest form u our interest is in numerical methods that can be used for more general problems, e.g. equations with non-constant coefficients, Also referred to as the Inverse Heat Conduction Problem (IHCP) [1]. or nonlinear problems, which occur in applications. For such problems one cannot use methods based on reformulating the problem as an integral equation of the first kind, since the kernel function k(t) is explicitly known only in the constant coefficient case. Instead, we propose to keep the problem in the differential equation form and solve it essentially as an initial value problem in the space variable ("space-marching", [1, 5, 6, 11, 12, 13, 22, 23]). In [11, 12] we have shown that it is possible to implement space-marching methods efficiently, if the time-derivative is approximated by a bounded op- erator. In this way we obtain a well-posed initial-boundary value problem for an ordinary differential equation (ODE). The initial value problem can be solved by a standard ODE solver, see Section 2.1. In this paper, we study three different methods for approximating the time derivative and their numerical implementation. First, the time derivative is approximated by a matrix representing differentiation of trigonometric interpolants, see e. g. [14, Sec. 1.4]. In Section 3 we first give some error estimates (for the non-discrete case, see also [24]), and then we discuss the numerical implementation of this method. Meyer wavelets have the property that their Fourier transform has compact support. This means that they can be used to prevent high frequency noise from destroying the solution 2 . In a previous paper [25] we studied the approximation of the time derivative in (1.3) using Meyer wavelets, and gave almost optimal error estimates. Here we discuss the numerical implementation of this method, and, in particular, we show how to compute the representation of the time derivative in wavelet basis. Daubechies wavelets themselves have compact support and, consequently, they cannot have a compactly supported Fourier transform. However, the Fourier transform decays so fast that in practice they can be used for solving the sideways heat equation in much the same way as Meyer wavelets. We discuss this in Section 4.5. We emphasize that although all theoretical results are for the model problem with constant coefficients, we only deal with methods that can be used for the more general problems (1.4)-(1.5). Also, the quarter plane assumption is made mainly for deriving theoretical results; it is not essential in the computations (provided that u x (1; \Delta) can be obtained, either from measurements of by assuming symmetry, cf. Section 5.2). A constructed 2 It is very common in ill-posed problems that the ill-posedness manifests itself in the blow-up of high frequency perturbations of the data. variable coefficient example is given in Section 5. There we also present a problem from an industrial application, with actual measured data. Ill-posedness and Stabilization The problem of solving the sideways heat equation is ill-posed in the sense that the solution, if it exists, does not depend continously on the data. The ill-posedness can be seen by solving the problem in the Fourier domain. In order to simplify the analysis we define all functions to be zero for t ! 0. Let g(t)e \Gammai-t dt; be the Fourier transform of the exact data function 3 . The problem (1.1) can now be formulated, in frequency space, as follows: bounded . The solution to this problem, in frequency space, is given by where i- denotes the principal value of the square root, ae In order to obtain this solution we have used the bound on the solution at infinity. Since the real part of i- is positive and our solution b u(x; -) is assumed to be in L 2 (R), we see that the exact data function, b g(-), must decay rapidly as - !1. Now, assume that the measured data function satisfies g m (R), is a small measurement error. If we try to solve the problem using g m as data we get a solution Since we can not expect the error b ffi(-) to have the same decay in frequency as the exact data bg(-) the solution bv(x; -) will not, in general, be in L 2 (R). 3 In this paper all Fourier transforms are with respect to the time variable. Thus, if we try to solve the problem (1.3) numerically, high frequency components in the error, ffi , are magnified and can destroy the solution. However, if we impose an apriori bound on the solution at and in addition, allow for some imprecision in the matching of the data, i.e. we consider the then we have stability in the folllowing sense: any two solutions of (2.4), u 1 and For this reason we call (2.4) the stabilized problem. The inequality (2.5) is sharp and therefore we can not expect to find a numerical method, for approximating solutions of (2.4) that satisfy a better error estimate. Different methods for approximating solutions of (2.4) exist. One difficulty is that for (2.4) we do not have uniqueness, so in order to get a procedure that can be implemented numerically, it is necessary to somehow modify the problem. Often, the dependence on ffl and M is included by choosing the value of some parameter in the numerical procedure. 2.1 Stabilization by Approximating the Time Derivative In this subsection we consider the "initial-value" problem ' u x @ '' u with initial-boundary values 4 The initial values for u x can be obtained (in principle and numerically) by solving the well-posed quarter plane problem with data u(1; x - 1. 4 For a discussion of the problem of setting numerical boundary values at to [12]. We can write the solution of (2.6)-(2.8) formally as ' u hm (t) @ Loosely speaking, since the operator B is unbounded, with unbounded eigen-values in the left half plane, the solution operator unbounded frequency noise in the data can be blown up and destroy the numerical solution. Even if the data are filtered [4, 23], so that high frequency perturbations are removed, the problem is still ill-posed: rounding errors introduced in the numerical solution will be magnified and will make the accuracy of the numerical solution deteriorate as the initial value problem is integrated. In a series of papers [11, 12, 25], we have investigated methods for solving numerically the sideways heat equation, where we have replaced the operator @=@t by a bounded operator. Thus we discretized the problem in time, using differences [11] or wavelets [25], so that we obtained an initial value problem U x x '' U U x where the matrix D is a discretization of the time derivative, and U are semi-discrete representations of the solution and its derivative, and Gm and Hm are vectors. Thus, (2.9) can be considered as a method of lines. The observation made in [11, 12] is that when D represents a discrete approximation of the time derivative, then D is a bounded operator, and the problem of solving (2.9) is a well-posed initial value problem for an ODE. In [11, 25] error estimates are given for difference and wavelet approximations. The coarseness of the discretization is chosen depending on some knowledge about ffl and M (actually their ratio). For solving (2.9) numerically, we can use a standard ODE solver. In [12] we show that, in the case of a finite difference approximation, it is often sufficient to use an explicit method, e.g. a Runge-Kutta code. In the rest of this paper, we will discuss error estimates and numerical implementation of approximations of (2.4) by discretized problems of the type (2.9), by a Fourier (spectral) method, and by Meyer and Daubechies wavelets. 5 Cf. (2.2), and also [12], where a discretized equation is considered. 3 A Fourier Method Here we consider how to stabilize the sideways heat equation by cutting off high frequencies in the Fourier space. Error estimates are given in Section 3.1. Similar results can be found in [24]. In Section 3.2 we then arque that in order to useful for more general equations (1.4)-(1.5), the Fourier method should be implemented as in (2.9). We start with the family of problems in Fourier space, bounded . parameterized by -. In Section 2 the solution was shown to be e Since the principal value of i- has a positive real part, small errors in high frequency components can blow up and completely destroy the solution for natural way to stabilize the problem is to eliminate all high frequencies from the solution and instead consider (3.1) only for Then we get a regularized solution. e where - max is the characteristic function of the interval [\Gamma- In the following sections we will derive an error estimate for the approximate solution (3.3) and discuss how to compute it numerically. 3.1 In this section we derive a bound on the difference between the solutions (3.2) and (3.3). We assume that we have an apriori bound on the solution, . The relation between any two regularized solutions (3.3) is given by the following lemma. Lemma 3.1 Suppose that we have two regularized solutions v 1 and v 2 defined by (3.3) with data g 1 and ffl. If we select then we get the error bound Proof: From the Parseval relation we get \Gamma- je 6 je Using From Lemma 3.1 we see that the solution defined by (3.3) depends continuously on the data. Next we will investigate the difference between the solutions (3.2) and (3.3) with the same exact data g(t). Lemma 3.2 Let u and v be the solutions (3.2) and (3.3) with the same exact data g, and let - Suppose that ku(0; \Delta)k 6 M . Then Proof: As in Lemma 3.1 we start with the Parseval relation, and using the fact that the solutions coincide for - 2 [\Gamma- Z je Z je Now we use the bound ku(0; \Delta)k 6 M , and as before we have - which leads to the error bound Now we are ready to formulate the main result of this section: Theorem 3.3 Suppose that u is given by (3.2) with exact data g and that v is given by (3.3) with measured data g m . If we have a bound ku(0; \Delta)k 6 M , and the measured function g m satisfies we choose then we get the error bound Proof: Let v 1 be the solution defined by (3.3) with exact data g. Then by using the triangle inequality and the two previous lemmas we get From Theorem 3.3 we find that (3.3) is an approximation of the exact solution, u(x; t). The approximation error depends continously on the measurement error and the error bound is optimal in the sense (2.5). 3.2 Numerical implementation of the Fourier-based method Here we discuss how to compute the regularized solution (3.3) numerically. Given a data vector Gm with measured samples from g m on a grid ft i g, a simple method is to approximate the solution operator directly. Thus we have the following algorithm: 1. b Gm := FGm . 2. b ae exp( 3. V (0; :) := F H b Vm (0; :), where F is the Fourier matrix. The product of F and a vector can be computed using the Fast Fourier Transform (FFT) which leads to an efficient way to compute the solution (3.3). When using the FFT algorithm we implicitly assume that the vector Gm represents a periodic function. This is not realistic in our application; and thus we need to modify the algorithm. We refer to Appendix A for a discussion on how to make the problem 'periodic'. The main disadvantage of this method is that it cannot be used for problems with variable coefficients (1.4), (1.5). Note also that by approximating the solution operator directly, we make explicit use of the assumption that our solution domain is the whole quarterplane, t ? 0 and x ? 0. An alternative, more widely applicable, approach is to approximate the time derivative and use the ordinary differential equation formulation given in Section 2.1, i.e. we take the matrix D in (2.9) to be where is a diagonal matrix which corresponds to differentiation of the trigonometric interpolant, but where the frequency components with are explicitly set to zero [14, Sec. 1.4]. This will filter the data and ensure that we remove the influence from the high frequency part of the measurement error in the solution. Thus in the ODE solver, multiplication by D F is carried out as a FFT, followed by multiplication by and finally an inverse FFT. Multiplication with D F thus requires O(n log(n)) operations. We conclude the section with the remark that this method can be considered as a Galerkin method (cf. Section 4.2) with trigonometric interpolants as basis and test functions. 4 Wavelet Methods 4.1 Multiresolution analysis Wavelet bases are usually introduced using a multiresolution analysis (MRA) (see Mallat [18], Meyer [21]). Consider a sequence of successive approximation spaces, and [ For a multiresolution analysis we also require that This means that all spaces V j are scaled versions of the space V 0 . We also require that there exists a scaling function, OE, such that the set fOE jk g j;k2Z , where OE jk is an orthogonal basis for V j . Since OE we find that the function OE must satisfy a dilation relation The wavelet function / is introduced as a generator of an orthogonal basis of the orthogonal complement W j of V j in V j+1 (V function / satisfies a relation similar to (4.2) where the filter coefficients fg k g are uniquely determined by the fh k g. Any function f 2 L 2 (R) can be written l-j where P j is the orthogonal projection onto V j , 4.2 Meyer Wavelets and a Galerkin Approach The Meyer wavelet / 2 C 1 (R)) is defined by its Fourier transform [7, e \Gammai-t where sin \Theta -j cos \Theta -j 0; otherwise, and j is a C k or C 1 function satisfying ae It can be proved that the set of functions for Zis an orthonormal basis of L 2 (R) [20, 7]. The corresponding scaling function is defined by its Fourier transform, cos \Theta -j 0; otherwise. The functions b OE jk have compact support (see e.g. [25]) for any k 2 Z. In [25] we presented a wavelet-Galerkin method, where starting from the weak formulation of the differential equation, (v with test functions from V j , and with the Ansatz c (-) (x)OE j- (t); we get the infinite-dimensional system of ordinary differential equations for the vector of coefficients c, ae c The initial values fl m are defined and the matrix D j is given by (D The well-posedness of the Galerkin equation (4.8) follows from the fact that the matrix D j is bounded with norm [25] In [25] we proved the following stability estimate for the wavelet-Galerkin method. Theorem 4.1 Let g m be measured data, satisfying kg that chosen so that log M Then the projection onto V j of the Galerkin solution v j+1 satisfies the error estimate Note that the result in Theorem 4.1 is suboptimal in two ways. Firstly, we get slower rate of convergence when ffl tends to zero than in the optimal estimate (2.5). Secondly, the error is not for v j itself, but only for the projection of v j+1 onto V j . 4.3 Discrete Wavelet Transform Discrete wavelet transforms can be defined in terms of discrete-time multiresolution analyses (see e. g. [26, Section 3.3.3]). We will use DMT as a short form of 'Discrete Meyer (wavelet) Transform'. Given a vector c 2 R n , is assumed to hold the coefficients of some function v in terms of the basis of V J . This is the fine level all contributions on yet finer levels are assumed to be equal to zero. The DMT of c is equivalent to a matrix-vector multiplication [16, p. 11] where G J n\Thetan is an orthogonal matrix. The subscript j indicates that the part of the vector ~ c that contains the coarse level coefficients has 2 j components; it holds the coefficients of the projection of v onto V j . We illustrate how the transformation by G J breaks up the vector into blocks of coefficients corresponding to different coarseness levels in Figure 4.1. ~ v: Figure 4.1: Schematic picture of the discrete wavelet transform ~ J \Gamma3 c. The rightmost part of the vector represents wavelet coefficients on the resolution level block coefficients on the level 2 J \Gamma2 , etc. The leftmost part represents the coarse level coefficients. Algorithms for implementing the discrete Meyer wavelet transform are described in the thesis of Kolaczyk [16]. These algorithms are based on the fast Fourier transform (FFT), and computing the DMT of a vector in R n requires O(n log 2 n) operations [16, p. 66]. The algorithms presuppose that the vector to be transformed represents a periodic function; we will return to this question in Appendix A. A further illustration is given in Figure 4.2. Here we have taken a data vector from R 256 , added a normally distributed perturbation of variance 0:5 extended the vector smoothly to size 512 in order to make it "periodic". Then we computed the DMT with Thus the partitioning of the transformed vector is the same as in Figure 4.1. We see that the wavelet transform can be considered as a low pass filter: the noise is removed and the leftmost part of the transformed vector is a smoothed version of the data vector. The noise is represented in the data vector Figure 4.2: The upper graph shows a data vector and the lower graph its DMT. finer level coefficients. Note also the locality properties of the fine level coefficients: only the first half of the data vector is noisy, and consequently, each segment of fine level coefficients has (noticeable) contributions only in its left half. We remark that we use periodicity only for computational purposes, since the codes (based on discrete Fourier transforms) are implemented for this case. The vectors used are considered as finite portions of sequences in 4.4 Numerical Implementation In the solution of the sideways heat equation in V j , we replace the infinite-dimensional ODE (4.8), by the finite-dimensional x represents the approximation of the solution in Gm and Hm are the projections of the data vectors on V j , and j is chosen according to Theorem (4.1). For simplicity we suppress the dependence on j in the notation for vectors. The matrix D d j represents the differentiation operator in V j , and since we are dealing with functions, for which only a finite number of coefficients are non-zero, D d j is a finite portion of the infinite matrix D j (we use superscript d to indicate this). In the numerical solution of (4.12) by an ODE solver, we need to evaluate matrix-vector products D d C. The representation of differentiation operators in bases of compactly supported wavelets are described in the literature, see, e.g. [2]. For such wavelets exact and explicit representations can be found. Also finite difference operators can be represented easily [19]. In our context of Meyer wavelets, which do not have compact support, the situation is different. The proof of (4.10) in [25] actually gives a fast algorithm for this. For \Gamma- t - define the function Extend \Delta periodically, and expand it in Fourier series. In [25, Lemma A.2] is is shown that where d k is the element in diagonal k of D 0 . From the definition of D j it is easily shown that D . Thus, we can compute approximations of the elements of D j by first sampling equidistantly the function \Delta, and then computing its discrete Fourier transform. 4.5 Daubechies Wavelets The Daubechies wavelets have compact support [7] and therefore only a finite number of filter coefficients (fh k g and fg k g in (4.2) and (4.3)) are nonzero. The filter coefficients fg k g are uniquely determined by the coefficients fh k g where L is the number of filter coefficients. In addition we want the wavelet to have a number of vanishing moments, that is Daubechies wavelets are defined in such a way that they have the largest Figure 4.3: The Daubechies db4 scaling function (left) and wavelet (right). possible number of vanishing moments given the number of non-zero filter coefficients; in fact, we have There is no explicit expression for the Daubechies wavelet, instead we can compute it from the filter coefficients fh k g k2Z . For example the filter coefficients associated with the Daubechies wavelet db4 are [7] Since we are interested in using a basis of Daubechies wavelets for solving the system of ODE:s (2.6) in a stable way, we need to find an approximation of the derivative operator in the approximation spaces V j . Here we will use the Galerkin approximation, D j , which is a Toeplitz matrix with elements given by (D where the last equality follows from the definition of OE jk . This means that it is sufficient to compute the derivative approximation on the space V 0 . Since the scaling functions OE jk have compact support, the matrix D j will be banded. For compactly supported wavelets it is possible to compute the matrix D j explicitly. This result is due to Beylkin [2]. Since the matrix D 0 is constant along diagonals and only a few diagonals are non-zero, we can insert the dilation relation (4.2) into (4.16) and we will get a small system of linear equations to solve for the elements in D 0 . In the paper by Beylkin, the elements of D 0 have been listed for different bases of Daubechies wavelets. As in Section 4.2 we want to use a Galerkin approximation of the differential equation. For the problem to be well-posed it is sufficient that kD j k is bounded. In the case of periodized wavelets, the differentiation matrix per 0 is a circulant and we have kD per [27]. The same estimate can be expected to hold for non-periodic wavelets. The implementation details are similar to those in Section 4.4. We have measured data g m (t) - u(1; t) on the finite interval A ). Thus we have to approximate (2.6) with a finite-dimensional system similar to (4.12) for the coarse level coefficients. For compactly supported wavelets it is possible to give a fast implementation of the wavelet transform by convolution with the filter coefficients g. Since the filters are of short length we can compute the wavelet transform of a vector in R n with O(n log(n)) operations[26]. When we solve numerically the discretized version of (2.6) we are interested in computing the product of D j and a vector. Since D j is banded and explicitly known this is simply a product with a sparse matrix and thus requires 5 Numerical Experiments 5.1 Experiments Here we present some numerical experiments to illustrate the properties of the methods presented in the previous sections. We have solved discretized versions of (2.6) using Matlab in IEEE double precision with unit roundoff 1:1\Delta10 \Gamma16 . The space marching was performed using a Runge-Kutta-Fehlberg method (ode45 in Matlab) with automatic step size control, where the basic method is of order 4 and the embedded method is of order 5. In all tests the required accuracy in the R-K method was 10 \Gamma4 . The tests were performed in the following way: First we selected a solution computed data functions u(1; and u x (1; by solving a well-posed quarter plane problem for the heat equation using a finite difference scheme. Then we added a normally distributed perturbation of variance 10 \Gamma3 to each data function, giving vectors m and hm . Our error estimates use the signal-to-noise ratio M=ffl and therefore we computed kfk From the perturbed data functions we to reconstructed u(0; t) and compared the result with the known solution. We conducted two tests: Test 1: We solved the model problem (1.1) using a discontinous function, f(t) as the exact solution. Test 2: We solved the more general problem (1.4) using the coefficient function The results from these tests are given in Figure 5.1. In both cases the length of the data vectors g m and hm were 1024. The regularization parameters were selected according to the recipes given in Theorems 3.3 and 4.1. In the Meyer case we used projection on the space V 5 in both tests and computed 64 coarse level coefficients. We have used the auxiliary function in the definition of the scaling function (4.5). In the Fourier case we used the 42 lowest frequency components when calculating the time derivative. Since we have no stability theory for the Daubechies method we chose to solve the problem in space V 5 in both cases. Thus we computed 70 coefficients on the coarsest level. We used Daubechies wavelets with filters of length 8 in the Galerkin formulation. In all cases the number of steps in the ODE-solver were between 6 and 11. Before presenting the results we recomputed our coarse level approximation on the finer scale, using the inverse wavelet transform. Figure 5.1: Solution of the sideways heat equation using the different meth- ods. From top we have the results from the Fourier method, the Meyer method and the Daubechies method. The results from Test 1 are to the left and the results from Test 2 are to the right. The dashed line illustrates the approximate solution, the solid line represents the exact solution and the dash-dotted line represents the data function, g m . 5.2 An Industrial Application In this section we present an example of an industrial problem where the methods presented in the previous sections can be useful. The viability of the methods is demonstrated by an experiment conducted in cooperation with the Department of Mechanical Engineering, Link-oping University. Consider a particle board, on which a thin lacquer coating is to be ap- plied. In order to reduce the time for the lacquer coating to dry, the particle board is initially heated. Since the temperature gradients on and close to the surface of the board influence the drying time and the quality of the lacquer coating, it is important to estimate the temperature and the temperature gradients close to the surface. Often it is difficult or impossible to measure the temperature directly on the surface of the board. Instead a hole was drilled from the other side of the board and a thermocouple was placed close to the surface, as seen in Figure 5.2. After the thermocouple had been placed, the hole was filled in using the same material as in the surrounding board. In the experiment a particle board, initially heated to Figure 5.2: The cross-section, in principle, of the particle board used in the experiment. The temperature g m is measured by a thermocouple, and we seek to recover the temperature, f m , on the surface of the board [15]. suddenly placed to cool in air of room temperature. The thermocouple placed inside the plate, at distance 2:9 mm from the surface, gave the temperature history, g m (t), for From the measured temperature we reconstructed the temperature history, f m (t), on the surface of the board. The experiment presented here does not include the lacquer coating. The heat equation in this case is given by Here the constant - represents the physical properties of the problem. We can formulate this as an initial value problem similar to (2.6). One difficulty is that the surface temperature is not determined by the measured data g m alone. We need also the temperature gradient at L. If the thickness of the particle board is large in comparison to the distance between the surface and the thermocouple, then it is reasonable to consider this to be a quarter plane problem. In that case we can solve the heat equation for x ? L and compute u x (L; \Delta). Another possibility is to assume that the temperature function u(x; t) is symmetric with respect to the center of the board. In this particular experiment this should be a very accurate assumption. In Figure 5.3 we present results obtained using both these assumptions. The computations were performed using the Meyer wavelet method presented in Section 4.2. Initially the length of the vector g m was 8192; we solved only for 128 coarse level coefficients. It is interesting to note that the solutions almost coincide the first three minutes. Temperature Time min Temperature gradient U x (0,t) Figure 5.3: The measured temperature vector, g m , sampled at 10Hz (solid) and the corresponding surface temperature, f m from a quarter plane assumption (dashed) and from a symmetry assumption (dash-dotted) In the rightmost plot we see the temperature gradients at the thermocouple computed from a quarter plane assumption (dashed) and from a symmetry assumption (solid). Time min Temperature Time min Figure 5.4: Attempt to reconstruct the surface temperature, f m , using a coarse level grid of size 512 (left) and 1024 (right), in the Wavelet-Galerkin method. This means that we use less regularization than was the case in Figure 5.3. Clearly these solutions are unphysical. Since the temperature inside the particle board must be between the initial temperature, 70 the temperature of the surrounding air, approximately C, we can easily get a bound, M , on the solution. The noise level, ffl, must be estimated from the measured data. It is often possible to get a rough estimate by inspecting the data vector visually. In our error estimates we use only the signal-to-noise ratio, ffl=M , and since we know only rough estimates of M and ffl, we solved the problem several times using different values for these parameters, i.e. different levels of regularization. By experimenting, it is often easy to find an appropriate level of regularization. In this particular experiment we know that the solution is monotonically de- creasing, and this information can be used to rule out unphysical solutions, see Figure 5.4. 6 Concluding Remarks We have considered three methods for solving the sideways heat equation, based on approximating the time derivative by a bounded operator (matrix), and then solving the problem in the space variable using a standard ODE solver. For the Fourier and the Meyer wavelet methods, there is a stability theory, which is used for choosing the level of approximation. In the case of Daubechies wavelets the theory is incomplete. From the theory for the Fourier and Meyer wavelet methods one can expect that these methods give results of comparable accuracy. Considering the fast decay in frequency of Daubechies wavelets, one can hope that this method is about as accurate as the other two. Our numerical experience confirms this. The Fourier method can be implemented in such a way that two FFT's are computed in each step of the ODE solver of Runge-Kutta type, and these are computed for a vector of the same length as the measured data vector. On the other hand, the two wavelet methods reduce the problem size, determined by the noise level, and multiplications by small dense (Meyer) or banded (Daubechies) matrices are performed in the ODE solver. In this respect, both wavelet methods, and in particular the Daubechies wavelet method, have an advantage over the Fourier method, if the data vectors are very long 6 . The quarter plane assumption is used mainly because it makes it easier to obtain stability estimates. The numerical methods can be used also for the case when the equation is defined only for a bounded interval in space, provided that also measurements for u x are available. In principle this requires measurements with two thermocouples. In certain symmetric configurations it is sufficient to use only one thermocouples. All three methods considered can easily be applied to problems with variable coefficients, as was seen in Section 5.1. Numerical methods for non-linear equations will be the studied in our future research. Acknowledgement The numerical experiments in Section 5.2 were based on measurements conducted at the Department of Mechanical Engineering, Link-oping university. We are grateful to Ricardo Garcia-Padr'on and Dan Loyd for performing the experiment and for several discussions on real world heat conduction problems. 6 In our implementation we have assumed that also intermediate results, for are of interest. If only the final solution, at needed, then, in principle, the Fourier method can be implemented with only two FFT's altogether, which could make it faster than the wavelet methods. Appendix A Periodization As we remarked in Section 4.3, the algorithms for computing the DMT for Meyer wavelets are based on the Discrete Fourier Transform (DFT) and use the FFT (Fast Fourier Transform) algorithm. In using the DFT, it is assumed that the sequence to be transformed is periodic (see, e.g. [3]). In our application it is not natural to presuppose that the data vectors are periodic: We have a function u(x; t) (temperature) equal to zero for t ! 0, and we consider it in the unit square [0; 1] \Theta [0; 1]. Further, we assume that there is a temperature change at represented by is diffused through some medium (in the interval 0 - x - 1), and recorded at 1. Thus we cannot assume that the functions that we are interested in, are equal to zero for t - 1. Nevertheless, for numerical reasons, in order to avoid wrap-around effects (cf. [3]) in the computation of the FFT, it is necessary to assume periodicity. Therefore we extended the (noisy) data function g m (t) to the interval 1 - first computing a smoothed version of the last 16 components of the original data vector, using a cubic smoothing spline. Then the last 7 components of the extended data vector were prescribed to be equal to zero, and a cubic spline function was constructed that interpolated 16 equidistant data points from the smooting spline and the 7 zero data points. Finally this interpolating spline was sampled at equidistant points, as many as in the original data vector. Thus the size of the data vector was doubled. The periodization of data vectors is illustrated in Figure 4.1. Therefore, in effect we solved the sideways heat equation for but the computed values for t ? 1 are not used. Of course, the computed values for t less than but close to 1 are affected by the artificial data for t ? 1. However, these values should not be trusted anyway, see the discussions in [8, 9, 10]. We used the same periodization for all three methods considered in this paper. --R Inverse Heat Conduction. On the representation of operators in bases of compactly supported wavelets. The DFT: An Owner's Manual for the Discrete Fourier Transform. Determining surface temperatures from interior obser- vations Space marching difference schemes in the nonlinear inverse heat conduction problem. Slowly divergent space marching schemes in the inverse heat conduction problem. Ten Lectures on Wavelets. The numerical solution of a non-characteristic Cauchy problem for a parabolic equation Hyperbolic approximations for a Cauchy problem for the heat equation. Numerical solution of the sideways heat equation. Numerical solution of the sideways heat equation by difference approximation in time. Solving the sideways heat equation by a 'method of lines'. A mollified space marching finite differences algorithm for the inverse heat conduction problem with slab symmetry. Time Dependent Problems and Difference Methods. Heat Transfer Wavelet Methods for the Inversion of Certain Homogeneous Linear Operators in the Presence of Noisy Data. Continuous data dependence Multiresolution approximation signal decomposition. Principe d'incertitude Ondelettes, fonctions splines et analyses gradu'ees. The mollification method and the numerical solution of the inverse heat conduction problem by finite differences. A stable space marching finite differences algorithm for the inverse heat conduction problem with no initial filtering procedure. Sideways heat equation and wavelets. Solving the sideways heat equation by a Wavelet-Galerkin method Prentice Hall PTR Spectral analysis of the differential operator in wavelet bases. --TR --CTR Fu Chuli , Qiu Chunyu, Wavelet and error estimation of surface heat flux, Journal of Computational and Applied Mathematics, v.150 n.1, p.143-155, January Chu-Li Fu, Simplified Tikhonov and Fourier regularization methods on a general sideways parabolic equation, Journal of Computational and Applied Mathematics, v.167 n.2, p.449-463, 1 June 2004 Xiang-Tuan Xiong , Chu-Li Fu, Determining surface temperature and heat flux by a wavelet dual least squares method, Journal of Computational and Applied Mathematics, v.201 n.1, p.198-207, April, 2007 Wei Cheng , Chu-Li Fu , Zhi Qian, A modified Tikhonov regularization method for a spherically symmetric three-dimensional inverse heat conduction problem, Mathematics and Computers in Simulation, v.75 n.3-4, p.97-112, July, 2007

cauchy problem;ill-posed;heat conduction;wavelet;inverse problem;fourier analysis

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Accuracy of Decoupled Implicit Integration Formulas.

Dynamical systems can often be decomposed into loosely coupled subsystems. The system of ordinary differential equations (ODEs) modelling such a problem can then be partitioned corresponding to the subsystems, and the loose couplings can be exploited by special integration methods to solve the problem using a parallel computer or just solve the problem more efficiently than by standard methods.This paper presents accuracy analysis of methods for the numerical integration of stiff partitioned systems of ODEs. The discretization formulas are based on the implicit Euler formula and the second order implicit backward differentiation formula (BDF2). Each subsystem of the partitioned problem is discretized independently, and the couplings to the other subsystems are based on solution values from previous time steps. Applied this way, the discretization formulas are called decoupled.The stability properties of the decoupled implicit Euler formula are well understood. This paper presents error bounds and asymptotic error expansions to be used in controlling step size, relaxation between subsystems and the validity of the partitioning. The decoupled BDF2 formula is analyzed within the same framework.Finally, the analysis is used in the design of a decoupled numerical integration algorithm with variable step size to control the local error and adaptive selection of partitionings. Two versions of the algorithm with decoupled implicit Euler and BDF2, respectively, are used in examples where a realistic problem is solved. The examples compare the results from the decoupled implicit Euler and BDF2 formulas, and compare them with results from the corresponding classical formulas.

Introduction . The numerical solution of stiff systems of ordinary differential equations (ODEs) requires implicit discretization formulas. The implicit algebraic problems are usually solved by a Newton-type iteration method which involves the solution of systems of linear equations that often turn out to be sparse. Attempts to parallelize a standard algorithm, as the one just outlined, often lead to disappointing efficiency because the parallel algorithm is too fine-grained or has too large a sequential fraction. The waveform relaxation method [1], although not developed with parallel computation in mind, leads to efficient parallel algorithms for the class of problems where it works well. Multirate integration can be considered an integral feature of the wave-form relaxation method. The relaxation part of the method is a potential source of computational inefficiency. One relaxation iteration is fairly expensive and convergence may be slow. The decoupled integration methods in this paper were developed as a response to the problems encountered in parallelizing standard integration methods and the problems with waveform relaxation. The decoupled integration methods, like the waveform relaxation method, exploit a partitioning of a system of ODEs into loosely coupled subsystems. The decoupled integration methods employ only one and occasionally two relaxation iterations. Multirate integration is not quite as natural as for the waveform relaxation method, although it is still possible, and the parallel implementations of decoupled integrations methods will be finer grained than parallel waveform relaxation methods. The decoupled integration methods may be more Department of Computer Science, University of Copenhagen, Universitetsparken 1, DK-2100 Copenhagen, Denmark, e-mail: stig@diku.dk efficient on a sequential computer than standard integration methods, just like the waveform relaxation method. A previous paper introduced the decoupled implicit Euler method [2]. The existence of a global error expansion was proved under very general choice of step size, thus permitting the use of Richardson extrapolation. A sufficient condition for stability of the discretization was given in [3]. This condition is called monotonic max-norm stability, and it guarantees contractivity. Partitioned systems of ODEs are in qualitative terms characterized as monotonically max-norm stable if each subsystem is stable and if the couplings from one subsystem to the others are weak. This paper is organized as follows. Section 2, "Partitioned systems of ODEs and decoupled discretization formulas", gives preliminaries and definitions, including the definition of monotonic max-norm stability and the presentation of the decoupled implicit Euler and the decoupled implicit second order backward differentiation formula (BDF2). Section 3, "Error bounds", gives error bounds for the classical and the decoupled implicit Euler formulas. The bounds are closely tied to the monotonic max-norm stability condition which applies only to the Euler formulas, and the bounds are not readily generalized to the BDF2 formula. Section 4, "Asymptotic error formulas and error estimation", includes four subsections treating explicit formulas, classical implicit formulas, decoupled implicit Euler, and finally decoupled BDF2 formulas. The explicit formulas are used in error estimation and for prediction in the decoupled formulas. The asymptotic errors of the classical formulas are the smallest achievable for the decoupled formulas and therefore of interest. The asymptotic errors of the decoupled formulas are given for various modes of operation, and error estimation techniques are presented. Section 5, "Integration algorithm", first presents general principles for decoupled integration algorithms derived from the previous analysis. Then the details of an implementation of the decoupled implicit Euler formula is given, followed by the minor modifications required for replacing the Euler formula with the BDF2 formula. The implementation employs variable step size to control the local error and adaptive selection of partitionings among two predetermined alternatives. These integration algorithms are used in section 6, "Examples: Chemical reaction kinetics", where a real problem is solved using both decoupled and classical versions of implicit Euler and BDF2. The examples show excellent performance of the decoupled formulas. 2. Partitioned systems of ODEs and decoupled discretization formulas. Define a system of ODEs, continuous in Y . Stable systems of differential equations are considered stiff when the step size of the discretization by an explicit integration method is limited by stability of the discretization and not by accuracy. Efficient numerical integration of stiff systems therefore requires implicit integration methods. Let the original problem (2.1) be partitioned as y 0y 0y 0 y 1;0 y 2;0 necessary, the partitioning of Y will be stated explicitly as in f r (t; y 2.1. Monotonic max-norm stability. The following stability condition introduced in [3] plays a crucial role for the stability of decoupled implicit integration methods. stability. The partitioned system (2.2) is said to be monotonically max-norm stable if there exist norms k \Delta k r and functions a rj (t; U; V ) such that a rj (t; U; V )ku and the following condition holds for the logarithmic max-norm -1 (\Delta) of the q \Theta q matrix (a rj The condition (2.4) states that the matrix (a rj ) should be diagonally dominant with non-positive diagonal elements. For a linear problem where A the (a rj ) matrix can be chosen as a Theorem 3 in [3] includes further results about monotonic max-norm stability, including possible choices of (a rj ). Monotonic max-norm stability admits arbitrarily stiff problems. 2.2. Decoupled implicit Euler: Stability, convergence. The decoupled implicit Euler method is defined by the following discretization of the subsystems by the implicit Euler formula [2]: y q;n ); and the variables ~ y i;n are convex combinations of values in fy i;k r. The convex combinations ~ y i;n will, in general, depend on subsystem index r, but in order to simplify notation this dependency will not be specified explicitly. The method is called "decoupled" because the algebraic system resulting from the discretization of (2.1) by Euler's implicit formula is decoupled into a number of independent algebraic problems. The decoupled implicit Euler formula can be used as the basis of parallel methods where (2.5) is solved independently and in parallel for different r-values. The method can be used in multirate mode with h r;n 6= h j;n for r 6= j, and the multirate formulation can be used in a parallel waveform relaxation method [1]. The sequential solution of (2.5) for on a single processor will, in general, be computationally cheaper than solving the complete system Therefore the decoupled implicit Euler method may also be an attractive alternative to the classical Euler formula on a sequential processor even when there is no multirate opportunity. Theorems 4 and 5 in [3] assure the stability of the discretization by (2.5) of a monotonically max-norm stable problem and the convergence of waveform relaxation. The stability condition poses no restrictions on the choice of the step sizes h r;n . The convexity of ~ y r;n is necessary in the stability theorem of the decoupled implicit Euler method. A convex combination would typically be a zero-order interpolation, ~ multirate organizations, where a linear interpolation is an option. The multirate aspect is discussed extensively in [2] and [3]. Most of the results and techniques in this paper can readily be extended to multirate algorithms. How- ever, presenting results for multirate algorithms would add substantially to notational complexity, and furthermore, the algorithms and the example presented in sections 5 and 6, respectively, do not exploit multirate formulation. Therefore, the multirate aspect will not be explored further. 2.3. Organization: Jacobi, Gauss-Seidel, relaxation. The definition of the decoupled implicit Euler formula in (2.5) suggests a Jacobi-type organization of the computation which is well suited for parallel computation. Define the function G J (t; Y; ~ Y ) from the partitioning of F given in (2.2), J;r (t; Y; ~ where y q ). Assuming the same step size hn for all subsystems, the decoupled implicit Euler formula can now be expressed in the compact form, Yn The definition of G J given above corresponds to the Jacobi organization of the computation. A similar GG\GammaS can be defined for the Gauss-Seidel organization, G\GammaS;r (t; Y; ~ y q The decoupled implicit Euler formula based on the Gauss-Seidel organization in general, be more accurate than (2.7). The Gauss-Seidel organization, although inherently sequential, may still be used in parallel if the partitioning (2.2) admits a red-black reordering or a similar reordering based on more colors (see, e.g. [4]). In the following the generic function G, meaning either G J or GG\GammaS , will be used. The decoupled implicit Euler formula can be used with relaxation iterations, Y [m+1] Yn . The computational cost of each iteration of (2.9) is approximately the same as the cost of computing Yn from (2.7). If the relaxation is carried to convergence, the resulting discretization is the classical implicit Euler discretization. When the numerical solution at t n has been computed and accepted, it is usually used for computing the next step, and it is denoted by Yn , where Yn := Y [m] n . In the relaxation (2.9), each subsystem r is solved in each sweep, usually by a Newton-type iteration carried to convergence. A relaxation iteration based on New- ton's method with Jacobi or Gauss-Seidel organization as shown below may converge just as fast and with substantially less computation per iteration. @y r;n y [m] q;n for y r;n . Again the potential for parallelisation is less with Gauss-Seidel organization than with Jacobi organization. 2.4. Decoupled BDF2. The BDF2 can also be used in a decoupled mode, where Stability results for this formula are only known for linear problems and constant step size [5]. The decoupled BDF2 can, of course, be relaxed just as the Euler formula in (2.9) or (2.10): Y [m+1] 3. Error bounds. 3.1. Decoupled implicit Euler. The error of the decoupled implicit Euler formula can be bounded as follows. Consider the local truncation error for subsystem r, and the decoupled implicit Euler formula using Jacobi organization, Y r where ~ Y r y q;n ). Assume that the monotonic max-norm stability condition (2.3), (2.4) is fulfilled with Y (t n ); ~ Y r , and introduce the simplified notation a n Y r Then subtraction leads to Y r a n A bound for the local error y r of the decoupled implicit Euler formula is then (cf. Lemma 2.2, Section III.2. in [6]). rr a n since rr by (2.4) and ' Y r 3.2. Classical implicit Euler. A similar local error bound can be established for the classical implicit Euler formula expressed as follows in a notation analogous to (2.5): Using the monotone max-norm stability condition, we obtain r where and m1 (t) is defined as A bound for the local error y r of the classical implicit Euler formula analogous to (3.2) is then are valid for all values of step sizes but may be most interesting and useful for values where L(t n The main difference between (3.2) and (3.3) is the last term in (3.2). The local truncation error norm L(t n ) is O(h 2 ). The order of the last term in (3.2) is O(h 2 since If the off-diagonal elements of (a ij ) are very small, the last term of (3.1) may be negligible, and the local error bound of the decoupled implicit Euler formula is almost equal to the local error bound of the classical implicit Euler formula. 4. Asymptotic error formulas and error estimation. 4.1. Explicit formulas. The numerical solution of stiff systems of ODEs requires implicit discretization formulas. A numerical integration algorithm typically also includes an explicit formula to be used in error estimation and for computing an initial value for the iterative (Newton-type) method used in solving the implicit discretization problem. The decoupled implicit Euler and BDF2 formulas furthermore require the computation of ~ Yn , where the use of an explicit formula is an option. The explicit Euler formula Y e is an obvious choice in connection with the implicit Euler formula, as with the implicit decoupled Euler formula. However, explicit formulas including F (t; Y ) should generally be avoided in connection with stiff problems when hk@F=@Y k AE 1. The bound kY e may be approached if Yn\Gamma1 is off the smooth solution, which is the case if solved approximately for Yn\Gamma1 . Therefore polynomial interpolation formulas are preferred as predictors [7]. The linear interpolation formula, where has the local error expansion for The second order polynomial predictor formula is Y p2 where The local error expansion for this formula is assuming that 4.2. Classical implicit Euler and BDF2. With the classical implicit Euler the local error Y can be expressed by for where all partial derivatives above and in the rest of section 4 are computed at Y (t n ). The e-terms are obtained by substituting the expansion for Yn (4.4) into the Euler formula (4.3), Taylor expanding Y (t at the point finally identifying e-terms of equal power of hn . The principal local error term h 2 can be estimated as is the divided difference of the values Yn\Gamma2 ; The classical BDF2 formula (cf. (2.11)) has the local error expansion assuming that The principal local error term of BDF2 can be estimated from The error estimates (4.5) and (4.7), based on divided differences, are asymptotically correct [8] if the step size varies according to and OE(t) is sufficiently smooth. This is essentially the step size variation attempted by the algorithms in section 5.2. 4.3. Decoupled implicit Euler. The local error of the decoupled implicit Euler formula (2.7) depends on the definition of ~ Yn and the number of relaxation iterations being performed (cf. (2.9)). Two different modes corresponding to different choices of ~ Yn are considered for different number of relaxation iterations. If the partitioned system (2.2) is monotonically max-norm stable, then mode 1 discretizations are stable while the mode 2 discretizations cannot be guaranteed to be stable. 4.3.1. Mode 1, ~ . The local error is expressed as follows: The e [m] r -terms of (4.8) are found analogously to the e r -terms of (4.4) from (2.7), Y [1] e [1] @ ~ Y e [1] @ ~ Y @ ~ The relation (@G=@Y exploited in the expression for e [1] 3 . Element i of the vector (@ 2 G=@ ~ evaluated as Y 0 The principal local error term h 2 can only be estimated using Y p (cf. )k. This inequality may be fulfilled when the subsystems are loosely coupled, but it is by no means guaranteed by the monotonic max-norm condition. Error estimation based on Y p should only be used if the quality of this estimate is somehow monitored continually. The e [2] r -terms of (4.8) are found from Y [2] like the analogous terms above: e [2] e [2] @ ~ Y The relation (@G=@Y exploited in the expression for e [2] 3 . For of the classical implicit Euler formula. Therefore the principal local error term h 2 2 can be estimated using (4.5) with Y [m] substituted for Yn . The e [m] 1 and e [m] 2 terms are unchanged for further relaxation iterations, m - 3, and e [m] which is identical to e 3 for the classical implicit Euler formula. 4.3.2. Mode 2, ~ . The local error is expressed as follows: Y [m] e [m] e [m] e [m] The - e [1] -terms for - Y [1] Y [1] are identical to e 1 and e 2 (classical implicit Euler), respectively, while e [1] @ ~ Y The computational cost of Y p n is in general much less than the cost of Y [1] n in mode 1, and although they lead to different e 3 -expressions, - e [1] 3 , one value will not be smaller than the other in general. However, the use of ~ n instead of ~ may compromise the stability of the decoupled implicit Euler formula; cf. section 2.2. As in mode 1, - e [m] are unchanged by further relaxation iterations, m - 2 and - e [m] 2. The development of the error terms e [m] 3 and - e [m] 3 is given to illustrate the influence of increasingly accurate values of Y [m] . If the decoupling of the original problem into subsystems is efficient, mode 1 or 2 of the decoupled implicit Euler formula gives results very close to those of the classical implicit Euler formula and 2: If the decoupling is poor, the differences in the higher order e [m] r - or - e [m] r -terms will be significant and kY [m] k. The error expansions do not include multirate integration. The derivation of error expansions analogous to the above covering multirate integration could have been done using the basic definition (2.5), but it would be notationally complicated, and the results are essentially the same. 4.4. Decoupled BDF2. The decoupled BDF2 (2.11) admits more "natural" choices for ~ Yn than the decoupled implicit Euler formula. Three modes using increasingly accurate ~ Yn are presented. Mode 1 is expected to possess the best stability properties among the three different modes although few theoretical results are available [5]. Mode 3 with its second order predictor value for ~ Yn is expected to be the weakest in terms of stability properties while mode 2 is somewhere in between. 4.4.1. Mode 1, ~ . The errors of the decoupled BDF2 are considered for Y [1] and subsequent relaxation iterations (2.12). The local error is expressed as follows for constant step size, The e [m] r -terms of (4.11) are found analogously to the e r -terms of (4.4): e [1] @ ~ Y e [1] @ ~ Y @ ~ Y @ ~ The full order of accuracy of the BDF2 is not reached so another relaxation iteration is performed: Y [2] The error terms are now e [2] e [2] @ ~ Y If kY (3) (t n )k AE k2(@G=@ ~ then the principal local error term can be estimated using formula (4.7). The inequality may be fulfilled when the subsystems are loosely coupled, but it is not guaranteed by the monotonic max-norm stability condition. Since the error resulting from (4.10) is O(h 2 ), this step might be replaced by the decoupled implicit Euler formula mode 1, the result of which is denoted by Y e1[1] n in this subsection. The BDF2 relaxation (4.12) is then replaced by Y [2] Y [2] The local error expansion for constant step size is Y [2] 9 h 3 @ ~ Y @ ~ Y The decoupled Euler-BDF2 combination is expected to be the O(h 3 )-error de-coupled formula with the best stability properties. This expectation is based on the fact that the decoupled implicit Euler formula mode 1 is stable when the partitioned system (2.2) is monotonically max-norm stable. Such a result does not exist for the decoupled BDF2 formula (4.10). Yet another relaxation iteration from Y [2] n or - Y [2] Y [3] leads to a local error expansion with the same principal local error term as the classical BDF2 (4.6), including the case of variable step size. The principal local error term can thus be estimated using formula (4.7). The computational cost of using mode 1 is rather high, so therefore the following modes are of practical interest. 4.4.2. Mode 2, ~ . Another possible choice for ~ Yn is ~ computed from (4.1). The corresponding local error expansion for constant step size is Y [1] 9 h 3 @ ~ Y assuming that The principal local error term can be estimated using formula (4.7) when the subsystems are loosely coupled. Another relaxation iteration would lead to the same principal local error as for the classical BDF2 so that the principal local error term can be estimated using (4.7). 4.4.3. Mode 3, ~ . Using this mode we obtain the same principal local error term as for the classical BDF2, and therefore we also have the same possibility of estimating the error using formula (4.7). The use of ~ n with the decoupled BDF2 may give rise to some concern about the stability of the discretization. 4.4.4. Summary of decoupled BDF2 formulas. The following table summarises the local error results of the decoupled BDF2 formula. Any combination of mode and number of relaxations having the error O(h 3 have the principal local error C 3 h 3 after one or more additional relaxations. ~ Yn Relax'ns 5. Integration algorithm. 5.1. General principles. The previous sections have presented the decoupled (section 2.2) and various iteration techniques for approaching the classical implicit Euler formula (section 2.3). Asymptotic local error expansions have been given for different modes of employment and corresponding local error estimation techniques (section 4.3). Finally, similar results are presented for the decoupled BDF2 (sections 2.4 and 4.4). All of these components can be used to construct numerous integration algorithms where the design decisions may be guided by the properties of the problem to be solved. The main objective of an integration algorithm is to solve a problem to a specified accuracy using as few arithmetic operations as possible. The following discussion only deals with the decoupled implicit Euler formula since the stability results and error bound only apply to this formula. The BDF2 formula is expected to have analogous properties, and the implementation of the decoupled BDF2 formula is very similar to the implementation of the decoupled Euler formula. 5.1.1. Mode. The local error bound (3.1) shows how an accurate value of ~ Yn reduces the influence of the partitioning on the local error. According to section 4.3, mode 2 is to be preferred over mode 1 because of the accuracy of ~ which can be computed at little additional cost. Although kY p 0, the reverse may be true for larger values of hn . Mode 1 should therefore be preferred if An alternative approach for improving the accuracy of ~ Yn is relaxation (2.9). After one relaxation iteration, ~ Yn can be considered having the value ~ Relaxation does not increase the mode, and it is attractive in this respect. However, relaxation is computationally expensive and should only be used when it is strictly necessary. 5.1.2. Partitioning error. The error due to the partitioning is described by the matrix (a rj ). An aggressive partitioning with few small subsystems and otherwise scalar equations may lead to an (a rj ) matrix with relatively large off-diagonal elements, and it may not be diagonally dominant (2.4). According to (3.1), the error term including kY (t n may therefore contribute significantly. A conservative partitioning will typically have some larger subsystems to accommodate strong couplings and to assure numerically small off-diagonal elements in (a rj ). The error bound (3.1) clearly shows how this may lead to a decoupled Euler formula with essentially the same error properties as the classical formula. The Gauss-Seidel organization (2.8) takes advantage of a non-symmetric structure of (a rj ). Assume that the partitioned system (2.2) is reordered symmetrically in equation number and variable number to make (a rj ) as close to lower triangular as possible, k(a rj ) r?j k AE k(a rj ) r!j k. Then (3.1) is modified as follows, j!r a n j?r a n and the larger lower triangular a n rj -values are multiplied with the smaller errors while the smaller upper triangular a n rj -values are multiplied with the larger errors. 5.1.3. Stability. Stability is assured by the matrix (a rj ) being diagonally dominant (2.4) and by ~ Yn being a convex combination of previous solution values. In mode 1, where ~ the latter condition is fulfilled. The diagonal dominance condition may be fulfilled by a sufficiently conservative partitioning. However, the monotonic max-norm stability condition in section 2.1 is a sufficient condition but not a necessary condition. Therefore, mode 2 may be used without encountering stability problems and also used when the diagonal dominance condition (2.4) is not fulfilled. A more conservative partitioning where (2.4) is fulfilled or closer to being fulfilled will not only improve stability but most likely also accuracy; cf. the previous section on partitioning error. However, a conservative partitioning leads to a more computationally expensive discretization than a more aggressive partitioning. Relaxation iterations in mode 1 do not compromise stability, and furthermore, the monotonic max-norm stability guarantees convergence of the process. 5.2. Implementation details. The algorithm will use the decoupled implicit Euler formula or BDF2 with variable step size and choose between two different partitionings: an aggressive partitioning and a conservative partitioning. The aggressive partitioning uses the smallest subsystems possible in order to computational cost. The conservative partitioning uses somewhat larger subsystems in order to maintain accuracy during transient solution phases. The decoupled implicit Euler formula is used in mode 2 ( ~ relaxation iteration (Y n := - Y [1] used with two relaxation iterations (Y n := Y [2] The quality of the partitioning is monitored using (4.9). The classical Euler solution should not be computed because of the incurred cost. In mode 1, Yn in the partitioning criterion above is therefore replaced by Y [2] for some oe ! 1. In mode 2, Y [1] are replaced by - Y [1] Y [2] mode 1 is used with two relaxation iterations, Y [2] n is always available, and the cost involved in monitoring the partitioning is negligible. Mode 2 is used with just one relaxation iteration. Therefore the cost of a step where the partitioning is monitored is double the cost of an ordinary mode 2 step since - Y [2] n is required. When - Y [2] n is available in mode 2, it seems obvious to return Y [2] n as the result of a step, Yn := - Y [2] n , since - Y [2] n supposedly is more accurate than Y [1] n . However, Yn := - Y [2] n is only returned occasionally, with Yn := - Y [1] n being the common result, and in the following example this generates oscillations in the local error estimate while failing to improve accuracy. Therefore, Yn := - Y [1] n is always returned from mode 2. The integration algorithm can now be outlined as follows. Initialisation, step 1: ffl choose conservative partitioning, N ffl no error estimation Step n: n from (4.1) and compute - Y [1] n from the decoupled backward Euler formula using mode 2 (Y [2] computed by mode 1 is only used when the predicted solution Y p n is inaccurate, as explained above) monitor then Monitor partitioning ffl estimate the principal local error term using (4.5), " est ffl new step size, " tol =" est ) (aggressive partitioning) then The step size formula averages 1 and " tol =" est to reduce the tendency of oscillations in step size selection. If the step size is decreasing, the solution may be entering a transient phase which requires the conservative partitioning. If the aggressive partitioning is being used, the number of steps until the next Monitor partitioning is reduced by decreasing N monitor . The partitioning is chosen using the following conditions. Monitor partitioning ffl in mode 2, relax the decoupled implicit Euler formula an extra time to compute Y [2] n . In mode 1, use Y [1] n in the following test Y [1] Y [2] Y [1] choose aggressive partitioning else choose conservative partitioning if shift from conservative to aggressive partitioning then A switch from aggressive to conservative partitioning reflects that the aggressive partitioning is not satisfactory. A switch in the opposite direction is tentative since it is only known that the conservative partitioning is satisfactory so the aggressive partitioning may also be satisfactory. Therefore the first step after a switch in this direction is monitored. The algorithm was developed and presented for the decoupled implicit Euler for- mula. The modifications, beyond the obvious, to accommodate the decoupled BDF2 are small. The Euler formula is replaced by the decoupled BDF2 (2.11) in mode 3, and the first order predictor is replaced by the second order predictor (4.2) except the first integration step which is still taken by the Euler formula and the second step which uses decoupled BDF2, mode 2. The principal local error is estimated using (4.7), and a normalization by fi of the local error estimate is introduced [6, Section III.2.], " est =fik. The step size selection scheme is modified slightly so that averaging is only used during increasing step sizes: est The step size averaging is useful for increasing step sizes to reduce the risk of instability while it prevents a very rapid reduction in step size at the transients. 6. Examples - Chemical reaction kinetics. The example problem is the mathematical model of the chemical reactions included in a three-dimensional trans- port-chemistry model of air pollution. The air pollution model is a system of partial differential equations where each equation models transport, deposition, emission, and chemical reactions of a pollutant. By the use of operator splitting, a number of sub-models are obtained including the following system nonlinear ODEs: The nonlinearities are mainly products, i.e. P i and L i are typically sums of terms of the form, c ilm (t)Y l Ym and d il (t)Y l , respectively, for l; m 6= i. The chemistry model is replicated for each node of the spatial discretization of the transport part. The numerical solution of a system of 32 ODEs is not very challenging as such, but the replication results in hundreds of thousands or millions of equations, and a very efficient numerical solution is crucial. The problem and a selection of solution techniques employed so far are described in [9] and [10]. The system of ODEs is very stiff with the real part of the eigenvalues of the Jacobian along the solution ranging from 0 to \Gamma8 step sizes in the range 100 to 1000. Therefore implicit integration schemes are required, and the resulting nonlinear algebraic problem is the main computational task involved in advancing the numerical solution one time step. A method based on partitioning the system called the Euler Backward Iterative method is described in [9]. It can be characterized as a discretization by the implicit Euler formula with block Gauss-Seidel iteration for the solution of the algebraic equations of the discretization. An ideal partitioning would involve subsystems of size one, i.e., s (cf. (2.2)). For this chemical reaction kinetics problem, it would be particularly advantageous since Y i is only included in L i (t; Y ) in very few equations and never in P i (t; Y ) (by definition). Therefore L i (t; Y )Y i is in general linear in Y i , and the solution of a scalar equation by an implicit integration formula can be performed without iterations. A partitioning into all scalar equations is not viable for this problem. However, the paper [9] identifies a total of 12 equations which should be solved in blocks of 4, 4, 2 and 2 equations. With the numbering of the chemical species used in Table B.1 in [9], the block Gauss-Seidel iteration proceeds as follows, where the parentheses denote the blocks of equations, f(1, 2, 3, 12), (4, 5, 19, 21), 6, 7, (8, 9), 10, 14, (15, remaining scalar equationsg. The partitioning in [9] specified above is used as the basis of the partitioning used for the results in [11]. In [9] the Euler Backward Iterative formula is relaxed until convergence to obtain the equivalent of the classical implicit Euler formula. The results in [11] are obtained from the decoupled implicit Euler formula mode 2 with one relaxation. Timing results in [11] show good efficiency for this approach. The aggressive partitioning and ordering used in this example for both decoupled implicit Euler and BDF2 is described by f12, (4, 5), 20 scalar equations, (1, 2, 15, 16), remaining scalar equationsg. The subsystems of two and four equations are enclosed in parentheses, and the rest of the equations are being treated as scalar equations. The conservative partitioning and ordering is specified by f12, (1, 2, 15, 16, 4, 5, remaining scalar equationsg. Except for the block of 10 equations, the partitioning is into scalar equations. A partitioning is considered more aggressive than another one if it has fewer equations appearing in blocks and/or the blocks are smaller. Concerning the two partitionings presented here, it is obvious which partitioning is the more aggressive since f(4, 5), (1, 2, 15, 16)g ae f(1, 2, 15, 16, 4, 5, 8, 10, 29, 30)g. The parentheses are only retained to facilitate reference to the partitioning and ordering specifications. The aggressive partitioning is clearly more aggressive than the partitioning presented in [9] since f(4, 5), (1, 2, 15, 16)g ae f(1, 2, 3, 12), (4, 5, 19, 21), (8, 9), (15, 16)g, while the relation to the conservative partitioning is not obvious since one has the larger block while the other has more equations appearing in smaller blocks. The partitioning used in [9] is presumably based on knowledge of the chemical re- actions, while the partitionings used here are obtained with a semiautomatic method described in paper [12]. None of these partitionings are monotonically max-norm stable although they come close, but despite this fact, no instability problems are en- countered. This should not be too surprising since the monotonic max-norm stability is a sufficient but not a necessary condition for the stability of the decoupled implicit Euler formula. The example used in this paper differs slightly from the example in [9], but the differences appear in the equations being treated as scalar in all the considered partitionings 6.1. Decoupled implicit Euler. Figure 6.1 shows the errors obtained using the classical implicit Euler formula (solid line) and the decoupled implicit Euler formula (dash-dot line) implemented as described in section 5. The two different partitionings mentioned above have been used adaptively. The step size is chosen by the decoupled Global integration error Time Fig. 6.1. Integration error for the classical implicit Euler (solid line) and decoupled implicit Euler (dash-dot line). implicit Euler algorithm to obtain a local error estimate of 10 \Gamma3 or to a minimum step size of 90, and the classical implicit Euler formula is applied with the same step size selection. The discrepancy between the errors is seen to be insignificant. A reference solution is computed using a variable step size variable order (max- imum implementation of the backward differentiation formulas [13] with a bound on the relative local error estimate of 10 \Gamma6 . The errors presented in the figures are the maximum relative deviations from the reference solution measured componentwise (the values of the components vary widely in magnitude). The time axis is in seconds, and the initial time corresponds to 6 a.m. The model includes the influence of the sun on some of the chemical reactions, and this leads to very distinct transients in the solution at sunrise and sunset. The minimum integration time step of 90 seconds is too large a step to integrate the transients accurately, and large spikes in the global integration error are seen around 7 p.m., 5 a.m. (t=105,000) next day, and 7 p.m. (t=155,000). The transient behaviour at sunrise and sunset can to some extent be considered a modelling artifact. Since the large local contribution to the global error does not influence the global error at large, it is essentially ignored by introducing the minimum time step. The observed behaviour of the global error is not uncommon for stiff systems of ODEs. Figure 6.2 shows the estimated principal local error. The step size is adjusted to keep the estimate at 10 \Gamma3 , and the algorithm is quite successful except at the transients where the minimum step size of 90 is used. Figure 6.3 shows the resulting step size selection (upper graph) and the selection of partitioning (lower graph). The low value indicates the aggressive partitioning, and the high value indicates the conservative partitioning. The values on the ordinate axis only pertain to the step size. The total number of steps is 737, and only 182 (25%) steps are using the conservative partitioning. e Estimated local error Fig. 6.2. Estimated principal local error for decoupled implicit Euler. Time Step size and partitioning Fig. 6.3. Integration step size selected by the decoupled implicit Euler formula (upper graph) and selection of partitioning (lower graph). During integration using the conservative partitioning, single steps that use the aggressive partitioning can be observed. At the first step after a switch to the aggressive partitioning, the quality is monitored, and if it is not satisfactory, the Monitor partitioning algorithm immediately returns to the conservative partitioning. This example demonstrates the application of an implementation of the decoupled Euler formula in solving a non-trivial practical problem. The computational cost is substantially less than that for the classical implicit Euler formula, and there is no trace of instability in the solution computed by the decoupled implementation. e Global integration error Fig. 6.4. Integration error for the classical BDF2 (solid line) and decoupled BDF2 (dash-dot line). Time Step size and partitioning Fig. 6.5. Integration step size selected by the decoupled BDF2 (upper graph) and selection of partitioning (lower graph). 6.2. Decoupled BDF2. The numerical integration of the previous section is repeated using the decoupled BDF2. The step size is controlled to keep the estimated local normalized error " est at 10 \Gamma3 as before. The resulting global integration error is shown in Figure 6.4 (dash-dot line) together with the corresponding error for the classical BDF2 using the same step size selection (solid curve). Some deviation is noticed, but the error of the decoupled BDF2 is comparable to the error of the classical BDF2 formula. Comparing with Figure 6.1, it is seen that the global error of the decoupled BDF2 formula is significantly smaller than the global error of the decoupled implicit Euler formula. The difference originates mainly from the transients, including the initial transient, where the minimum stepsize of 90 seconds is used. Finally, Figure 6.5 shows step size and partitioning selection similar to Figure 6.3. The maximum step size of the decoupled BDF2 is greater than 1700 which is three times the maximum step size of the Euler formula. The necessary number of integration steps is 311 which is 42% of the number of steps needed by the Euler formula. The conservative partitioning is only used for 73 steps out of 311 (23%). There is a substantial pay-off to using the decoupled BDF2 instead of the decoupled Euler formula, since the amount of work per step is essentially the same for the two decoupled formulas. The performance of the decoupled BDF2 algorithm in mode 3 is very convincing, and there is no trace of instability, although the use of the second order polynomial predictor for computing ~ Yn is somewhat risky from a stability point of view. Acknowledgements . The comments by the editor and the anonymous referees have helped choosing asymptotically correct error estimates and improving the presentation. --R The waveform relaxation method for time-domain analysis of large scale integrated circuits Methods for parallel integration of stiff systems of ODEs Stability of backward Euler multirate methods and convergence of waveform relaxation A connection between the convergence properties of waveform relaxation and the A-stability of multirate integration methods Solving Ordinary Differential Equations I The control of order and steplength for backward differentiation methods Estimation of errors and derivatives in ordinary differential equations A photochemical kinetics mechanism for urban and regional computer modelling Exploiting the natural partitioning in the numerical solution of ODE systems arising in atmospheric chemistry Partitioning techniques and stability of decoupled integration formulas for ODEs INTGR for the Integration of Stiff Systems of Ordinary Differential Equations --TR

parallel numerical integration;absolute stability;multirate formulas;backward differentiation formulas;euler's implicit formula;partitioned systems

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Theory of dependence values.

A new model to evaluate dependencies in data mining problems is presented and discussed. The well-known concept of the association rule is replaced by the new definition of dependence value, which is a single real number uniquely associated with a given itemset. Knowledge of dependence values is sufficient to describe all the dependencies characterizing a given data mining problem. The dependence value of an itemset is the difference between the occurrence probability of the itemset and a corresponding maximum independence estimate. This can be determined as a function of joint probabilities of the subsets of the itemset being considered by maximizing a suitable entropy function. So it is possible to separate in an itemset of cardinaltiy k the dependence inherited from its subsets of cardinality (k 1) and the specific inherent dependence of that itemset. The absolute value of the difference between the probability p(i) of the event i that indicates the prescence of the itemset {a,b,... } and its maximum independence estimate is constant for any combination of values of Q &angl0; a,b,... &angr0; Q. In1p addition, the Boolean function specifying the combination of values for which the dependence is positive is a parity function. So the determination of such combinations is immediate. The model appears to be simple and powerful.

INTRODUCTION A well known problem in data mining is the search for association rules, a powerful and intuitive conceptual tool to represent the phenomena that are recurrent in a data set. A number of interesting solutions of that problem has been proposed in the last five years together with as many powerful algorithms [Agrawal et al. 1993b; Agrawal et al. 1995; Agrawal and Srikant 1994; A.Savasere et al. 1995; Han and Fu 1995; Park et al. 1995; H.Toivonen 1996; Brin et al. 1997; I.Lin and M.Kedem 1998]. They are used in many application fields, such as analysis of basket data of supermarkets, failures in telecommunication networks, medical test results, lexical features of texts, and so on. An association rule is an expression of the form X ) Y, where X and Y are sets of items which are often found together in a given collection of data. For example, the expression f milk, coffee g ) f bread, sugar g might mean that a customer purchasing milk and coffee is likely to also purchase bread and sugar. The validity of an association rule has been based on two measures. The first measure, called support, is the percentage of transactions of the database containing both X and Y. The second one, called confidence, is the probability that, if X is purchased, also Y is purchased. In the case of the previous example, a value of 2% of support and a value of 15% of confidence would mean that 2% of all the customers buy milk, coffee, bread and sugar, and that 15% of the customers that buy milk and coffee also buy bread and sugar. Recently, Silverstein, Brin and Motwani [Silverstein et al. 1998] have presented a critique of the concept of association rule and the related framework support- confidence. They have observed that the association rule model is well-suited to the market basket problem, but that it does not address other data mining problems. In place of association rules and the support-confidence framework, Silverstein, Brin and Motwani propose a statistical approach based on the chi-squared measure and a new model of rules, called dependence rules. This work can be viewed as a continuation of the line of the rules, even if the model and the tools here proposed are rather different and in particular the concept of dependence rules has been replaced by the concept of dependence values. This paper is organized as follows. Section 2 contains a summary of the main results of earlier work, the emphasis being placed on the framework support- confidence, the critique of this model by Silverstein, Brin and Motwani and the concept of dependence rules in opposition to the one of association rules. Section 3 contains the definition of dependence value and other basic definitions of the model here proposed as well as the theorems following from these definitions. These theorems suggest an easy and quick way to determine the dependence values, which is described in Section 5, whereas Section 4 discusses the use of the well known concept of entropy as a tool to evaluate the relevance of a dependence rule. Finally, Section 6 draws the conclusions. 2. ASSOCIATION RULES AND DEPENDENCE RULES As mentioned, this Section contains a summary of earlier work on association rules. For ease of reference, the notation used by Silverstein, Brin and Motwani in their paper will be adopted here. Theory of Dependence Values \Delta 3 2.1 Association Rules be a set of k elements, called items. Basket of items is any subset of I. For example, in the market basket application, I =fmilk, coffee, bread, sugar, tea, . g contains all the items stocked by a supermarket and a basket of items such as f milk, coffee, bread, sugarg is the set of purchases from one register transaction. As a second example, in the document basket application, I is the set of all the dictionary words and each basket is the set of all the words used in a given document. An association rule X ) Y, where X and Y are disjoint subsets of I, was defined by Agrawal, Imielinski and Swami [Agrawal et al. 1993b] as follows. is a subset of at least s% (the support) of all the baskets, and of all the baskets containing all the items of X at least c% (the confidence) contain all the items of Y. The concept of association rules and the related support-confidence framework are very powerful and useful, but they suffer from some limitation, especially when the absence of items is considered. An interesting example proposed by Silverstein, Brin and Motwani is the following. Consider the purchase of tea (t) and coffee (c) in a grocery store and assume the following probabilities: where c and t denote the events "coffee not purchased" and "tea not pur- chased", respectively. According to the preceding definitions, the potential rule tea ) coffee has a support equal to 20% and a confidence equal to 80%, and therefore can be considered as a valid association rule. However, a deeper analysis shows that a customer buying tea is less likely to also buy coffee than a customer not buying tea (80% against more than 90%). We would write tea ) coffee, but, on the contrary, the strongest positive dependence is between the absence of coffee and the presence of tea. 2.2 Dependence Rules Silverstein, Brin and Motwani propose a view of basket data in terms of boolean indicator variables, as follows. be a set of k boolean variables called attributes. A set of baskets bng is a collection of the n k-tuples from fTRUE, FALSEg k which represent a collection of value assignments to the k attributes. Assigning the value TRUE to an attribute variable I j in a basket represents the presence of item i j in the basket. The event a denotes A=TRUE, or equivalently, the presence of the corresponding item a in a basket. The complementary event a denotes A=FALSE, or, the absence of item a from a basket. The probability that item a appears in a random basket will be denoted by P(a)=P(A=TRUE). Likewise, will be the probability that item a is present and item b is absent. Silverstein, Brin and Motwani have proposed the following definitions of independence and dependence of events and variables. Definition 1. Two events x and y are independent if P(x " Definition 2. Two variables A and B are independent if for all possible values hv a , v b 2 fTRUE, FALSEgi. Definition 3. Events, or variables, that are not independent are dependent. Definition 4. Let I be a set of attribute variables. We say that the set I is a dependence rule if I is dependent. The following Theorem 1 is based on the preceding Definitions 1-4. Theorem 1. If a set of variables I is dependent, so is every superset of I. Theorem 1 is important in the dependence rule model, because it makes it possible to restrict the attention to the set of minimally dependent itemsets, where a minimally dependent itemset I is such if it is dependent, but none of its subsets is dependent. Silverstein, Brin and Motwani have proposed using the X 2 test for independence to identify dependence rules. X 2 statistic is upward-closed with respect to the lattice of all possible itemsets, as well as dependence rules. In other terms, if a set I of items is deemed dependent at significance level ff, then all supersets of I are also dependent at the same significance level ff and, therefore, they do not need to be examined for dependence or independence. 3. DEPENDENCE VALUES In this Section the new model based on the concept of dependence values is presented and discussed. A Theorem proved in this section will provide the basic tools to evaluate the dependence rules of a certain itemset. To simplify the presentation, we shall procede from the most simple cases towards the most complex ones, in the order of increasing cardinality of itemsets. In other terms, we shall discuss dependence rules first for pairs of items, then for triplets of items, and finally for m-plets of arbitrary cardinality m. 3.1 Dependence Rules for Pairs of Items Assume we know the occurrence probabilities of all the items: The evaluation of such probabilities is the first problem of data mining, but it is seldom considered because of its simplicity. Generally, the maximum likelihood estimate is adopted according which P(a) is assumed equal to O(a)/n, where O(a) Theory of Dependence Values \Delta 5 is the number of baskets containing a and n is the total number of baskets. However, more complex computations based on Bayes's Theorem, might also be used. In the absence of specific determinations, if we know only formulate the following conjectures: These conjectures are equivalent to the assumption that variables A and B are independent. Assume that the exact determination of P(a,b), evaluated as O(a,b)/n (where O(a,b) is the number of baskets containing both a and b), is different from the conjecture It is easy to prove the following Theorem. Theorem 2 (Unicity of the value for second-order probabilities). If P(A=TRUE) and P(B=TRUE) are known, determination of a single value is sufficient to evaluate all the second-order joint probabilities P(a; b); P(a; b); P(a; b). Proof. The proof is contained in the following simple relationships: Analogously, and The fact that a single datum \Delta contains the whole information pertaining joint probabilities of pair fA, Bg suggests the following Definitions. Definition 5. (Dependence value of a pair) The dependence value of the pair fA,Bg will be defined as the difference Definition 6. (Dependence state of a pair) If the absolute value of exceeds a given threshold th, A and B are said "dependent". If \Delta ? th dependence is defined as "positive"; otherwise, it is defined as "negative". The following notations will be adopted to indicate a positive dependence, a negative one or no dependence, respectively. Figure 1 shows that the difference between the joint probability P(a (with a a or a a and b b) and the corresponding "a-priori" estimate P(a has always the same absolute value but a different sign in the various cells of the Karnaugh's map of variables A and B. To represent this fact, we need another definition and a new Theorem. A A Fig. 1. The joint probabilities of P(a,b) in the cells of Karnaugh's map of fA,Bg. Definition 7. (Dependence function of two variables) The boolean function of variables A and B, whose minterms correspond to the values hv A ,v B i for which P(A=v A will be called the dependence function of variables A and B. Theory of Dependence Values \Delta 7 Theorem 3 (Parity of two variables dependence functions). If the dependence function of variables A and B is: which is the parity function with parity odd (Figure 2). If D 2 the dependence function of variables A and B is: which is the parity function with parity even (Figure 3). Fig. 2. The dependence function of variables A and B if D 2 (A,B)AE 0. Fig. 3. The dependence function of variables A and B if D 2 (A,B) 0. As a simple example, consider the case of purchases of coffee (c) and tea (t), which was discussed in [Silverstein et al. 1998] to show the weakness of the traditional support-confidence framework (Subsection 2.1). If P(c,t)=0.2 then Therefore P(c)\DeltaP(t)=0.225 and which shows that dependence is negative (D 2 (C,T) 0). One might wonder whether the usual notation X adopted in the well known papers on data mining does make still sense and how to indicate a negative dependence like The answer is simple: simply, D 2 (c; t) 0 contain all information on the second-order dependencies. However, one might argue that \Delta is more significant for the events having a lower probability. In the case of coffee and tea, is the lowest probability in the cells of the dependence function; therefore, it is not completely unreasonable to write: C ) T. 3.2 Dependence Rules for Triplets of Items This Subsection is devoted to the generalization of Definitions and Theorems presented in previous Subsection 3.1 to the case of triplets of items. As we shall see, such generalization implies some new problems. Consider the case of a triplet of the boolean variables A, B and C, and assume we know the first- and second-order joint probabilities such as and others. We are interested in determining the third-order joint probabilities of triplets such as P(a,b,c), P(a,b,c), and so on, from which also the third-order conditional probabilities such as directly. The following Theorem shows that the knowledge of a single third-order probability is sufficient to determine all the third-order probabilities. Theorem 4 (Unicity of the value for third-order probabilities). All the third-order joint probabilities can be calculated as functions of first- and second-order joint probabilities and a single datum such as a third-order joint probability. Proof. Assume, for example, we know P(a,b,c). The other joint probabilities can be determined as follows: Theorem 4 may be viewed as an extension of Theorem 2 on the unicity of the value for second-order probabilities shown in previous Subsection 3.1. However, Theorem 2 makes reference to the differences between the determined P(a and the estimated P(a which correspond to the conjecture of independence of a and b. In the case of triplets, the condition of independence is more difficult to be identified. Our proposal is contained in the following considerations. The relationships written in the proof of Theorem 4 can also be formulated in the following form: Theory of Dependence Values \Delta 9 They express the values of all the third-order joint probabilities as functions of the known second-order probabilities P(a,b), and the unknown third-order probability Now consider the entropy This function of the unknown x is the average amount of information needed to know a, b and c. The maximum value of E(x) is reached when a, b and c are at the maximum level of independency compatible with the dependencies imposed by the second-order joint probabilities. This consideration explains the following Definition 8. Definition 8. (Maximum independence estimate for third-order probabilities) If first- and second-order joint probabilities are known but no information is available on the third-order probabilities, the conjecture x of P(a,b,c) maximizing the joint entropy of A, B, C: P(a; b; c) \Delta log P(a; b; c) (where the sum is to be extended to all the combinations of values of a, b and c) will be defined as the maximum independence estimate. Such maximum independence estimate will be denoted with the symbol P(a,b,c)MI . Analogously, for any combination x i of values of a, b, c, we shall define P(a )MI as the value of P(a ) for which E(x ) is maximum. Notice that in virtue of Theorem 4, for any combination of values ha ,b ,c i of a, b, c, P(a ,b ,c )MI can be computed in terms of second-order joint probabilities and P(a,b,c)MI by applying the following relationships The meaning of Definition 8 is rather important for the model presented in this paper. If D 2 (A,B) or D 2 (A,C) or D 2 (B,C) are AE 0 or 0, A, B, C are not in- dependent, but they could own only the dependence inherited from the second-order dependencies or their dependence might be stronger. In the former case, P(a ,b ,c ) is equal to P(a ,b ,c )MI and there is no real third-order depen- dence. In the latter, there is an evidence of a third-order dependence whose value and sign depend on the differences between P(a ,b ,c ) and P(a ,b ,c )MI , as shown in the following analysis. Notice that in the case of the pairs of items the Definition 8 of maximum independence coincides with the more known definitions of independence cited in previous Subsection 3.1. Indeed, in this case, as is shown in Figure 1, the joint entropy of A and B is where x=P(a,b). It is easy to prove that function E has a maximum for By applying the same algorithm, it is easy to prove the analogous results: Unfortunately, in the case of triplets and k-plets the determination of the maximum independence estimates is not so simple. However, as will be seen later, it is not necessary to know all the estimates P(i 1 but one of them is sufficient to determinate all the other ones. Besides, the numerical evaluation of this estimate can be performed very quickly by applying the method which will be described in Section 5. The definition of maximum independence estimate is applied in the following Theorem, which can be viewed as a specification of Theorem 4 on the unicity of value for third-order probabilities and as the natural extension of the Theorem 2 proved in previous Subsection 3.1. Theorem 5. If the first- and second-order joint probabilities and the third-order maximum independence estimate are known, a single number \Delta defined as the difference P(a,b,c) - P(a,b,c)MI is sufficient to specify all the third-order joint probabilities. Proof. Theorem 5 is a direct consequence of Theorem 4 on the unicity of the value. Indeed, from the knowledge of the first- and second-order joint probabilities we can obtain P(a,b,c)MI and from this according to Theorem 4, the knowledge of P(a,b,c) is sufficient to determine all the third-order joint probabilities. Theory of Dependence Values \Delta 11 In virtue of Theorem 5, we can state the following Definitions which are an extension of Definitions 5 and 6. Definition 9. (Dependence value of a triplet) The dependence value of the triplet fA,B,Cg will be defined as the difference Definition 10. (Dependence state of a triplet) If the dependence value of fA,B,Cg exceeds a given threshold th, A, B and C are defined as "connected by a third-order dependence". If \Delta ? th, dependence is defined as otherwise, it is defined as "negative". The following notations will be used to indicate the existence or not of a third-order dependence and its sign. Notice that in the model proposed by Silverstein, Brin and Motwani, the existence of one or more second-order dependencies implies the existence of the third-order dependence, whereas in our model D 2 (A,B), D 2 (A,C), D 2 (B,C) and D 3 (A,B,C) are independent, in the sense that any combination of their values is possible. For example, even if all the three second-order dependencies are positive, D 3 (A,B,C) might be zero or negative. In Subsection 3.3 an example about the purchase of a triplet of items is discussed and the differences with respect to the other models are discussed. The following Definition 11 on the dependence function of three variables and Theorem 6 extend the statements of Definition 7 on the dependence function for pairs and Theorem 3 on the parity function to third-order dependencies. Definition 11. (Dependence function of three variables) The boolean function of variables A, B and C, whose minterms correspond to the values hv A ,v B ,v C i for which P(A=v A "C=v C )MI will be called the dependence function of variables A, B and C. Theorem 6 (Parity of three variables dependence functions). If the dependence function of variables A, B and C is that is the parity function with parity even (Figure 4). If D 3 (A,B,C) 0, the dependence function is that is the parity function with parity odd and the complementary function of the preceding one (Figure 5). Proof. By definition Fig. 4. The dependence function when D 3 (A,B,C) AE 0. Fig. 5. The dependence function when D 3 (A,B,C) 0. From analogous computations the values presented in Figure 6 follow. From these ones, it is immediate to derive the two maps of Figure 4 and 5, when D 3 or D 3 A P(a,b,c)MI P(a,b,c)MI P(a,b,c)MI P(a,b,c)MI A P(a,b,c)MI P(a,b,c)MI P(a,b,c)MI P(a,b,c)MI Fig. 6. The dependence function for the three variables A, B and C. 3.2.1 Justification of the maximum independence definition. The idea of maximum independence introduced in this paper is not intuitively obvious and needs some further justification. First consider the simple case of two variables A and B. In this case, as shown above, the definition of maximum independence coincides with the well known definition of absolute independence, according which A and B are independent if, and only if, P(A=v A ,B=v B any combination of values of A and B. It is well known that the joint entropy of A and B E(A,B)=E(A)+E(BjA)= E(B)+E(AjB) where E(BjA) and E(AjB) are the equivocation of B with respect to A and the equivocation of A with respect to B, respectively. Therefore, the maximum value of E(A,B) is reached when E(BjA) (or E(AjB)) is maximum. When A and B are independent, the amount of information needed to know B, if A is known, Theory of Dependence Values \Delta 13 or to know A, if B is known, is maximum. Notice that in this case E(AjB)=E(A) and E(BjA)=E(B). These equalities will not hold in the case of three variables. Now consider the case of three variables A, B and C. In general, if the probabilities of A, B and C, and the second-order joint probabilities P(A,B), P(B,C) and P(A,C) have been assigned, there is no assignment of the probability P(A,B,C) for which A, and C are independent, that is, P(A=v A ,B=v B ,C=v C for any combination of values of v A and v C However, it makes sense to search the value of P(A,B,C) for which the joint entropy E(A,B,C) is maximum and to define that condition as the one of the maximum level of independence compatible with the dependencies imposed by the second-order joint probabilities. Indeed, Therefore, since E(A,B), E(A,C) and E(B,C) depend only on the values of second-order probabilities, E(A,B,C) reaches its maximum for that assignment of P(A,B,C) for which also E(CjA,B), E(BjA,C) and E(AjB,C) reach their maximum values. In other terms, the maximum independency level corresponds to the condition in which the maximumamount of information is needed to know the value of a variable, the other two being known. However, in general, since A, B and C are not independent, and this is different from the case of pairs of variables for which the concepts of maximum independence and absolute independence coincide. 3.3 The lattice of dependencies Since the knowledge of the dependence value of an itemset of cardinality k, together with the values of the joint probabilities of all its subsets of cardinality k-1, is sufficient to know the probabilites of all the combinations of its values, the lattice of the itemsets can be adopted to describe the whole system of dependencies of a given database. Of course, in such a lattice every node should be labelled with its associated dependence value. Besides, the nodes at the top of the lattice representing the itemsets of cardinality 1 will be labelled with the values of the differences between the probability estimates, P(a)=O(a)/n, so on, and the corresponding starting estimates (typically, and in absence of other estimates, equal to 0.5). By way of example, Figure 7 represents the dependence lattice relative to the sample reported by Silverstein, Brin and Motwani in their paper. The following are the data of purchases of coffee (c), tea (t), and doughnuts (d) and their combinations proposed by those authors: c t d ct cd td ctd Fig. 7. The lattice relative to the purchases of coffee (c), tea (t) and doughnuts (d). P(c,t,d)=0.4 The dependence values of the nodes of the lattice have been calculated as follows. \Delta(c)=P(c)-P(c) MI=O(c)/n-0.5=0.93-0.5=+0.43 \Delta(t)=P(t)-P(t) MI=O(t)/n-0.5=0.21-0.5=-0.29 \Delta(d)=P(d)-P(d) MI=O(d)/n-0.5=0.51-0.5=+0.01 \Delta(c,t)=P(c,t)-P(c,t) MI=O(c,t)/n-P(c)\DeltaP(t)=0.18-0.19=-0.01 \Delta(c,d)=P(c,d)-P(c,d) MI=O(c,d)/n-P(c)\DeltaP(d)=0.48-0.47=+0.01 \Delta(t,d)=P(t,d)-P(t,d) MI=O(t,d)/n-P(t)\DeltaP(d)=0.09-0.1=-0.01 \Delta(c,t,d)=P(c,t,d)-P(c,t,d) MI=O(c,t,d)/n-0.078=0.08-0.078=+0.002 P(c,t,d)MI has been computed maximizing the entropy E(x) with x=P(c,t,d) as suggested in Definition 8 on the maximum independence estimate. Notice that from the value of \Delta(c,t,d) and from Definition 10 on the state of dependencies it follows, for example, that the dependence of itemset fc,t,dg is positive, whereas, by adopting the model proposed by Silverstein, Brin and Mot- wani, the same dependence, evaluated as P(a;b;c) P(a)\DeltaP(b)\DeltaP(c) would be negative. This is due to the fact that in the model by Silverstein, Brin and Motwani, the dependencies which the subset fc,t,dg has inherited by the subsets fc,tg, fc,dg and ft,dg are not distinguished from the specific inherent dependence. The complete dependence table showing the sign of dependence function for all the values of hc,t,di is shown in Figure 8. The dependence lattice can also be viewed as an useful tool to display the results of a data mining investigation on a given database. Of course, it will be convenient Theory of Dependence Values \Delta 15 TD TD TD TD Fig. 8. The sign of dependence function for the example of the purchases of coffee (variable C), tea (variable T) and doughnuts (variable D). to display only the sub-lattice of the nodes having sufficient support and positive or negative dependencies - anyway different from zero in a significant way. Often, the dependence value is not necessary, it being sufficient to introduce the indication of the dependence state in the lattice produced. 3.4 Dependence Rules for k-plets of Items of Arbitrary Cardinality The case of triplets discussed in previous Subsection 3.2 is absolutely general. How- ever, for the sake of completeness, the Definitions and the Theorems presented in Subsection 3.2 will be extended in this Subsection to the more general case of k- plets of arbitrary cardinality. For the sake of brevity, the proofs of the Theorems will be omitted, with the exception of Theorem 7 which needs a specific proof. Consider the case of a k-plet of boolean variables I 1 , I assume we know all the joint probabilities up to the order (k-1): We want to determine the k-th-order joint probabilities like P(i 1 on. The following Theorem shows that the knowledge of a single k-th-order joint probability is sufficient to determine all the k-th-order probabilities. Theorem 7 (Unicity of the value). All the k-th-order joint probabilities can be calculated as functions of the joint probabilities of the orders less than k and a single k-th-order joint probability. Proof. Assume, for example, we know P(i 1 , First, we determine Analogously, we determine all the other joint probabilities related to elementary conditions in which a single literal is complemented: and so on. Then, we compute all the joint probabilities referring to elementary conditions in which two literals appear complemented: and so on. In general, in order to determine all the joint probabilities related to elementary conditions containing m complemented literals, we apply the following relationship in which a 1 6= a P(i a 1 ,i a 2 P(i a 1 ,i a 3 ,i a2 ,i a 3 where at most (m-1) complemented literals appear in the right size. Definition 12. (Maximum independence estimate) If the joint probabilities up to order k-1 are known but no information is available on the joint probabilities of order k, then the conjecture on P(i maximizing the joint entropy of I 1 ,I will be considered as the maximum independence estimate. For any fi the maximum independence estimate will be indicated with the symbol Definition 13. (Dependence value) The difference will be defined as the dependence value of the itemset fi 1 , g. Theorem 8 (Unicity of the value). If the joint probabilities up to the order are known, the knowledge of the dependence value P(i 1 , is sufficient to describe all the k-th order joint probabilities. Definition 14. (Dependence state) If the absolute value of the dependence value exceeds a given threshold th, I 1 , I 2 ,: : :, I k are defined as connected by a dependence of order k. If \Delta ? th, the dependence is defined as positive; otherwise, it is defined as negative. The following notations: will be used to indicate the existence or not of a dependence of order k and its sign. Definition 15. (Dependence function) The boolean function of variables I 1 , I I k , whose minterms correspond to the values hv I will be called the dependence function of variables I 1 , I Theorem 9 (Parity of dependence functions). If D k 0, the dependence function of variables I 1 , I , that is the parity function with even parity. Theory of Dependence Values \Delta 17 the dependence function of variables I 1 , I I 1 , that is the parity function with odd parity. In both cases, and for all the values of I 1 , I I k the difference P(i 1 , has an absolute value equal to \Delta. 4. ENTROPY AND DEPENDENCIES A less intuitive but for some aspects more effective approach to determine dependencies can be based on the concept of entropy. In this Section only a summary of a possible entropy based theory of dependencies is presented, the task being left to the reader of developing such a theory following the scheme of Section 3. First consider the case of the pairs of items. Assume P(a), P(a), P(b), P(b) are known. The entropy of A is the measure of the average information content of the events a j A=TRUE and a j A=FALSE. An analogous meaning can be attributed to Consider now the mutual information a ;b P(a , b )\Delta logP(b ja ). is a measure of the average information content carried by b on the value of A, and viceversa, and therefore, it can be assumed as an indication of independence of A and B. Unfortunately, mutual information I(A;B) is always positive; so it is necessary to verify whether P(a,b) ? P(a)\DeltaP(b) or not, in order to determine the sign of dependence. Therefore, we propose the following definition: Definition 16. (Entropy based second-order dependence) If I(A;B) exceeds a given threshold, we shall state that D 2 (A,B) AE 0 or D 2 (A,B) 0 according to whether not. If I(A;B) does not exceed that threshold, we shall state that D 2 (A,B) 0. The extension of such definition to triplets is not immediate, since the ternary mutual information I(A;B;C) defined in information theory does not own the meaning we need now. Therefore we suggest the following one: Definition 17. (Entropy based third-order dependence) If both E(AjB) - E(AjB,C) and E(AjC) - E(AjB,C) exceed a given threshold, we state that D 3 (A,B,C) AE 0 when P(a,b,c) is larger than all the following estimates: P(a)\DeltaP(b,c) P(b)\DeltaP(a,c) or D 3 (A,B,C) 0 when P(a,b,c) is less than all the three preceding estimates; otherwise, D 3 (A,B,C) 0. Definition 17 might seem asymmetric with respect to variables A, B and C. In order to understand the reasons for which the relationships written in Definition 17 are symmetric, remember that, for example, if E(AjB)-E(AjB,C)?th then also E(CjB)-E(CjA,B)?th. Besides, E(AjB,C) != E(AjB) and E(AjB,C) != E(AjC). When, for example, E(AjB,C), E(AjB,C) is maximum and, therefore, (maximum independence condi- tion). It follows that also E(BjA,C) and E(CjA,B) take their maximumvalues, equal to E(BjA) or E(BjC) and to E(CjA) or E(CjB). An analogous definition can be introduced to evaluate dependencies in k-plets with arbitrary k. Definition 18. (Entropy based general dependence) Let E k MIN the minimum of all the conditional entropies are the subsets of (k-2) variables of the set of (k-1) variables. MIN exceeds a given threshold, larger than the maximum of the estimates we state that D k (A, B, If the same difference E k MIN exceeds the given threshold and smaller than the minimum of the estimates we state that D k (A,B,C,: : :,Z) 0. If none of the two above specified conditions holds, we state that D k (A,B,C,: : :,Z) Theory of Dependence Values \Delta 19 Notice that the computation of entropies can be simplified by applying the following Theorem. Theorem 10. If the entropies of order k-1 are known, a single entropy needs to be determined in order to calculate all the entropies of order k. Proof. Assume we know E(I 1 j I 2 , I 3 First we determine E(I 2 j I 1 , I 3 observing that: where only the last entropy is unknown. A similar method can be applied to determine all the others conditional entropies of the type E(I j I j+1 (with 2 j k). Finally, any other entropy can be easily calculated in terms of the already calculated ones. For example: E(I 1 I 4 I 4 5. THE DETERMINATION OF THE DEPENDENCE VALUES The analysis developed in this paper refers essentially to the concept of confidence and does not concern the principles of support. Almost all the algorithms so far proposed for data mining are based on a first step aimed at determining the k-plets having a sufficient support, namely, a sufficient statistical relevance. Such solutions are compatible with the following algorithm for determining all the relevant dependencies up to a certain order. (1) Determination of the k-plets having a sufficient support. Most algorithms for determining the k-plets having a sufficient support proceed in the order of increasing cardinalities. In other terms, they first determine the single items, then the pairs of items, the triplets, and so on. Such algorithms are well suited to the following procedure. Other algorithms should be modified in order to examine a k-plet P after the (k-1)-plets contained in P. The program which has been specifically developed to verify the ideas described in this paper is based on an algorithm for the determination of the k-plets (also called itemsets) having sufficient support [Meo 1999] which has been chosen in virtue of its speed, but which produces the list of itemsets organized in a family of trees. However, this data structure, as any other, can be transformed into a lattice suitable to the above described computations in a relatively short time. Notice that it is not necessary that the complete lattice of all the itemsets having sufficient support is represented in the main memory at the same time. What is really needed is that at the starting point every node of the structure, that is every analyzed itemset, is represented by two sets of data: a) the values of the joint probabilities describing that itemset; b) the pointers to its parents. For the sake of simplicity, the program here described is characterized, as concerns preceding point a), by the choice of describing an itemset with a single datum, as is possible in virtue of Theorem 7 on the unicity of the value. The chosen datum is the number of occurrences n ab:::z of that itemset, which is proportional to its probability P(a,b,: : :,z) (see Figure 9 where ptr x denotes the pointer to itemset x). abc abc ptr ab ptr ac ptr bc bc ptr c bc ac ptr a ptr c ac ab ptr a ab a a c c Fig. 9. The data structure of the itemsets. This choice makes it possible to store millions of itemsets in the main memory at the same time and to perform all the following computations without storing any partial results in the mass memory. (2) Determination of all the joint probabilities of an itemset. The computation of the joint probabilities P(i k ) for all the combinations of values of i k can be performed recursively applying the relationships presented in the proof of Theorem 7 on the unicity of the value. Of course, recursion proceeds towards the parents and the grandparents. For example, in the case of Figure 9, where only the probabilities directly connected to the numbers of occurrences introduced in Figure 9 appear. Theory of Dependence Values \Delta 21 (3) Determination of maximum independence estimates. The determination of the value x for which the joint entropy takes its maximum value can be performed numerically, at the desired level of accuracy, with conventional interpolation techniques. (4) Computation of the dependence value and states. A direct application of Definitions 13 and 14 leads to the final results. 5.1 Experimental Evaluation The proposed approach has been verified with an implementation in C++ using the Standard Template Library. The program has been run on a PC Pentium II, with a 233 Mhz clock, 128 MB RAM and running Red Hat Linux as the operating system. We have worked on a class of databases that has been taken as benchmark by most of data mining algorithms on association rules. It is the class of synthetic databases that project Quest of IBM has generated for its experiments (see [Agrawal and Srikant 1994] for detailed explanations). We have made many experiments on several databases with different values of the main parameters and of the minimum support, but the obtained results are all similar to the ones here proposed. In particular, the experiments have been run with the value of minimum support equal to 0.2% and with a precision in the computation of the dependence values equal to 10 \Gamma6 . In the generation of the databases we have adopted the same parameter settings proposed for synthetic databases: D, the number of transactions in the database has been fixed to 100 thousands; N, the total number of items, fixed to 1000. T, the average transaction length, has been fixed to 10, since its value does not influence the program behaviour. On the contrary, I, the average length of the frequent itemsets, has been varied, since its value really determines the depth of the lattice to be generated. Each database contains itemsets with sufficient support having a different average length (3,4,: : :,8). The extreme values of the interval [2-10] have been discarded for the reasons that follow. The low value has been discarded because it does not make much sense to maximize the entropy related to itemsets having only two items: the direct approach, that compares the probability of such an itemset with the product of the probabilities of the two items, has been adopted in this case. The high values have been avoided in consideration of the fact that longer are the itemsets the more probability they have to occur under the threshold of minimum support. In this case too few itemsets reveal to be over the threshold and the comparison of the different experiments is no more fair. Thus, it happens that even if the "nominal" average itemsets length is increased the actual average length of the itemsets with sufficient support reveals to be significantly lower. Table 10 reports the total number of itemsets with sufficient support and their nominal and actual average lengths in the experiments. In Figure 11 two execution times are shown. T 1 is the CPU execution time needed for the identification of all the itemsets with sufficient support; T 2 is the time for the computation of the dependence values of the itemsets previously identified. Both 22 \Delta R. Meo Total number nominal average actual average of itemsets itemset length itemset length Fig. 10. The number of itemsets and their average lengths in the experiments. the times have been normalized with respect to the total number of itemsets, since this one changes considerably in the different experiments.0.0050.0150.0250.0350.0452.81 3.28 3.67 4.05 4.82 4.96 CPU time[s] Average itemset length Execution times per itemset Fig. 11. Experimental results. You can notice that time T 1 decreases with the actual average itemset length. This is a particularity of the algorithm adopted (called Seq) for the first step, because this one builds, during its execution, temporary data structures that does not depend on the itemset length. Furthermore, Seq has been proved to be suitable to very long databases and to searches characterized by very low values of resolution. On the contrary, time T 2 increases with the average itemset length, since the depth of the resulting lattice increases. Thus, this fact is not surprising. On the other side, you can observe that the increments are moderate with the exception of the experiment having the average itemset length equal to 4.82 (corresponding to a nominal average itemset length equal to 6). In that experiment, as Table 10 reports, the total number of itemsets that exceed the minimumsupport threshold, compared Theory of Dependence Values \Delta 23 to the itemsets length, grows suddenly: in these conditions, the generated lattice is very heavily populated. Therefore, the high dimension of the output explains the result. Finally, these experiments demonstrate that this new approach to the discovery of knowledge on the itemsets dependencies is feasible and suitable to the high resolution researches that are typical of data mining. 6. CONCLUSIONS In this paper it has been shown that a single real - the dependence value - contains all the information on dependencies relative to a given itemset. Besides, by virtue of the Theorem stating that dependence functions are always parity func- tions, the determination of the combinations of values for which dependencies are positive or negative is immediate. Furthermore, the feasibility of this new theory is demonstrated in practical cases on a set of experiments on different databases. Some themes are worth being developed further. The first one concerns the maximum independence estimates. Is it possible to find a closed formula giving the maximum independence probability of a given itemset as a function of the lower level probabilities? The second theme to be developed is the definition of the confidence levels. Which percentage of the probability P(a,b,: : :) must be exceeded by the dependence value in order to state that dependence is strong? This question has been discussed adequately neither in the known framework support-confidence, but probably the model here introduced is more suitable to a sound theoretical analysis. A third area of investigation concerns the algorithms for determining the dependence values. The method proposed in this paper assumes that the itemset having sufficient support have been determined by adopting one of the known methods and on these ones performs the computation of the dependence values. However, it is likely that an integrated method combining the two steps is more rapid and effective. Theoretical analysis based on probability and information theory and the development of new algorithms should be combined and integrated in this area of research. --R Online generation of association rules. Database mining: A performance perspective. Mining association rules between sets of items in large databases. Fast discovery of association rules. Fast algorithms for mining association rules in large databases. An efficient algorithm for mining association rules in large databases. Knowledge discovery in databases: An attribute-oriented approach Sampling large databases for association rules. From file mining to database mining. A statistical perspective on kdd. Practitioner problems in need of database research: Research directions in knowledge discovery. A new approach for the discovery of frequent itemsets. An effective hash based algorithm for mining association rules. Mining quantitative association rules in large relational tables. Beyond market baskets: generalizing association rules to dependence rules. Mining generalized association rules. --TR Practitioner problems in need of database research Mining association rules between sets of items in large databases An effective hash-based algorithm for mining association rules Mining quantitative association rules in large relational tables Dynamic itemset counting and implication rules for market basket data Fast discovery of association rules Beyond Market Baskets Database Mining Pincer Search Online Generation of Association Rules Set-Oriented Mining for Association Rules in Relational Databases Knowledge Discovery in Databases Fast Algorithms for Mining Association Rules in Large Databases Discovery of Multiple-Level Association Rules from Large Databases An Efficient Algorithm for Mining Association Rules in Large Databases Mining Generalized Association Rules Sampling Large Databases for Association Rules A New Approach for the Discovery of Frequent Itemsets --CTR Alexandr Savinov, Mining dependence rules by finding largest itemset support quota, Proceedings of the 2004 ACM symposium on Applied computing, March 14-17, 2004, Nicosia, Cyprus Elena Baralis , Paolo Garza, Associative text categorization exploiting negated words, Proceedings of the 2006 ACM symposium on Applied computing, April 23-27, 2006, Dijon, France Peter Fule , John F. Roddick, Experiences in building a tool for navigating association rule result sets, Proceedings of the second workshop on Australasian information security, Data Mining and Web Intelligence, and Software Internationalisation, p.103-108, January 01, 2004, Dunedin, New Zealand Elena Baralis , Silvia Chiusano, Essential classification rule sets, ACM Transactions on Database Systems (TODS), v.29 n.4, p.635-674, December 2004

entropy;dependence rules;variables independence;association rules

364001

Discontinuous enrichment in finite elements with a partition of unity method.

We present an approximate analytical method to evaluate efficiently and accurately the call blocking probabilities in wavelength routing networks with multiple classes of calls. The model is fairly general and allows each source-destination pair to service calls of different classes, with each call occupying one wavelength per link. Our approximate analytical approach involves two steps. The arrival process of calls on some routes is first modified slightly to obtain an approximate multiclass network model. Next, all classes of calls on a particular route are aggregated to give an equivalent single-class model. Thus, path decomposition algorithms for single-class wavelength routing networks may be readilt extended to the multiclass case. This article is a first step towards understanding the issues arising in wavelength routing networks that serve multiple classes of customers.

Introduction The modeling of evolving discontinuities with the finite element method is cumbersome due to the need to update the mesh topology to match the geometry of the discontinuity. In this paper, we present a technique to model discontinuities in the finite element framework in a general fashion. The essential feature is the incorporation of enrichment functions which contain a presently Assistant Professor of Civil and Environmental Engineering, Duke University y Research Associate in Mechanical Engineering, Northwestern University z Walter P. Murphy Professor of Mechanical Engineering, Northwestern University discontinuous field. In the application of the technique to fracture mechanics, functions spanning the appropriate near-tip crack field can also be included to improve accuracy. The enrichment of the finite element approximation in this manner provides for both the modeling of discontinuities and accurate moment intensity factors with minimal computational resources. The concept of incorporating crack fields in a finite element context is not new, see for example [1]. In addition, there are several well established techniques for modeling cracks and crack growth such as boundary element methods, finite elements with continuous remeshing [2], and meshless methods [3]. Recently, the trend has focused on the development of finite element methods which model discontinuities independently of element boundaries. These include the incorporation of a discontinuous mode in an assumed strain framework [4], and enrichment with near-tip fields for crack growth with minimal remeshing [5]. The latter method has recently been extended by enriching with a discontinuous function behind the crack tip [6], [7], such that no remeshing is necessary. In this paper, the method of discontinuous enrichment is cast in a general framework, and we illustrate how both two-dimensional and plate formulations can be enriched to model cracks and crack growth. The enrichment of the approximation with discontinuous near-tip fields requires a mapping technique, and so an alternative near-tip function is developed. The present method offers several advantages over competing techniques for modeling crack growth. In contrast to traditional finite element methods, this technique incorporates the discontinuity of the crack independently of the mesh, such that the crack can be arbitrarily located within an element. The present technique has a distinct advantage over boundary element methods as it is readily applicable to non-linear problems, anisotropic materials, and arbitrary geometries. The method does not require any remeshing for crack growth, and as it is an extension of the finite element method, it can exploit the large body of finite element technology and software. Specific examples of augmenting well established two-dimensional and plate elements with both discontinuous and asymptotic near-tip functions are presented. The present technique exploits the partition of unity property of finite elements first cited by [8], which allows global enrichment functions to be locally incorporated into a finite element approximation. A standard approximation is 'enriched' in a region of interest by the global functions in conjunction with additional degrees of freedom and the local nodal shape functions. The application of this idea to capture a specific frequency band in dynamics can be found in [9]. The utility of the method has found application in solving the scalar Laplacian problem for domains with re-entrant corners [10]. In the context of fracture mechanics, the appropriate enrichment functions are the near-tip asymptotic fields and a discontinuous function to represent the jump in displacement across the crack line. In contrast to the work of [4], the enrichment is not through an assumed-strain method, so the displacement field is continuous along either side of the crack. This paper is organized as follows. Following this introduction, we review the construction of an enriched approximation and we develop a discontinuous near-tip function which does not require a mapping. For specific applications we consider two-dimensional linear elastic fracture mechanics and the fracture of Mindlin-Reissner plates in Section 3. Numerical results to verify the accuracy of the formulation are given in Section 4, with a summary and some concluding remarks provided in the last section. 2 Construction of a Finite Element Approximation With Discontinuities In this section, we present the construction of a finite element approximation with discontinuous enrichment. Emphasis is placed on modeling cracks, in which a standard approximation is enriched with both the asymptotic near-tip functions and a discontinuous 'jump' function. The incorporation of discontinuous near-tip functions requires a mapping for kinked cracks, and so an alternative near-tip function is presented. The manner in which nodes are selected for enrichment and the modifications to the numerical integration of the weak form are also given. 2.1 General Form To introduce the concept of discontinuous enrichment, we begin by considering the domain\Omega bounded by \Gamma with an internal boundary \Gamma c as shown in Fig. 1a. We are interested in the construction of a finite element approximation to the field u 2\Omega which can be discontinuous along \Gamma c . Consider the uniform mesh of N nodes for the domain shown in Fig. 1b which does not model the discontinuity. The discrete approximation u h to the function u takes the form I I ((x))u I (1) where N I is the shape function for node I in terms of the parent coordinates ((x)), and u I is the vector of nodal degrees of freedom. The nodal shape function N I is non-zero over the support of node I, defined to be union of the elements connected to the node. We now pose the question of how to best incorporate the discontinuity in the field along \Gamma c . The traditional approach is to change the mesh to conform to the line of discontinuity as shown in Fig. 1c, in which the element edges align with \Gamma c . While this strategy certainly creates a discontinuity in the approximation, it is cumbersome if the line \Gamma c evolves in time, or if several different configurations for \Gamma c are to be considered. In this paper we propose to model the discontinuity along \Gamma c with extrinsic enrichment [11], in which the standard approximation (1) is modified as I ((x))@u I a Il G l (x)A (2) where G l (x) are enrichment functions, and a Il are additional nodal degrees of freedom for node I. In the above, the total number of enriched degrees of freedom for a node is denoted by nE (I). If the enrichment functions G l are discontinuous along the boundary \Gamma c , then the finite element mesh does not need to model the discontinuity. For example, the uniform mesh in Fig. 1d is capable of modeling a jump in u when the circled nodes are enriched with functions which are discontinuous across \Gamma c . The above form of a finite element approximation merits some discussion. We note that the enrichment functions G l are written in terms of the global coordinates x, but that they are multiplied by the nodal shape functions N I . In this fashion the additional enrichment takes on a local character. This concept of multiplying global functions by the finite element partition of unity was first suggested in [8]. The change in the form of the approximation from (1) to (2) is only made locally in the vicinity of a feature of interest, such as a discontinuity. We now turn to the precise form of the enrichment functions used to model discontinuous fields, with the goal of modeling cracks and crack growth. Three distinct regions are identified for the crack geometry, namely the crack interior and the two near-tip regions as shown in Fig.2. In the set I of all nodes in the mesh, we distinguish three different sets which correspond to each of these regions. The set J is taken to be the set of nodes enriched for the crack interior, and the sets K 1 and K 2 are those nodes enriched for the first and second crack tips, respectively. The precise manner in which these sets are determined from the interaction of the crack and the mesh geometry is given in Section 2.3. The enriched approximation takes the form I I u I l l where b J and c 1 Kl , c 2 are nodal degrees of freedom corresponding to the enrichment functions H(x), F 1 l (x) and F 2 l (x), respectively. The function H(x) is discontinuous across the crack line, and the sets F 1 l (x) and l (x) consist of those functions which span the near-tip asymptotic fields. For two-dimensional elasticity, these are given by fF l (r; ')g 4 r sin( ' r cos( ' r sin( ' r cos( ' where (r; ') are the local polar coordinates for the crack tip [12]. Note that the first function in (40), rsin( '), is discontinuous across the crack faces whereas the last three functions are continuous. The form of the near-tip functions for plates is similar and is developed in Section 3. The jump function H(x) is defined as follows. The crack is considered to be a curve parametrized by the curvilinear coordinate s, as in Fig. 3. The origin of the curve is taken to coincide with one of the crack tips. Given a point x in the domain, we denote by x the closest point on the crack to x. At x , we construct the tangential and normal vector to the curve, e s and with the orientation of e n taken such that e s " e . The function H(x) is then given by the sign of the scalar product In the case of a kinked crack, the cone of normals at x needs to be considered (see [6]). Roughly speaking, the function H(x) takes the value of 1 'above' the crack, and \Gamma1 'below' the crack. 2.2 An Alternative Near-Tip Function The jump function H(x) is in general not capable of representing the discontinuity in the displacement field along the entire crack geometry. For example, if the crack tip is not aligned with an element edge, then a near-tip function must also be used (see [6]). For cracks which are not straight, a mapping is required to align the near-tip discontinuities with the crack edges. Due to the local form of the enrichment, the mapping procedure is only necessary in those elements with nodes enriched with the near-tip functions. In this section, we review the mapping procedure and present an alternative near-tip function. The crack is modeled as a series of straight line segments connecting ver- tices, with new crack segments added as the crack grows. The discontinuities in the near-tip fields are aligned with each segment by using a procedure developed in [12] and [5]. In this procedure, the discontinuity in the near-tip functions are aligned with the crack by a mapping technique that rotates each section of the discontinuity onto the crack model. A key step in technique is the modification of the angle ' in F l (r; '). Given a point define an angle terms of the angle of the segment ' R (see Fig 4) and the sampling point angle ff(x The coordinates of the sampling point are mapped to coordinates in the crack tip frame ("x; " y) as shown in Fig. 5: r cos( r sin( where l is the distance between r is as shown in the figure. The variables r and ' in the enrichment functions are then computed in terms of the local ("x; " y) coordinates. This procedure is repeated similarly for each segment of the crack and the sequence of mappings leaves the length of the crack invariant. In [5], the entire crack was modeled with the near-tip fields and the above mapping procedure. The use of the discontinuous function H(x) eliminates the need for the mapping on the crack interior, so that the above procedure is only necessary at each crack tip. In the following, we propose the use of a smooth 'ramp' function in conjunction with the function H(x) to model the near-tip region. Consider the following function defined in terms of the crack tip coordinates x l c l c where the length l c is taken to be the characteristic length of the element containing the crack tip. The above function and its derivative vanishes at the crack tip. When this smooth ramp function is multiplied by the function H(x), i.e., ~ a near-tip function which is discontinuous across the crack edges and vanishes in front of the crack tip results. This function in turn does not require any mapping to align the discontinuity with the crack edges. The function ~ R is shown in Fig. 6. in the vicinity of a crack tip and two consecutive segments. It is clear from the figure that the resulting near-tip function is continuous in the domain\Omega and discontinuous across the crack line. The above near-tip function is useful on several levels. In the first in- stance, in conjunction with the function H(x) for the crack interior it is perhaps the simplest means to model the entire crack discontinuity. In addi- tion, for non-linear problems the exact near-tip functions may not be known. The concept of multiplying a smooth function which vanishes at the crack tip by the jump function H(x) can also be extended to three-dimensional problems. In linear elastic fracture mechanics, however, the incorporation of the asymptotic near-tip fields is still useful to obtain greater accuracy at the crack tip. To some extent, this advantage can be maintained by simply replacing the first function in (4) with the function ~ R: fF l (r; ')g 4 r cos( ' r sin( ' r cos( ' The last three functions need not be mapped into the crack faces, as they are all continuous in the domain\Omega\Gamma 2.3 Node Selection for Enrichment In the preceding development, three distinct regions were identified for en- richment, corresponding to the nodal sets J , K 1 and K 2 . In this section we define these sets precisely, and present the methodology by which nodes are identified for inclusion in each set. We begin with some preliminary notations. The support of node I is denoted by ! I , with closure ! I . Essentially, a node's support is the open set of element domains connected to the node, and the closure is the closed set which includes the outer boundary. The distinction as it applies to nodal selection will be discussed shortly. We also denote the location of crack tips respectively, and by C the geometry of the crack. With these definitions, the sets J , K 1 and K 2 are defined as follows. The sets K 1 and K 2 consist of those nodes whose support closure contains crack tip 1 or 2, respectively. The set J is the set of nodes whose support is intersected by the crack and do not belong to K 1 or Note that the set J consists of nodes whose support, as opposed to support closure, is intersected by the crack. This distinction implies that if the crack only intersects the boundary of a node's support, the node will not be enriched with the function H(x). This prevents any nodes from being enriched with a constant function (either -1 or +1) over their entire support, which is important in order to avoid creating a linear dependency in the approxi- mation. We note that for the alternative near-tip function proposed in the previous section, the support closure must be changed to the open set for similar reasons. In practice the above sets are determined as follows. All elements intersected by the crack are first determined. From this set of elements, we distinguish three disjoint sets of 'tip elements' (for either tip 1 or 2) and 'interior elements'. The set of tip elements are given by those which contain either crack tip. The nodes of the 'tip elements' correspond to either set K 1 or K 2 . The nodes of the 'interior elements' in turn correspond to the set J . Fig. 7 illustrates the nodes that are selected as tip nodes and interior nodes for the cases of a uniform mesh and an unstructured mesh. An additional step is taken to remove those nodes from the set J whose support closure, but not support, is intersected by the crack. For this pur- pose, subpolygons which align with both the crack and element boundaries are generated as shown in Fig. 8b. These subpolygons are generated easily enough by triangulating the polygons formed from the intersection of the crack and element boundaries. The generation of the subtriangles allows the computation of the amount of a node's support `above' and 'below' the crack, which can then be compared against a tolerance. In two-dimensional analysis, we denote the area of a nodal support by A ! , which is calculated from the sum of the areas of each element connected to the node. With the aid of the subtriangles, we also calculate the nodal support area above A ab and below A be ! the crack. We then calculate the ratios A ab (13a) A be If either of these ratios are below a tolerance (we have used 0.01, or 1%), the nodes are removed from the set J . 2.4 Numerical Integration of the Weak Form For elements cut by the crack and enriched with the jump function H(x), we make a modification to the element quadrature routines for the assembly of the weak form. As the crack is allowed to be arbitrarily oriented in an element, standard Gauss quadrature may not adequately integrate the discontinuous field. For those nodes in the set J , it is important that the quadrature scheme accurately integrate the contributions to the weak form on both sides of the discontinuity. If the integration of the discontinuous enrichment is indistinguishable from that of a constant function, spurious singular modes can appear in the system of equations. In this section, we present the modifications made to the numerical integration scheme for elements cut by a crack. The discrete weak form is normally constructed with a loop over all ele- ments, as the domain is approximated by where m is the number of elements, and\Omega e is the element subdomain. For elements cut by a crack, we define the element subdomain to be a union of a set of subpolygons whose boundaries align with the crack geometry: s s denotes the number of subpolygons for the element. The subtriangles shown in Fig. 8 already generated for the selection of the interior nodes also work well for integration. It is emphasized that the subpolygons are only necessary for integration purposes; no additional degrees of freedom are associated with their construction. In the integration of the weak form, the element loop is replaced by a loop over the subpolygons for those elements cut by the crack. 3 Application to Fracture Mechanics In this section, we review the pertinent equations for linear elastic fracture mechanics. In this paper, emphasis is placed on plate fracture, although some two-dimensional plane-strain studies are discussed. A key difference in the plate formulation is the enrichment of the displacement components with different sets of near-tip functions. After reviewing the governing equations for Mindlin-Reissner plates, we examine the form of the asymptotic crack tip fields. A domain form of the J-integral for plates is derived for the calculation of the energy release rate and the moment intensity factors. Finally, the enriched finite element approximation is presented. 3.1 Mindlin-Reissner Plate Formulation Two main formulations exist to model a plate: the classical theory or Kirchhoff plate theory and the Mindlin-Reissner plate theory. Allowing three boundary conditions instead of two for the Kirchhoff theory, the Mindlin theory gives a more realistic shear and moment distribution around a crack tip (see [13] and [14]). A summary of the Mindlin theory follows. 3.1.1 Governing Equations There are several different ways to introduce the Mindlin theory. As we are also interested in examining problems in two-dimensional elasticity, the theory is presented here as a degeneration of the three-dimensional elasticity problem using the principal of virtual work with the appropriate kinematic assumptions. Consider a plate of thickness t whose mid-plane lies in the x plane. The conventions used throughout this paper are shown in Fig. 9. The main assumptions of the Mindlin theory state that the in plane dis- placements, u 1 and u 2 vary linearly through the thickness with the section rotations / 1 and / 2 . In addition, the normal stress oe 33 is assumed to vanish in the domain. For the sake of simplicity, we make the additional assumptions that the surface of the plate and any crack faces are traction-free. In the (e 1 is the unit normal vector to the plate, the deformation components at a point are given by where w is the transverse displacement and / 1 and / 2 are the rotations about the x 2 and x 1 axes, respectively. The above can be expressed in a more compact form as The strain is given by2 (ru with the bending contribution and a shear contribution We note that the x 3 related components are zero for both ffl b and ffl s . The virtual internal work is defined by Z d\Omega (21) where oe is the symmetric stress tensor, and ffiu is an arbitrary virtual displacement from the current position. After a few manipulations, we obtain the relation where the superscript indicates a reduction of the operator to the in plane component and oe s is the shear stress vector oe Making the substitution (22) into (21) and integrating through the thickness gives the work expression Z A where the moment M and shear Q are defined by Z t=2 \Gammat=2 Z t=2 \Gammat=2 oe s dx 3 (24) The virtual external work is composed of the action of the bending and twisting moments gathered in a couple vector C, and of the shear traction T . We assume there is no external pressure acting on the plate. The virtual external work is then given by Z Z Equating the internal and external virtual work, and applying the divergence theorem yields the equilibrium equations in\Omega r and the traction boundary conditions on \Gamma where n is the unit outward normal to the boundary. The constitutive relationships are obtained by energetic equivalence between the plate and the three-dimensional model. Assuming the plate is made of an isotropic homogeneous elastic material of Young's modulus E and of Poisson's ratio , the constitutive relations are given by4 M 11 22 ffl b22 and Ekt 5=6 is a correction factor which accounts for the parabolic variation of the shear stresses through the plate thickness. These are rewritten in a more compact form using the fourth order bending stiffness tensor D b and the second order shear stiffness tensor 3.1.2 Weak Form Let the boundary \Gamma be divided into a part \Gamma u on which displacement boundary conditions are imposed and a part \Gamma t on which loads are applied with the restrictions The kinematics constraints are given by a prescribed transverse displacement w and prescribed rotations / while the loads come from the prescribed couples C and prescribed shear tractions T . As in Section 2, we also designate c as an internal boundary across which the displacement field is allowed to be discontinuous. be the space of kinematically admissible transverse displacements and rotations: where V is a space of sufficiently smooth functions on \Omega\Gamma The details on this matter when the domain contains an internal boundary or re-entrant corner may be found in [15] and [16]. We note that the space V allows for discontinuous functions across the crack line. The space of test functions is defined similarly as: The weak form is to find (w; /) 2 V g such that Z (D Z (D s s(w; /)) \Delta s(ffiw; ffi/) Z Z It can be shown that the above is equivalent to the equilibrium equations (26) and traction boundary conditions (27). When the space V is discontinuous along \Gamma c , the traction-free conditions on the crack faces are also satisfied. In contrast to boundary element techniques, this enables the method to be easily extended to non-linear problems. In the finite element method, the space V is approximated with a finite dimensional space V h ae V. The space V h is typically made discontinuous across \Gamma c by explicitly meshing the surface, as in Fig. 1c. In the present method, the approximating space is constructed with discontinuous enrichment 3.2 Plate Fracture Mechanics Consider the problem of a through crack in a plate as shown in Fig. 10, where for convenience we adopt a local polar coordinate system centered at the crack tip. In contrast to the stress intensity factors obtained in classical linear elasticity, in plate theory the quantities of interest are moment and shear force intensity factors. The moment intensity factors are denoted by K I and K II , while the shear force intensity factor is denoted by K III . These are defined as The relationship between these factors and the energy release rate G is similar to the three-dimensional theory: \Theta I II 5Et III (37) The form of the asymptotic near-tip displacement fields differs significantly from the three-dimensional theory. In particular, the transverse displacement w is only singular when a K III mode is present. The asymptotic displacement fields in Mindlin-Reissner plate theory are given in [17], and they are provided here for the sake of completeness. 5h \Gamma3 (38a) \Gammasin( \Gamma2cos( (38c) For the purposes of defining the near-tip enrichment functions in the plate theory, we consider only the terms proportional to r for the rotations / 1 and / 2 . For the transverse displacement, we consider terms proportional to both r and r 3=2 . With these restrictions, the near-tip fields are contained in the span of the sets where fg I (r; ')g j ae p oe (40a) ff I (r; ')g j ae p oe (40b) The discrete approximation for the plate which incorporates the above near-tip functions is presented in Section 3.4. 3.3 Domain Form of the J-integral Several different domain and path-independent integrals have been developed for the extraction of mixed mode moment and shear force intensity factors in plates [17]. These integrals typically consist of a contour integral enclosing the crack-tip singularity. With finite elements, the numerical evaluation of these integrals usually involves some kind of smoothing technique, as the required field quantities are discontinuous at element interfaces. In this section, we illustrate the use of a weighting function q to recast these line integrals into their equivalent domain form. The development presented here closely follows that given for two-dimensional elasticity in [18]. The domain forms of crack contour integrals are particularly well suited for use with finite elements, as the same quadrature points used for the integration of the weak form can be used to calculate the domain integral. The construction of additional quadrature points or the use of a smoothing procedure is not required. Consider the open contour \Gamma surrounding a through crack as shown in Fig. 11. In the following, we use indicial notation where the Greek indices (ff; fi) range over the values and a comma denotes a derivative with respect to the following argument. The contour integral proposed by [17] in the absence of an externally applied pressure is given by I where W is the strain energy density of the plate. This is defined as We now introduce a weight function q 1 which is defined over the domain of interest. Consider the simply connected curve shown in Fig. 11. The function q 1 is defined to be sufficiently smooth in the area A enclosed by C, and is given on the surfaces by: ae 1 on \Gamma on C We then use this function to rewrite (41) as I Z where we have used \Gamman i on \Gamma, and on the crack faces. The last integral above vanishes for traction free crack faces. Applying the divergence theorem to the closed integral, we then obtain Z A Z which is the equivalent domain form of the J 1 integral proposed by [17]. The measure number J 1 is domain independent and its magnitude is equivalent to the energy release rate (37). Therefore, under pure mode I loading, the moment intensity factor K I is given by r In more general mixed-mode conditions, the values K I ,K II and K III cannot be separated so easily. While the comparison of J 1 to analytical values is adequate to assess the effectiveness and qualities of the enrichment strategy, crack growth laws are typically expressed in terms of the mixed-mode intensity factors. In classical linear elasticity, the interaction integral approach [19] has proven effective to extract mixed mode stress intensity factors. The application of this method to plate fracture is currently under development 3.4 Enrichment of the MITC4 plate element When discretizing the plate equations (26), some care must be taken to avoid shear locking. As the plate becomes very thin (i.e. t ! 0), the following relationship must be satisfied to keep the strain energy in the plate bounded: In other words, the shear strain ffl s must vanish as t ! 0. Standard displacement based elements, such as the four node isoparametric element, have difficulty satisfying this constraint. The consequence is a structure which exhibits an overly stiff response, often referred to as shear locking. To discretize the plate displacements (16), we begin with the MITC4 element. To avoid shear locking, the MITC formulation modifies the approximation for the section rotations / in the expression for the shear stiffness (see [21]). In the following, we express this modification using the notation ~ I , where it is understood that only those expressions relating to the shear components are modified. The enriched discretization takes the form I w I c w Kl G l (r; ') (48a) ~ I / I ~ c / Kl F l (r; ') (48b) where N I are the standard bilinear shape functions. In the above, we have collapsed the sums over each crack tip into one for compactness. The sets of near-tip functions G l and F l are derived from (40) in the following fashion. We take G l to be only those functions in g l which are proportional to r 3=2 . The set F l is taken to be equivalent to f l . In addition to having four additional degrees of freedom for each displacement component, this choice for G l and F l satisfies the following relation such that a linear combination of the near-tip enrichment functions can satisfy (47). We note that this relationship does not ensure that the enriched formulation will be completely free of shear locking. However, the numerical examples presented in the next section indicate that the above formulation performs well for a wide range of plate thicknesses. In this section we present several different numerical calculations. We first examine some problems in two-dimensional elasticity, including a robustness test and the simulation of crack growth. Then a benchmark and an additional study are presented for Mindlin-Reissner plates. 4.1 Two-dimensional problems We begin with a simple example of an edge crack to demonstrate the robustness of the discretization scheme, and then present results for more complicated geometries. In all of the following examples, the material is taken to be isotropic with Young's modulus and plane strain conditions are assumed. The calculation of the stress intensity factors is performed with the domain form of the interaction integral, and the maximum hoop stress law is used to govern crack growth ( see [19] and [5]). 4.1.1 Robustness tests Consider the geometry shown in Fig. 12: a plate of width w and height L with an edge crack of length a, subjected to a far-field stress oe o . We analyze the influence of the location of the crack with respect to the mesh on the K I stress intensity factor when the position of the crack is perturbed by ffix in the X direction and ffiy in the Y direction. The geometry is discretized with a uniform mesh of 24x48 4-noded quadrilateral elements. In this study, several different discretizations are obtained depending on the position of the crack with respect to the mesh. Two cases are shown in Fig. 13. In this investigation, we wish to examine the performance of the modified tip function ~ R(x), and the accuracy of the formulation when it is used in conjunction with the other near-tip functions as in (9). The exact solution for this problem is given by [22] a (50) where C is a finite-geometry correction factor: a a a a The numerical results normalized by the exact solution when only the function ~ R(x) is used to model the near-tip region are given in Table 1. Depending on the location of the crack tip, the total number of degrees of freedom varies from 2483 to 2503. The results vary by approximately 4% over all crack tip locations tested. When the near-tip functions are added, the accuracy improves as shown in Table. 2. These results are consistent with those reported in [6] in that the best results are obtained when the crack is aligned with mesh boundaries. We note that the results are not as accurate as when the exact asymptotic function rsin(') is used, in which case the error is less than 2% (see [6]). 4.1.2 Crack Growth from a Fillet This example shows the growth of a crack from a fillet in a structural mem- ber, and serves to illustrate how the present method can be used as an aid to design against failure. The configuration to be studied is shown in Fig. 14, with the actual domain modeled as indicated. The setup is taken from experimental work found in [23]. In this example, we investigate the effect of the thickness of the lower I-beam on crack growth. Only the limiting cases for the bottom I-beam of a rigid constraint (very thick beam) and flexible constraint (very thin beam) are considered. In addition, the welding residual stresses between the member and the I-beam are neglected. The structure is loaded with a traction of and the initial crack length is taken to be a 5mm. The geometry is discretized with 8243 three-node triangular elements. To model a rigid constraint, the displacement in the vertical direction is fixed along the entire bottom of the domain. A flexible constraint is idealized by fixing the vertical displacement at both ends of the bottom of the domain. For both sets of boundary conditions, an additional degree of freedom is fixed to prevent a rigid body rotation. For each load case, we simulate crack growth with a step size of \Deltaa = 5mm for a total of 14 steps. Fig. 15 shows the mesh in the vicinity of the fillet and compares the crack paths for the cases of a thick I-beam (upper crack) and a thin I-beam (lower crack). It is emphasized that the same mesh is used throughout the simulation, and that no remeshing is required. As new crack segments are added, additional enriched degrees of freedom are generated for each new segment. The results shown are consistent with both the experimental [23] and previous numerical results [24]. 4.2 Plate examples In this section, we present some examples using the enriched MITC4 plate formulation developed in Section 3.4. We first examine the accuracy of the method as a function of plate thicknesses for a benchmark problem, and then present a more general example. Throughout this section, the material properties are assumed to be isotropic with Young's modulus of GPa, and Poisson's ratio As a benchmark problem we consider a through crack in an infinite plate subjected to a far-field moment M o . The crack is oriented at an angle fi with respect to the x 1 axis as shown in Fig. 16. Recently, very accurate calculations were carried out by [25] for various plate thicknesses for the case when . In this case, the loading is purely mode I, and the domain form of the J-integral for plates (45) is used in conjunction with (46) to determine the moment intensity factor K I . In the finite element model, only one-half of a square plate is modeled, with symmetry conditions along the x 2 axis. To approximate the infinite plate, the plate width w is taken to be 10 times the half crack length a. The crack length for all of the results presented in this section is taken to be Fig. 17 shows the normalized K I for four discretizations, two standard and two enriched. The lower curve corresponds to a non-enriched formulation, and the values for K I are within 5% of the exact for the entire range of plate thicknesses t. These values are improved when the mesh is refined for a total of 2463 degrees of freedom as shown. We observe that the enriched solution with only 755 degrees of freedom is as accurate as the solution with 2463 degrees of freedom without enrichment. The last curve for the enriched case with 3087 degrees of freedom exhibits less than 1% error. The enriched solutions show good correlation with the analytical solution for the full range of plate thicknesses tested. As a last example, moment intensity factors are calculated for a finite plate as a function of crack length for various plate thicknesses. The geometry of the plate is taken to be the same as the previous example, and the results are compared to those given in [26]. In this study, the mesh does not model the crack discontinuity; the jump in the rotations and transverse displacement is created entirely with enrichment. Table 3 gives the results for four different plate width to thickness ratios for the case when the plate is modeled with 1424 MITC4 elements. These results show excellent correlation for the cases when w=8, in which the maximum error is 1.2%. For the remaining cases the maximum difference between the numerical solutions and those given in [26] is 9.4%. We note, however, that the reference [26] is not as current as [25]. In the latter, the moment intensity factors are shown to be significantly greater than the classical results as the thickness t ! 0. The results shown in Table 3 are consistent with these findings. Summary A method of constructing finite element approximations with enrichment functions was presented which allows for the simulation of evolving discontinuities in a straightforward fashion. The specific examples of cracks and crack growth in two-dimensional elasticity and Mindlin-Reissner plate theory were examined. By incorporating the appropriate asymptotic near-tip fields, accurate moment and stress intensity factors were obtained for coarse meshes. A new near-tip function was also developed to remove the need for a mapping in the case of kinked cracks. The methodology for the construction of the discrete approximation from the interaction of the crack geometry and the mesh was provided, and numerical tests served to illustrate the algo- rithm's robustness. Additional numerical studies for Mindlin-Reissner plates demonstrated the extent to which stress intensity factors can be calculated accurately for a wide range of plate thicknesses. The present method has a lot of potential to extend the finite element method for the modeling of evolving interfaces and free surfaces. A key feature of the enrichment in conjunction with numerical integration is the capability of modeling geometrical features which are independent of the mesh topology. As was shown in this paper, several different crack configurations can be considered for a single mesh of a component, simply by changing the enrichment scheme according to the crack geometry. Future work will focus on the application of the method to three-dimensional and dynamic fracture, as well as other areas of mechanics in which moving interfaces are of importance. Acknowledgements The support of the Office of Naval Research and Army Research Office, to Northwestern University, is gratefully acknowledged. The authors are grateful for the support provided by the DOE Computational Science Graduate Fellowship program, to John Dolbow. --R A hybrid-element approach to crack problems in plane elasticity Modeling mixed-mode dynamic crack propagation using finite elements: Theory and applications Modelling strong discontinuities in solid mechanics via strain softening constitutive equations. Elastic crack growth in finite elements with minimal remeshing. A finite element method for crack growth without remeshing. An extended finite element method with discontinuous enrichment for applied mechanics. Multiple scale finite element methods. Meshless methods: An overview and recent developments. Enriched methods for singular fields On the bending of an elastic plate containing a crack. Mechanics of fracture 3: Plates and shells with cracks. Elliptic Problems in Nonsmooth Domains. Computation of stress intensity factors for plate bending via a path-independent integral Crack tip and associated domain integrals from momentum and energy balance. Modeling fracture in Mindlin- Reissner plates with the extended finite element method Displacement and stress convergence of our MITC plate bending elements. Fracture Mechanics. Morphological aspects of fatigue crack propagation. The Element-free Galerkin Method for Fatigue and Quasi-Static Fracture Bending of a thin Reissner plate with a through crack. Internal and edge cracks in a plate of finite width under bending. --TR --CTR Kenjiro Performance assessment of generalized elements in the finite cover method, Finite Elements in Analysis and Design, v.41 n.2, p.111-132, November 2004 L. B. Tran , H. S. Udaykumar, A particle-level set-based sharp interface cartesian grid method for impact, penetration, and void collapse, Journal of Computational Physics, v.193 n.2, p.469-510, 20 January 2004 H. S. Udaykumar , L. Tran , D. M. Belk , K. J. Vanden, An Eulerian method for computation of multimaterial impact with ENO shock-capturing and sharp interfaces, Journal of Computational Physics, v.186 n.1, p.136-177, 20 March

partition-of-unity;fracture;discontinuous enrichment

364268

Recognizing Action Units for Facial Expression Analysis.

AbstractMost automatic expression analysis systems attempt to recognize a small set of prototypic expressions, such as happiness, anger, surprise, and fear. Such prototypic expressions, however, occur rather infrequently. Human emotions and intentions are more often communicated by changes in one or a few discrete facial features. In this paper, we develop an Automatic Face Analysis (AFA) system to analyze facial expressions based on both permanent facial features (brows, eyes, mouth) and transient facial features (deepening of facial furrows) in a nearly frontal-view face image sequence. The AFA system recognizes fine-grained changes in facial expression into action units (AUs) of the Facial Action Coding System (FACS), instead of a few prototypic expressions. Multistate face and facial component models are proposed for tracking and modeling the various facial features, including lips, eyes, brows, cheeks, and furrows. During tracking, detailed parametric descriptions of the facial features are extracted. With these parameters as the inputs, a group of action units (neutral expression, six upper face AUs and 10 lower face AUs) are recognized whether they occur alone or in combinations. The system has achieved average recognition rates of 96.4 percent (95.4 percent if neutral expressions are excluded) for upper face AUs and 96.7 percent (95.6 percent with neutral expressions excluded) for lower face AUs. The generalizability of the system has been tested by using independent image databases collected and FACS-coded for ground-truth by different research teams.

Introduction Recently facial expression analysis has attracted attention in the computer vision literature [3, 5, 6, 9, 11, 13, 17, 19]. Most automatic expression analysis systems attempt to recognize a small set of prototypic expressions (i.e. joy, surprise, anger, sadness, fear, and disgust) [11, 17]. In everyday life, however, such prototypic expressions occur relatively infrequently. Instead, emotion is communicated by changes in one or two discrete facial features, such as tightening the lips in anger or obliquely lowering the lip corners in sadness [2]. Change in isolated features, especially in the area of the brows or eyelids, is typical of paralinguistic displays; for instance, raising the brows signals greeting. To capture the subtlety of human emotion and paralinguistic communication, automated recognition of fine-grained changes in facial expression is needed. Ekman and Friesen [4] developed the Facial Action Coding System (FACS) for describing facial expressions. The FACS is a human-observer-based system designed to describe subtle changes in facial features. FACS consists of 44 action units, including those for head and eye positions. AUs are anatomically related to contraction of specific facial muscles. They can occur either singly or in combinations. AU combinations may be additive, in which case combination does not change the appearance of the constituents, or nonadditive, in which case the appearance of the constituents changes (analogous to co-articulation effects in speech). For action units that vary in intensity, a 5-point ordinal scale is used to measure the degree of muscle contraction. Although the number of atomic action units is small, more than 7,000 combinations of action units have been observed [12]. FACS provides the necessary detail with which to describe facial expression. Automatic recognition of action units is a difficult problem. AUs have no quantitative definitions and as noted can appear in complex combinations. Several researchers have tried to recognize AUs [1, 3, 9]. The system of Lien et al. [9] used dense-flow, feature point tracking and edge extraction to recognize 6 upper face AUs or AU combinations (AU1+2, AU1+4, AU4, AU5, AU6, and AU7) and 9 lower face AUs and AU combinations (AU12, AU25, AU26, AU27, AU12+25, AU20+25, AU15+17, AU17+23+24, AU9+17). Bartlett et al. [1] recognized 6 individual upper face AUs (AU1, AU2, AU4, AU5, AU6, and but none occurred in combinations. The performance of their feature-based classifier on novel was 57%; on new images of faces used for training, the rate was 85.3%. By combining holistic spatial analysis and optical flow with local features in a hybrid system, Bartlett et al. increased accuracy to 90.9% correct. Donato et al. [3] compared several techniques for recognizing action units including optical flow, principal component analysis, independent component analysis, local feature analysis, and Gabor wavelet representation. Best performances were obtained by Gabor wavelet representation and independent component analysis which achieved a 95% average recognition rate for 6 upper face AUs and 6 lower face AUs. In this report, we developed a feature-based AU recognition system. This system explicitly analyzes appearance changes in localized facial features. Since each AU is associated with a specific set of facial muscles, we believe that accurate geometrical modeling of facial features will lead to better recognition results. Furthermore, the knowledge of exact facial feature positions could benefit the area-based [17], holistic analysis [1], or optical flow based [9] classifiers. Figure 1 depicts the overview of the analysis system. First, the head orientation and face position are detected. Then, subtle changes in the facial components are measured. Motivated by FACS action units, these changes are represented as a collection of mid-level feature parameters. Finally, action units are classified by feeding these parameters to a neural network. Because the appearance of facial features is dependent upon head orientation, we develop a multi-state model-based system for tracking facial features. Different head orientations and corresponding variation in the appearance of face components are defined as separate states. For each state, a corresponding description and one or more feature extraction methods are developed. We separately represent all the facial features into two parameter groups for upper face and lower face because facial actions in the upper and lower face are relatively independent [4]. Fifteen parameters are used to describe eye shape, motion, and state, and brow and cheek motion, and upper face furrows for upper face. Nine parameters are used to describe the lip shape, lip motion, lip state, and lower face furrows for lower face. After the facial features are correctly extracted and suitably represented, we employ a neural network to recognize the upper face AUs (Neutral, AU1, AU2, AU4, AU5, AU6, and AU7) and lower face AUs (Neutral, AU9, AU 10, AU 12, AU 15, AU 17, AU 20, AU 25, AU 26, AU 27, and AU23+24) respectively. Seven basic upper face AUs and eleven basic lower face AUs are identified regardless of whether they occurred singly or in combinations. For the upper face AU recognition, compared to Bartlett's [1] results by using the same database, our system achieves recognition accuracy with an average recognition rate of 95% with fewer parameters and in the more difficult case in which AUs may occur either individually or in additive and nonadditive combinations. For the lower face AU recognition, a previous attempt for a similar task [9] recognized 6 lower face AUs and combinations(AU 12, AU12+25, AU20+25, AU9+17, AU17+23+24, and AU15+17) with 88% average recognition rate by separate hidden Markov Models for each action unit or action unit combination. Compared to the previous results, our system achieves recognition accuracy with an average recognition rate of 96.71%. Difficult cases in which AUs occur either individually or in additive and nonadditive combinations are handled also. Figure 1. Feature based action unit recognition system. 2. Multi-State Models for Face and Facial Components 2.1. Multi-state face model Head orientation is a significant factor that affects the appearance of a face. Based on the head orien- tation, seven head states are defined in Figure 2. To develop more robust facial expression recognition system, head state will be considered. For the different head states, facial components, such as lips, appear very differently, requiring specific facial component models. For example, the facial component models for a front face include F rontLips, F rontEyes (left and right), F rontCheeks(left and right), NasolabialFurrows, and Nosewrinkles. The right face includes only the component models SideLips, Righteye, Rightbrow, and Rightcheek. In our current system, we assume the face images are nearly front view with possible in-plane head rotations. 2.2. Multi-state face component models Different face component models must be used for different states. For example, a lip model of the front face doesn't work for a profile face. Here, we give the detailed facial component models for the nearly front-view face. Both the permanent components such as lips, eyes, brows, cheeks and the transient components such as furrows are considered. Based on the different appearances of different components, different geometric models are used to model the component's location, shape, and appearance. Each (a) Head state. (b) Different facial components used for each head state. Figure 2. Multiple state face model. (a) The head state can be left, left-front, front, right-front, right, down, and up. (b) Different facial component models are used for different head states. component employs a multi-state model corresponding to different component states. For example, a three-state lip model is defined to describe the lip states: open, closed, and tightly closed. A two-state eye model is used to model open and closed eye. There is one state for brow and cheek. Present and absent are use to model states of the transient facial features. The multi-state component models for different components are described in Table 1. Table 1. Multi-state facial component models of a front face Component State Description/Feature Opened Closed (xc, yc) Tightly closed Lip corner1 Lip corner2 Eye Open (xc, yc) h2 Closed corner2 corner1 Brow Present Cheek Present Furrow Present Eye's inner corner line furrows nasolabial Absent 3. Facial Feature Extraction Contraction of the facial muscles produces changes in both the direction and magnitude of the motion on the skin surface and in the appearance of permanent and transient facial features. Examples of permanent features are the lips, eyes, and any furrows that have become permanent with age. Transient features include any facial lines and furrows that are not present at rest. We assume that the first frame is in a neutral expression. After initializing the templates of the permanent features in the first frame, both permanent and transient features can be tracked and detected in the whole image sequence regardless of the states of facial components. The tracking results show that our method is robust for tracking facial features even when there is large out of plane head rotation. 3.1. Permanent features features: A three-state lip model is used for tracking and modeling lip features. As shown in Table 1, we classify the mouth states into open, closed, and tightly closed. Different lip templates are used to obtain the lip contours. Currently, we use the same template for open and closed mouth. Two parabolic arcs are used to model the position, orientation, and shape of the lips. The template of open and closed lips has six parameters: lip center (xc, yc), lip shape (h1, h2 and w), and lip orientation ('). For a tightly closed mouth, the dark mouth line connecting lip corners is detected from the image to model the position, orientation, and shape of the tightly closed lips. After the lip template is manually located for the neutral expression in the first frame, the lip color is obtained by modeling as a Gaussian mixture. The shape and location of the lip template for the image sequence is automatically tracked by feature point tracking. Then, the lip shape and color information are used to determine the lip state and state transitions. The detailed lip tracking method can be found in paper [15]. Eye features: Most eye trackers developed so far are for open eyes and simply track the eye locations. However, for recognizing facial action units, we need to recognize the state of eyes, whether they are open or closed, and the parameters of an eye model, the location and radius of the iris, and the corners and height of the open eye. As shown in Table 1, the eye model consists of "open" and "closed". The iris provides important information about the eye state. If the eye is open, part of the iris normally will be visible. Otherwise, the eye is closed. For the different states, specific eye templates and different algorithms are used to obtain eye features. For an open eye, we assume the outer contour of the eye is symmetrical about the perpendicular bisector to the line connecting two eye corners. The template, illustrated in Table 1, is composed of a circle with three parameters and two parabolic arcs with six parameters This is the same eye template as Yuille's except for two points located at the center of the whites [18]. For a closed eye, the template is reduced to 4 parameters for each of the eye corners. The default eye state is open. Locating the open eye template in the first frame, the eye's inner corner is tracked accurately by feature point tracking. We found that the outer corners are hard to track and less stable than the inner corners, so we assume the outer corners are on the line that connects the inner corners. Then, the outer corners can be obtained by the eye width, which is calculated from the first frame. Intensity and edge information are used to detect an iris because the iris provides important information about the eye state. A half-circle iris mask is used to obtain correct iris edges. If the iris is detected, the eye is open and the iris center is the iris mask center In an image sequence, the eyelid contours are tracked for open eyes by feature point tracking. For a closed eye, we do not need to track the eyelid contours. A line connects the inner and outer corners of the eye is used as the eye boundary. The detailed eye feature tracking techniques can be found in paper [14]. Brow and cheek features: Features in the brow and cheek areas are also important to facial expression analysis. For the brow and cheek, one state is used respectively, a triangular template with six parameters is used to model the position of brow or cheek. Both brow and cheek are tracked by feature point tracking. A modified version of the gradient tracking algorithm [10] is used to track these points for the whole image sequence. Some permanent facial feature tracking results for different expressions are shown in Figure 3. More facial feature tracking results can be found in http://www.cs.cmu.edu/-face. 3.2. Transient features Facial motion produces transient features. Wrinkles and furrows appear perpendicular to the motion direction of the activated muscle. These transient features provide crucial information for the recognition of action units. Contraction of the corrugator muscle, for instance, produces vertical furrows between the brows, which is coded in FACS as AU 4, while contraction of the medial portion of the frontalis muscle (AU 1) causes horizontal wrinkling in the center of the forehead. Some of these lines and furrows may become permanent with age. Permanent crows-feet wrinkles around the outside corners of the eyes, which is characteristic of AU 6 when transient, are common in adults but not in infants. When lines and furrows become permanent facial features, contraction of the corresponding muscles produces changes in their appearance, such as deepening or lengthening. The presence or absence of the furrows in a face image can be determined by geometric feature analysis [9, 8], or by eigen-analysis [7, 16]. Kwon and Lobo [8] detect furrows by snake to classify pictures of people into different age groups. Lien [9] detected whole face horizontal, vertical and diagonal edges for face expression recognition. In our system, we currently detect nasolabial furrows, nose wrinkles, and crows feet wrinkles. We define them in two states: present and absent. Compared to the neutral frame, the wrinkle state is present if the wrinkles appear, deepen, or lengthen. Otherwise, it is absent. After obtaining the permanent facial features, the areas with furrows related to different AUs can be decided by the permanent facial feature locations. We define the nasolabial furrow area as the area between eye's inner corners line and lip corners line. The nose wrinkle area is a square between two eye inner corners. The crows feet wrinkle areas are beside the eye outer corners. We use canny edge detector to detect the edge information in these areas. For nose wrinkles and crows feet wrinkles, we compare the edge pixel numbers E of current frame with the edge pixel numbers E 0 of the first frame in the wrinkle areas. If E=E 0 large than the threshold T , the furrows are present. Otherwise, the furrows are absent. For the nasolabial furrows, we detect the continued diagonal edges. The nasolabial furrow detection results are shown in Fig. 4. 4. Facial Feature Representation Each action unit of FACS is anatomically related to contraction of a specific facial muscle. For instance, AU 12 (oblique raising of the lip corners) results from contraction of the zygomaticus major muscle, AU 20 (lip stretch) from contraction of the risorius muscle, and AU 15 (oblique lowering of the lip corners) from contraction of the depressor anguli muscle. Such muscle contractions produce motion in the overlying skin and deform shape or location of the facial components. In order to recognize the (a) (b) (c) (d) Figure 3. Permanent feature tracking results for different expressions. (a) Narrowing eyes and opened smiled mouth. (b) Large open eye, blinking and large opened mouth. (c) Tight closed eye and eye blinking. (4) Tightly closed mouth and blinking. Figure 4. Nasolabial furrow detection results. For the same subject, the nasolabial furrow angle(between the nasolabial furrow and the line connected eye inner corners) is different for different expressions. subtle changes of face expression, we represent the upper face features and lower face features into a group of suitable parameters respectively because facial actions in the upper face have little influence on facial motion in lower face, and vice versa [4]. For defining these parameters, we first define the basic coordinate system. Because the eye's inner corners are the most stable features in the face and are relatively insensitive to deformation by facial expressions, we define the x-axis as the line connecting two inner corners of eyes and the y-axis as perpendicular to x-axis. In order to remove the effects of the different size of face images in different image sequences, all the parameters except those about wrinkles' states are calculated in ratio scores by comparison to the neutral frame. 4.1. Upper Face Feature Representation We represent the upper face features as 15 parameters. Of these, 12 parameters describe the motion and shape of eyes, brows, and cheeks. 2 parameters describe the state of crows feet wrinkles, and 1 parameter describes the distance between brows. Figure 5 shows the coordinate system and the parameter meanings. The definitions of upper face parameters are listed in Table 2. Table 2. Upper face feature representation for AU recognition Permanent features (Left and right) Inner brow Outer brow Eye height motion (r binner ) motion (r bouter ) (r eheight ) r binner r bouter r eheight If r binner >0, If r bouter >0, If r eheight >0, Inner brow Outer brow Eye height move up. move up. increases. Eye top lid Eye bottom lid Cheek motion motion (r top ) motion (r btm ) (r cheek ) r top r btm r cheek . =\Gamma h2\Gammah2 0 . =\Gamma c\Gammac 0 If r top ? 0, If r btm ? 0, If r cheek ? 0, Eye top lid Eye bottom lid Cheek move up. move up. move up. Other features Distance Left crows Right crows of brows feet wrinkles feet wrinkles (D brow D brow If W lef . Left crows feet Right crows feet wrinkle present. wrinkle present. Figure 5. Upper face features. hl(hl1 are the height of left eye and right eye; D is the distance between brows; cl and cr are the motion of left cheek and right cheek. bli and bri are the motion of the inner part of left brow and right brow. blo and bro are the motion of the outer part of left brow and right brow. f l and fr are the left and right crows feet wrinkle areas. 4.2. Lower Face Feature Representation We define nine parameters to represent the lower face features from the tracked facial features. Of these, 6 parameters describe the permanent features of lip shape, lip state and lip motion, and 3 parameters describe the transient features of the nasolabial furrows and nose wrinkles. We notice that if the nasolabial furrow is present, there are different angles between the nasolabial furrow and x-axis for different action units. For example, the nasolanial furrow angle of AU9 or AU10 is larger than that of AU12. So we use the angle to represent its orientation if it is present. Although the nose wrinkles are located in the upper face, but we classify the parameter of them in the lower face feature because it is related to the lower face AUs. The definitions of lower face parameters are listed in Table 3. These feature data are affine aligned by calculating them based on the line connected two inner corners of eyes and normalized for individual differences in facial conformation by converting to ratio scores. The parameter meanings are shown in Figure 6. Figure 6. Lower face features. h1 and h2 are the top and bottom lip heights; w is the lip width; D lef t is the distance between the left lip corner and eye inner corners line; D right is the distance between the right lip corner and eye inner corners line; n1 is the nose wrinkle area. 5. Facial Action Unit Definitions Ekman and Friesen [4] developed the Facial Action Coding System (FACS) for describing facial expressions by action units (AUs) or AU combinations. are anatomically related to Table 3. Representation of lower face features for AUs recognition Permanent features Lip height Lip width Left lip corner r height r width r lef t . =\Gamma D left \GammaD left0 If r height >0, If r width >0, If r lef t >0, lip height lip width left lip corner increases. increases. move up. Right lip corner Top lip motion Bottom lip right r right r top r btm right \GammaD right0 D right0 . =\Gamma Dtop \GammaD top0 . =\Gamma D btm \GammaD btm0 If r right >0, If r top >0, If r btm >0, right lip corner top lip bottom lip move up. move up. move up. Transient features Left nasolibial Right nasolibial State of nose furrow angle furrow angle wrinkles (Ang Left nasolibial Left nasolibial If S nosew furrow present furrow present nose wrinkles with angle Ang lef t . with angle present. Ang right . contraction of a specific set of facial muscles. Of thses, 12 are for upper face, and are for lower Action units can occur either singly or in combinations. The action unit combinations may be additive such as AU1+5, in which case combination does not change the appearance of the constituents, or nonadditive, in which case the appearance of the constituents does change such as AU1+4. Although the number of atomic action units is small, more than 7,000 combinations of action units have been observed [12]. FACS provides the necessary detail with which to describe facial expression. Table 4. Basic upper face action units or AU combinations Inner portion of Outer portion of Brows lowered the brows is the brows is and drawn raised. raised. together Upper eyelids Cheeks are Lower eyelids are raised. raised. are raised. AU 1+4 AU 4+5 AU 1+2 Medial portion Brows lowered Inner and outer of the brows is and drawn portions of the raised and pulled together and brows are raised. together. upper eyelids are raised. AU 1+2+4 AU1+2+5+6+7 AU0(neutral) Brows are pulled Brow, eyelids, and Eyes, brow, and together and cheek are raised. cheek are upward. relaxed. 5.1. Upper Face Action Units Table 4 shows the definitions of 7 individual upper face AUs and 5 non-additive combinations involving these action units. As an example of a non-additive effect, AU4 appears differently depending on whether it occurs alone or in combination with AU1, as in AU1+4. When AU1 occurs alone, the brows are drawn together and lowered. In AU1+4, the brows are drawn together but are raised by the action of AU 1. As another example, it is difficult to notice any difference between the static images of AU2 and AU1+2 because the action of AU2 pulls the inner brow up, which results in a very similar appearance to AU1+2. In contrast, the action of AU1 alone has little effect on the outer brow. 5.2. Lower face action units Table 5 shows the definitions of 11 lower face AUs or AU combinations. 6. Image Database 6.1. Image Database for Upper Face AU Recognition We use the database of Bartlett et al. [1] for upper face AUs recognition. This image database was obtained from 24 Caucasian subjects, consisting of 12 males and 12 females. Each image sequence consists of 6-8 frames, beginning with a neutral or with very low magnitude facial actions and ending with a high magnitude facial actions. For each sequence, action units were coded by a certified FACS coder. For this investigation, 236 image sequences from 24 subjects were processed. Of these, 99 image sequences contain only individual upper face AUs, and 137 image sequences contain upper-face AU combinations. Training and testing are performed on the initial and final two frames in each image sequence. For some of the image sequences, lighting normalizations were performed. To test our algorithm on the individual AUs, we randomly generate training and testing sets from the image sequences, as shown in Table 6. In T rainS3 and TestS3, we ensure that the subjects do not appear in both training and testing sets. To test our algorithm on the both individuall AUs and AU combinations, we generate a training set (T rainC1) and a testing set (T estC1) as shown in Table 6. Table 5. Basic lower face action units or AU combination The infraorbital The infraorbital The lips and the triangle and triangle is lower portion of center of the pushed upwards. the nasolabial upper lip are Upper lip is furrow are pulled pulled upwards. raised. Nose pulled back Nose wrinkling wrinkle is absent. laterally. The is present. mouth is elongated. The corner of The chin boss Lip corners are the lips are is pushed pulled obliquely. pulled down. upwards. AU 25 AU 26 AU27 Lips are relaxed Lips are relaxed Mouth stretched, and parted. and parted; open and the mandible is mandible pulled lowered. downwards. AU 23+24 neutral Lips tightened, Lips relaxed narrowed, and and closed. pressed together. Table 6. Data distribution of each data set for upper face AU recognition. Single AU Data Sets AUs AU0 AU1 AU2 AU4 AU5 AU6 AU7 Total TrainS1 TrainS2 76 20 28 228 TrainS3 52 AU Combination Data Sets 6.2. Image Database for Lower Face AU Recognition We use the data of Pitt-CMU AU-Coded Face Expression Image Database for lower face AU recogni- tion. The database currently includes 1917 image sequences from 182 adult subjects of varying ethnicity, performing multiple tokens of 29 of primary FACS action units. Subjects sat directly in front of the camera and performed a series of facial expressions that included single action units (e.g., AU 12, or smile) and combinations of action units (e.g., AU 6+12+25). Each expression sequence began from a neutral face. For each sequence, action units were coded by a certified FACS coder. Total 463 image sequences from 122 adults (65% female, 35% male, 85% European-American, 15% African-American or Asian, ages to 35 years) are processed for lower face action unit recognition. Some of the image sequences are with more action unit combinations such as AU9+17, AU10+17, AU12+25, AU15+17+23, AU9+17+23+24, and AU17+20+26. For each image sequence, we use the neutral frame and two peak frames. 400 image sequences are used as training data and 63 different image sequences are used as test data. The training and testing data sets are shown in Table 7. Table 7. Training data set for lower face AU recognition neutral AU9 AU10 AU12 AU15 AU17 AU20 AU25 AU26 AU27 AU23+24 Total Train Set 400 38 Test 7. Face Action Units Recognition 7.1. Upper Face Action Units Recognition We used three-layer neural networks with one hidden layer. The inputs of the neural networks are the parameters shown in Table 2. Three separate neural networks were evaluated. For comparison with Bartlett's results, the first NN is for recognizing individual AUs only. The second NN is for recognizing AU combinations when only modeling 7 individual upper face AUs. The third NN is for recognizing AU combinations when separately modeling nonadditive AU combinations. The desired number of hidden units to achieve a good recognition was also investigated. 7.1.1 Upper Face Individual AU Recognition The NN outputs are 7 individual upper face AUs. Each output unit gives an estimate of the probability of the input image consisting of the associated action units. From experiments, we have found 6 hidden units are sufficient. In order to recognize individual action units, we used the training and testing data that include individual AUs only. Table 8 shows results of our NN on the T rainS1, TestS1 training and testing sets. A 92.3% recognition rate was obtained. When we increase the training data by using T rainS2 and test by using TestS2, a 92% recognition rate was obtained. For detecting the system's robustness to new faces, we tested our algorithm on the T rainS3/TestS3 training/testing sets. The recognition results are shown in Table 9. The average recognition rate is 92.9% with zero false alarms. For the misidentifications between AUs, although the probability of the output units of the labeled AU is very close to the highest probability, it was treated as an incorrect result. For Table 8. AU recognition for single AUs on T rainS1 and TestS1. The rows correspond to NN outputs, and columns correspond to human labels. Average Recognition rate: 92.3% example, if we obtain the probability of AU1 and AU2 with AU1=0.59 and AU2=0.55 for a labeled AU2, it means that AU2 was misidentified as AU1. When we tested the NN trained on single AU image sequences on data set containing AU combinations, we found the recognition rate decreases to 78.7%. Table 9. AU recognition for single AUs when all test data come from new subjects who were not used for training. Average Recognition rate: 92.9% 7.1.2 Upper Face AU Combination Recognition When Modeling 7 Individual AUs This NN is similar to the one used in the previous section, except that more than one output units could fire. We also restrict the output to be the first 7 individual AUs. For the additive and nonadditive AU combinations, the same value is given for each corresponding individual AUs in training data set. For example, for AU1+2+4, the outputs are AU1=1.0, AU2=1.0, and AU4=1.0. From experiments, we found we need to increase the number of hidden units from 6 to 12. Table shows the results of our NN on the (T rainC1)/(TestC1) training/testing set. A 95% average recognition rate is achieved, with a false alarm rate of 6.4%. The higher false alarm rate comes from the AU combination. For example, if we obtained the recognition results with AU1=0.59 and AU2=0.55 for a labeled AU2, it was treated as AU1+AU2. This means AU2 is recognized but with AU1 as a false alarm. Table 10. AU recognition for AU combinations when modeling 7 single AUs only. AU No. Correct false Missed Confused Recognition rate Total 94 False alarm: 6.4% 7.1.3 Upper Face AU Combination Recognition When Modeling Nonadditive Combinations For this NN,we separately model the nonadditive AU combinations. The 11 outputs consist of 7 individual upper face AUs and 4 non-additive AU combinations (AU1+2, AU1+4, AU4+5, and AU1+2+4). The non-additive AU combinations and the corresponding individual AUs strongly depend on each other. Table 11 shows the correlations between AU1, AU2, AU4, AU5, AU1+2, AU1+2+4, AU1+4, and AU4+5 used in the training set. We set the values based on the appearances of these AUs or combinations. Table 12 shows the results of our NN on the (T rainC1)/(T estC1) training/testing set. An average recognition rate of 93.7% is achieved, with a slightly lower false alarm rate of 4.5%. In this case, modeling separately the nonadditive combinations does not improve recognition rate due to the fact that Table 11. The correlation of AU1, AU2, AU4, AU5, AU1+2, AU1+2+4, AU1+4, and AU4+5. the AUs in these combinations strongly depend on each other. Table 12. AU recognition for AU combinations by modeling the non-additive AU combinations as separate AUs. AU No. Correct false Missed Confused Recognition rate Total 111 104 5 7 - 93.7% False alarm: 4.5% 7.2. Lower Face Action Units Recognition We used a three-layer neural network with one hidden layer to recognize the lower face action units. The inputs of the neural network are the lower face feature parameters shown in Table 3. 7 parameters are used except two parameters of the nasolabial furrows. We don't use the angles of the nasolabial furrows because they are varied much for the different subjects. Generally, we use them to analyze the different expressions of same subject. Two separate neural networks are trained for lower face AU recognition. The outputs of the first NN ignore the nonadditive combinations and only models 11 basic single action units which are shown in Table 5. We use AU 23+24 instead of AU23 and AU24 because they almost occur together. The outputs of the second one separately models some nonadditive combinations such as AU9+17 and AU10+17 besides the basic single action units. The recognition results for modeling basic lower face AUs only are shown in Table 13 with recognition rate of 96.3%. The recognition results for modeling non-additive AU combinations are shown in Table 14 with average recognition rate of 96.71%. We found that separately model the nonadditive combinations slightly increase lower action unit recognition accuracy. All the misidentifications come from AU10, AU17, and AU26. All the mistakes of AU26 are confused by AU25. It is reasonable because both AU25 and AU26 are with parted lips. But for AU26, the mandible is lowered. We did not use the jaw motion information in current system. All the mistakes of AU10 and AU17 are caused by the image sequences with AU combination AU10+17. Two combinations AU10+17 are classified to AU10+12. One combination of AU10+17 is classified as AU10 (missing AU17). The combination AU 10+17 modified the single AU's appearance. The neural network needs to learn the modification by more training data of AU 10+17. There are only ten examples of AU10+17 in 1220 training data in our current system. More data about AU10+17 is collecting for future training. Our system is able to identify action units regardless of whether they occurred singly or in combinations. Our system is trained with the large number of subjects, which included African-Americans and Asians in addition to European-Americans, thus providing a sufficient test of how well the initial training analyses generalized to new image sequences. For evaluating the necessity of including the nonadditive combinations, we also train a neural network using 11 basic lower face action units as the outputs. For the same test data set, the average recognition rate is 96.3%. 8. Conclusion and Discussion We developed a feature-based facial expression recognition system to recognize both individual AUs and AU combinations. To localize the subtle changes in the appearance of facial features, we developed a multi-state method of tracking facial features that uses convergent methods of feature analysis. It has Table 13. Lower face action unit recognition results for modeling basic lower face AUs only. AU No. Correct false Missed Confused Recognition rate 9 Total Table 14. Lower face action unit recognition results for modeling non-additive AU combinations. AU No. Correct false Missed Confused Recognition rate 9 26 14 9 - 5 (AU25) 64.29% Total high sensitivity and specificity for subtle differences in facial expressions. All the facial features are represented in a group of feature parameters. The network was able to learn the correlations between facial feature parameter patterns and specific action units. Although often correlated, these effects of muscle contraction potentially provide unique information about facial expression. Action units 9 and 10 in FACS, for instance, are closely related expressions of disgust that are produced by variant regions of the same muscle. The shape of the nasolabial furrow and the state of nose wrinkles distinguishe between them. Changes in the appearance of facial features also can affect the reliability of measurements of pixel motion in the face image. Closing of the lips or blinking of the eyes produces occlusion, which can confound optical flow estimation. Unless information about both motion and feature appearance are considered, accuracy of facial expression analysis and, in particular, sensitivity to subtle differences in expression may be impaired. A recognition rate of 95% was achieved for seven basic upper face AUs. Eleven basic lower face action units are recognized and 96.71% of action units were correctly classified. Unlike previous methods [9] which build a separate model for each AU and AU combination, we build a single model that recognizes AUs whether they occur singly or in combinations. This is an important capability since the number of possible AU combinations is too large (over 7000) for each combination to be modeled separately. Using the same database, Bartlett et al. [1] recognized only 6 single upper face action units but no combinations. The performance of their feature-based classifier on novel faces was 57%; on new images of a face used for training, the rate was 85.3%. After they combined holistic spatial analysis, feature measures and optical flow, they obtained their best performance at 90.9% correct. Compared to their system, our feature-based classifier obtained a higher performance rate about 92.5% on both novel faces and new images of a face used for training for individual AU recognition. Moreover, our system works well for a more difficult case in which AUs occur either individually or in additive and nonadditive AU combinations. 95% of upper face AUs or AU combinations are correctly classified regardless of whether these action units occur singly or in combination. Those disagreements that did occur were from nonadditive AU combinations such as AU1+2, AU1+4, AU1+2+4, AU4+5, and AU6+7. As a result, more analysis of the nonadditive AU combinations should be done in future. From the experimental results, we have the following observations: 1. The recognition performance from facial feature measurements is comparable to holistic analysis and Gabor wavelet representation for AU recognition. 2. 5 to 7 hidden units are sufficient to code 7 individual upper face AUs. 10 to 16 hidden units are needed when AUs may occur either singly or in complex combinations. 3. For upper face AU recognition, separately modeling nonadditive AU combinations affords no increase in the recognition accuracy. In contrast, separately modeling nonadditive AU combinations affords slightly increase in the recognition accuracy for lower face AU recognition. 4. After using sufficient data to train the NN, recognition accuracy is stable for recognizing AUs of new faces. In summary, the face image analysis system demonstrated concurrent validity with manual FACS coding. The multi-state model based convergent-measures approach was proved to capture the subtle changes of facial features. In the test set, which included subjects of mixed ethnicity, average recognition accuracy for 11 basic action units in the lower face was 96.71%, for 7 basic action units in the upper face was 95%, regardless of these action units occur singly or in combinations. This is comparable to the level of inter-observer agreement achieved in manual FACS coding and represents advancement over the existing computer-vision systems that can recognize only a small set of prototypic expressions that vary in many facial regions. Acknowledgements The authors would like to thank Paul Ekman, Human Interaction Laboratory, University of California, San Francisco, for providing the database. The authors also thank Zara Ambadar, Bethany Peters, and Michelle Lemenager for processing the images. This work is supported by NIMH grant R01 MH51435. --R Measuring facial expressions by computer image analysis. Facial expression in hollywood's portrayal of emotion. Classifying facial actions. 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Shih, Rapid and Brief Communication: Recognizing facial action units using independent component analysis and support vector machine, Pattern Recognition, v.39 n.9, p.1795-1798, September, 2006 Jia-Jun Wong , Siu-Yeung Cho, Facial emotion recognition by adaptive processing of tree structures, Proceedings of the 2006 ACM symposium on Applied computing, April 23-27, 2006, Dijon, France Robust feature detection for facial expression recognition, Journal on Image and Video Processing, v.2007 n.2, p.5-5, August 2007 Jiatao Song , Zheru Chi , Jilin Liu, A robust eye detection method using combined binary edge and intensity information, Pattern Recognition, v.39 n.6, p.1110-1125, June, 2006 Seong G. Kong , Jingu Heo , Besma R. Abidi , Joonki Paik , Mongi A. Abidi, Recent advances in visual and infrared face recognition: a review, Computer Vision and Image Understanding, v.97 n.1, p.103-135, January 2005 Matthew Turk, Computer vision in the interface, Communications of the ACM, v.47 n.1, January 2004 Alice J. O'Toole , Joshua Harms , Sarah L. Snow , Dawn R. Hurst , Matthew R. Pappas , Janet H. Ayyad , Herve Abdi, A Video Database of Moving Faces and People, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.27 n.5, p.812-816, May 2005 Hatice Gunes , Massimo Piccardi , Tony Jan, Face and body gesture recognition for a vision-based multimodal analyzer, Proceedings of the Pan-Sydney area workshop on Visual information processing, p.19-28, June 01, 2004 Mu-Chun Su , Yi-Jwu Hsieh , De-Yuan Huang, A simple approach to facial expression recognition, Proceedings of the 2007 annual Conference on International Conference on Computer Engineering and Applications, p.456-461, January 17-19, 2007, Gold Coast, Queensland, Australia Dong Liang , Jie Yang , Zhonglong Zheng , Yuchou Chang, A facial expression recognition system based on supervised locally linear embedding, Pattern Recognition Letters, v.26 n.15, p.2374-2389, November 2005 Congyong Su , Li Huang, Spatio-temporal graphical-model-based multiple facial feature tracking, EURASIP Journal on Applied Signal Processing, v.2005 n.1, p.2091-2100, 1 January 2005 Yongmian Zhang , Qiang Ji, Active and Dynamic Information Fusion for Facial Expression Understanding from Image Sequences, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.27 n.5, p.699-714, May 2005 Shyi-Chyi Cheng , Ming-Yao Chen , Hong-Yi Chang , Tzu-Chuan Chou, Semantic-based facial expression recognition using analytical hierarchy process, Expert Systems with Applications: An International Journal, v.33 n.1, p.86-95, July, 2007 Philipp Michel , Rana El Kaliouby, Real time facial expression recognition in video using support vector machines, Proceedings of the 5th international conference on Multimodal interfaces, November 05-07, 2003, Vancouver, British Columbia, Canada Benjamn Hernndez , Gustavo Olague , Riad Hammoud , Leonardo Trujillo , Eva Romero, Visual learning of texture descriptors for facial expression recognition in thermal imagery, Computer Vision and Image Understanding, v.106 n.2-3, p.258-269, May, 2007 Iain Matthews , Jing Xiao , Simon Baker, 2D vs. 3D Deformable Face Models: Representational Power, Construction, and Real-Time Fitting, International Journal of Computer Vision, v.75 n.1, p.93-113, October 2007 Yan Tong , Yang Wang , Zhiwei Zhu , Qiang Ji, Robust facial feature tracking under varying face pose and facial expression, Pattern Recognition, v.40 n.11, p.3195-3208, November, 2007 Maria Shugrina , Margrit Betke , John Collomosse, Empathic painting: interactive stylization through observed emotional state, Proceedings of the 4th international symposium on Non-photorealistic animation and rendering, June 05-07, 2006, Annecy, France Haisong Gu , Yongmian Zhang , Qiang Ji, Task oriented facial behavior recognition with selective sensing, Computer Vision and Image Understanding, v.100 n.3, p.385-415, December 2005 Yanxi Liu , Karen L. Schmidt , Jeffrey F. Cohn , Sinjini Mitra, Facial asymmetry quantification for expression invariant human identification, Computer Vision and Image Understanding, v.91 n.1-2, p.138-159, July Tao Xiang , Shaogang Gong, Model Selection for Unsupervised Learning of Visual Context, International Journal of Computer Vision, v.69 n.2, p.181-201, August 2006 Chuang , Christoph Bregler, Mood swings: expressive speech animation, ACM Transactions on Graphics (TOG), v.24 n.2, p.331-347, April 2005 Zhang , Zicheng Liu , Dennis Adler , Michael F. Cohen , Erik Hanson , Ying Shan, Robust and Rapid Generation of Animated Faces from Video Images: A Model-Based Modeling Approach, International Journal of Computer Vision, v.58 n.2, p.93-119, July 2004 R. W. Picard , S. Papert , W. Bender , B. Blumberg , C. Breazeal , D. Cavallo , T. Machover , M. Resnick , D. Roy , C. Strohecker, Affective Learning A Manifesto, BT Technology Journal, v.22 n.4, p.253-269, October 2004 Aleix M. Martnez, Recognizing Imprecisely Localized, Partially Occluded, and Expression Variant Faces from a Single Sample per Class, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.24 n.6, p.748-763, June 2002 Scott Brave , Clifford Nass, Emotion in human-computer interaction, The human-computer interaction handbook: fundamentals, evolving technologies and emerging applications, Lawrence Erlbaum Associates, Inc., Mahwah, NJ, 2002

neural network;facial expression analysis;action units;multistate face and facial component models;computer vision;facial action coding system;AU combinations

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Scale effects in steering law tasks.

Interaction tasks on a computer screen can technically be scaled to a much larger or much smaller sized input control area by adjusting the input device's control gain or the control-display (C-D) ratio. However, human performance as a function of movement scale is not a well concluded topic. This study introduces a new task paradigm to study the scale effect in the framework of the steering law. The results confirmed a U-shaped performance-scale function and rejected straight-line or no-effect hypotheses in the literature. We found a significant scale effect in path steering performance, although its impact was less than that of the steering law's index of difficulty. We analyzed the scale effects in two plausible causes: movement joints shift and motor precision limitation. The theoretical implications of the scale effects to the validity of the steering law, and the practical implications of input device size and zooming functions are discussed in the paper.

INTRODUCTION This research addresses the following questions: Can we successfully accomplish the two steering tasks in Figure 1 in the same amount of time? Can a large input device be substituted with a small one without significantly impacting user performance? Does size matter to input control quality? Can a small-sized input area be compensated by higher control gain (i.e. control-display ratio)? What are the scale effects in movement control, if any? How sensitive are the scale effects? There are many practical reasons to ask these questions. One concerns the miniaturization of the computing devices. We are indeed stepping into the long-awaited era of inexpensive, powerful and portable computers. In the rush towards minia- turization, input devices are expected to adapt to the system physical constraints: trackballs now come in a much smaller Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. SIGCHI'01, March 31-April 4, 2001, Seattle, WA, USA. Figure 1: The two circular tunnels are equivalent in steering law difficulty but they differ in movement scale. Does it take the same amount of time to steer through the two tunnels? diameter than before and touchpads are designed with a fairly small contact surface, for instance. It is not clear whether these reduced-size input devices still maintain the same level of performance as their predecessors. If we push the question of scale to the extreme, the answer is obvious: of course size matters. Humans can not do well in movement scales that are either greater than their arm's reach or smaller than their absolute motor precision toler- ance. Within these extremes, however, the question is much more difficult to answer. RELATED WORK AND LITERATURE One might imagine that the scale effects in input control should be a well documented topic in the human-machine system literature. In reality, however, the results were scattered and controversial. The scale effects were often studied and reported under two related concepts: control gain and control-display (CD) ratio. When the display (output) size is fixed, these two concepts correspond to control movement scale. Major handbooks [5, 6, 15] tend to suggest that human performance is an inverted U-shaped function of control gain or CD ratio: it reaches the highest point in a medium range of the control gain and deteriorates in both directions away from this range. Such a U function was usually found in studies that involved control systems with higher order dynamics (e.g. rate control system, or systems with inertia or lag). Hess [11] is a common source regarding the U-shaped func- tion. In his experiment, subjects performed tracking tasks by manipulating a near-isometric joystick in rate control. A U-shaped function was found between participants' subjective rating and the system control gain 1 . Gibbs [9] provided the most comprehensive set of data on control gain. He studied control gain in both positional and rate control systems and found that the target acquisition time follows the function: 0:02G 0:106 0:003 G (1) where G is the control gain and L is the system lag. The function produced U-shaped curve when L was greater than zero. When L was zero (no system lag), the performance- gain function produces a straight line - the greater the gain was (which means the smaller the movement scale was), the worse the performance was. Buck [7] called into question the views on the significance of CD gain. Based on results from a target alignment exper- iment, he argued that target width on both the control device and the display were important, but their ratio was not. Arnaut and Greenstein [3] conducted a rather convoluted study in which control input magnitude (movement scale), display output magnitude, display target width, control target width and Fitts' index of difficult were varied in two experi- ments. They found that a greater movement scale increased gross movement time but decreased fine adjustment time. Gross and fine movements were defined by the initial entry point in the target. The total completion time, in the case of a tablet, was a U-shaped function. In the case of a track- ball, however, the greater movement scale increased the total completion time monotonically. They concluded that a combination of gain and Fitts' index of difficulty could be a more useful predictor than either of them alone. Jellinek and Card studied users' performance as a function of the control gain in a computer mouse [12]. They found a U-shaped performance-gain function, but argued against its status as a basic human performance characteristic. They believed the performance loss in case of a large control gain was due to the loss of relative measurement resolution (i.e. a quantization effect). If there were not a resolution limit, and as long as the control gain was in a "moderate" range, a user's performance should have stayed constant, so they argued. It is necessary to clarify that the most basic construct, in our view, should be the "control movement scale". Other vari- ables, such as control gain and CD ratio, are derivative or sec- ondary. We think the concept of C-D ratio (or gain) in itself is partially responsible for the contradictions in the literature. By definition, C-D ratio is a compound variable between the display scale and the control movement scale. The same relative C-D ratio could have very different implications on input 1 Note that the notion of control gain is related but not always interchangeable with movement scale or C-D ratio. Control gain, a term originated in feedback control theory, exists in both zero order (position control) and higher order systems. Control display ratio and movement scale only exist in position control systems. For example, since a force joystick was used, there was no control movement scale per se in Hess' study [11]. control, depending on the display size. Control movement scale, on the other hand, is absolute and can be compared to human body measurements. Furthermore, between the display and the control movement scale, the former is more relevant to perception and the latter is more directly relevant to control performance. One implication of movement scale is the limb segments (or motor joints) involved in executing a task. Although limb segments rarely work in isolation, a large movement (e.g. m), tends to be carried out primarily by the arm (shoulder and elbow joints), a medium range by the hand (wrist joint), a small range (e.g. 10 mm) by the fingers. Langolf et al. [13] demonstrated that Fitts' law gave different slopes and intercepts in finger, wrist, and forearm scales 2 . They came to the conclusion that the smaller the scale, the greater the aiming performance, which in terms of primary limb segments means that: fingers > wrist > forearm (2) This performance order was confirmed by Balakrishnan et al. [4], who found that a combined use of multiple fingers resulted in higher performance than other limb segments. However they noted that "the finger(s) do not necessarily perform better than the other segments of the upper limb" when a single finger was involved 3 . In a six degrees of freedom docking task, Zhai et al. [16] showed that relative performance of 6-DOF devices did depend on the muscle groups used. More specifically, they demonstrated that the user performance was superior with the fingers involved (together with the wrist and the arm) in operating the control device than without (wrist and arm only). It is natural to ask the question of scale in light of the well-known Fitts' law [8, 14]), which predicts that the time T to select a target of width W that lies at distance A is: A where a and b are empirically determined constants. The logarithmic transformation of the ratio between A and W is called the index of difficulty of the task. Some researchers argue that control scale should not matter in view of Fitts' law [12]. If a reaching task is scaled by a factor of two, both the distance A and the width W will be twice as large and hence cancel each other in the index of difficulty measure. On the other hand, the impact of scale could be reflected in a and b, as shown in [13]. The validity of index of difficulty as the sole determinant of aimed movement has been recently called into question by Guiard [10]. He argued that the way Fitts' law was studied and applied in the past was problematic; both difficulty and scale should be viewed as the basic dimensions of aimed movement. Some objections have been raised about this study, suggesting a faulty experimental design [4]. But the finding that the index of performance varies with movement scale is widely accepted. 3 Note that in the Balakrishnan study, however, the finger movement was controlled in the lateral direction, which does not occur frequently in natural movement We recently established a movement law that models human performance in a different class of tasks: trajectory-based tunnel steering [1]. It is both theoretically and practically necessary to study the scale effects in relation to the steering law. Theoretically, it is important to investigate how the steering law prediction is affected by movement scale. Prac- tically, the steering law may serve as a platform based on a new class of tasks for studying the control movement scale effects, which may guide the design and selection of inter-action devices and techniques. Some input devices, such as tablet, are primarily designed for trajectory-based tasks. To move a stylus tip or a cursor through a tunnel or path (see Figure 1 for examples) without crossing the boundaries is a steering task. One common steering task in HCI is traversing multi-layered menus. In a recent study [1], we proposed and validated a theoretical model for the successful completion of steering tasks. This model, called the steering law, comes in both an integral and a local form. Integral form The integral form of the steering law states that the difficulty for steering through a generic tunnel C can be determined by integrating the inverse of the path width along the tunnel (see [1] for details). Formally, we define the index of difficulty ID C for steering though C by: Z ds where the integration variable s stands for the curvilinear abscissa along the path. As in Fitts' law, the steering task difficulty ID C predicts the time T needed to steer through tunnel C in a simple linear form: where a and b are constants. Finally, by analogy to Fitts' law, we define the index of performance IP in a steering task by This quantity is usually used for comparing steering performance between experimental conditions. The steering law also has a local formulation, which states that the instantaneous speed at any point in a steering movement is proportional to the permitted variability at that point: where v(s) is the velocity of the limb at the point of curvilinear abscissa s, W (s) is the width of the path at the same point, and is an empirically determined time constant. Types of tunnels Equation 4 allows the calculation of steering difficulty for a wide range of tunnel shapes. In [1], three shapes were tested: straight, narrowing and spiral tunnels. It was suggested [2] that the properties of a great variety of tunnel shapes could be captured by two common tunnel shapes: a linear tunnel and a circular one (Figure 2). For both of the two steering tasks, the steering law can be reduced to the following simple form: where A is the tunnel length in the case of linear tunnels and the perimeter of the center circle in the case of circular tunnels (Figure 2). In both cases W stands for the path width. a and b are experimentally determined constants. They were found to be different for linear and circular tunnels, due to the very different nature of steering in the two cases. PSfrag replacements A (a) Linear tunnel PSfrag replacements A (b) Circular tunnel Figure 2: Two steering tasks Influence of scale One will notice from Equation 7 that the argument against scale effects based on Fitts' law can also be found here: in both the linear circular path, the steering difficulty depends on the length/width ratio, such that dividing both the length and the width of the steering path by a factor k gives the same index of difficulty. In other words, although the speed in a tunnel of width W k will be k times slower than in a tunnel of width W , this decrease in speed should be fully compensated by the shortened steering length by the same ratio, such that the movement time remains the same. It is thus pertinent to ask whether the steering law still holds over very different scales and, if not, how significant the scale impact is. The experimental task was steering through linear and circular tunnels at five different scales. The scales were chosen to cover a broad range of movement amplitudes so as to guarantee that different combinations of motor joints were tested. The input device used in the experiment was a graphics tablet, which, in comparison to other input devices, provided the most direct interaction, hence allowing us to focus on more fundamental human performance characteris- tics. Depending on the movement scale the movement of a tablet stylus may be controlled by the fingers, the wrist, or the arm joints. When operating the stylus, multiple fingers work in conjunction, which should be much better than a single finger working in isolation [4]. Ten volunteers participated in the experiment. All were right-handed and had no or little experience using graphics tablets. Apparatus The experiment was conducted on a PC running Linux, with a 24-inch GW900 Sony monitor (19201200 pixels resolu- tion), and equipped with a Wacom UD-1218E tablet (455 303 mm active area, 12761277 dpi resolution). The computer system was sufficiently fast that the input or feedback lag was not perceptible. The size of the active view of the monitor was set exactly equal to the size of the active area of the tablet, which gives an approximate 107100 dpi screen resolution. Different portions of the tablet area were mapped onto the screen depending on the movement scale currently being tested (mappings are detailed in the design section). All experiments were done in full-screen mode, with the background color set to black. Procedure Subjects performed two types of steering tasks: linear tunnel and circular tunnel steering (Figure 2). At the beginning of each trial, the path to be steered was presented on the screen, in green color. After placing the stylus on the tablet (to the left of the start segment) and applying pressure to the stylus tip, the subject began to draw a blue line on a screen, showing the stylus trajectory. When the cursor crossed the start segment, left to right, the line turned red, as a signal that the task had begun and the time was being recorded. When the cursor crossed the end segment, also left to right, all drawings turned yellow, signaling the end of the trial. Crossing the borders of the path resulted in the cancellation of the trial and an error being recorded. Releasing pressure on the stylus after crossing the start segment and before crossing the second, but without crossing the tunnel border, resulted in an invalid trial, but no error was recorded 4 . Subjects were asked to minimize errors. Finally, linear tunnels were all oriented horizontally and were to be steered left to right; as for circular steering, it had to be done clockwise. Design A fully-crossed, within-subject factorial design with repeated measures was used. Independent variables were movement scale detailed below), test phase first and second block), task type linear and circular tunnels), tunnel length width on the screen pixels). The tunnel lengths and widths define 6 different IDs, ranging from 4 to 33. The order of testing of the five scales (S conditions) was balanced between five groups of subjects according to a Latin square pattern. Within each S condition, subjects performed a practice session, consisting of 1 trial in each of the 6 ID conditions, in both linear and circular steering. The practice session was followed by two identical sets of the 12 T -A- conditions presented in random order, during which data was actually collected. Subjects performed 3 trials in each S-P -T -A-W condition. The five scales were chosen considering the maximummove- ment amplitude for each arm segment and in order to cover the maximum number of motor "strategies". They were: very large scale (S =1): the whole active area of the tablet (455303 mm) was used, which corresponded to standard 4 Subjects sometimes released the pressure by mistake, but this could not be attributed to the constraints imposed on movement variability. A3 format. This scale involves movement amplitudes typically around 20 cm, which require mainly forearm movements large scale (S =2): the active tablet area was 227151 mm, which was one half of the tablet in both dimensions. This was equivalent to a A5-sized tablet. In this scale, movement amplitudes are typically around 10 cm, which require mainly wrist movements but involve to a certain extent the use of the forearm. medium scale (S =4): with an active area of 11476mm of the tablet), movement amplitudes in this scale condition are around 5 cm, which require mainly finger and wrist movements and prevent the use of the forearm. This scale was somewhat equivalent to a A6-format tablet. small scale the tablet active area size was 57 of the tablet). Typical movement amplitudes in that condition are ' 2 cm, which require finger movements and to some extent wrist movements. This was the size of a touchpad used in some notebook computers. very small scale active area of the tablet ( 1 =16 of the tablet) implied very small movements amplitudes, around 1 cm, which require finger movements exclusively, with the wrist and forearm joints stabilized on the tablet surface. Note that this smallest scale tested was still orders of magnitude above the tablet resolution, hence preventing the possible machinery quantization effect in previous studies. Figure 3 illustrates the relative size of active areas of the tablet for the different movement scales. The outermost box, labeled S=1, corresponds to the whole tablet active area. PSfrag replacements Figure 3: Relative active tablet sizes at different scales Table 1 shows the movement amplitudes and path widths in input space for each scale condition: for instance, the tunnel to be steered on the graphics tablet when S =4, W=60 and A=250 has a width of 5 mm and a length of 14:8 mm. Finally, in light of the movement scale vs. C-D ratio and display scale discussion, the visual stimuli were kept in the same size over all five movement scales, so that no visual perception effect could influence the results. The experimental software was identical for all scales; only the tablet scale settings were changed. Table 1: Movement amplitudes and path widths in input space for each scale condition (in millimeters). The results of the experiment include steering time, steering speed and error rates. Steering time As expected, movement amplitude and tunnel width significantly influenced steering time and F there was also a strong interaction between movement amplitude and tunnel width which is consistent with the fact that the steering time depends on the ratio of amplitude and width. As in [2], steering type (linear vs. circular) proved to be also a significant factor influencing steering time As for the studied variable, movement scale, it had a significant influence on steering performance While the significant impact of test phase shows a strong learning effect, the non-significant interaction between test phase and movement scale suggests that the influence of scale is likely not to vary much with practice. Paired t-tests between scale levels classified the scales into three groups. The first group includes scales 2 and 4, the second includes scales 1 and 8, and the last one is only composed of scale 16. The differences were insignificant between the two scales of the first group (p > :08) and the scales of the second group (p > :31). The scales of the first group outperformed significantly the scales of the second group (with p< :0001 for all compared pairs), while the last group is outperformed by both the first group (p< :0001) and the second one (p< :0001). The ranking between movement scales in terms of time performance is: This grouping of scales and the ranking between groups held in both linear and circular steering. Figure 4 summarizes the average steering time depending on the movement scale and steering task. There was also a strong interaction between movement scale and tunnel type which suggests that the circular tasks were more sensitive to changes in movement scale than the linear ones (see Figure 4). We also found a significant interaction between scale and amplitude 3:2, p< :01), as well as between scale and width :0001). The movement scale effects was greater when the movement amplitude was greater or the tunnel width was Scale Steering time linear tunnel circular tunnel Figure 4: Steering time as a function of scale smaller (Figure 5). This was especially true for the scale-16 conditions: very long amplitudes or very narrow tunnels in this case were very difficult to steer, such that often subjects could achieve the trial only after a couple of attempts (see error rates analysis further). However, a significant interaction between test phase and movement amplitude p< :001) suggests that subjects tend to deal with long amplitudes much better with practice. As for the fitness to the steering law model, the integral form of the steering law [1] proved to hold at all studied scales with very good regression fitness (see Figure 6). The models of steering time were, for linear steering (in ms): and for circular steering: The slope of linear regressions was significantly influenced by tunnel type movement scale :05). The intercept was not significantly affected by tunnel type or by movement scales were rather small comparing to the total times involved, which is consistent with previous results [1, 2]. Similar to the results found on steering times, the relationship between the steering law index of performance and the movement scale was an (inverted) U-shaped function (Fig- ure 7), with the best performance in the medium scale 2 and Movement amplitude Steering time (ms) scale 1 scale 2 scale 4 scale 8 scale circular steering linear steering (a) Interaction between movement amplitude and movement scale Tunnel width Steering time (ms) scale 1 scale 2 scale 4 scale 8 scale circular steering linear steering (b) Interaction between tunnel width and movement scale Figure 5: Steering time against scale depending on movement amplitude and tunnel width slightly higher than scale 4 in linear steering) and lower performance in scale 1 and 8, and lowest in scale 16. All scales resulted in almost equivalent performance for very easy tasks, but significant differences appeared for very difficult tasks; this was characterized by the statistical interaction between scale and index of difficulty of the tasks In conclusion, the time performance was the highest when the movement scale was between scale 2 and scale 4. In terms of tablet size, this corresponds to A4/A5-sized tablets. In terms of motor joints involved, it was when the wrist and the fingers were the primary movement carriers. A3-sized tablets seemed to be too large and require too much movement efforts, while tablets smaller than A6 format were likely to amplify noise beyond reasonable rates. Steering time (ms) scale 1 scale 2 scale 4 scale 8 scale (a) Linear steering Steering time (ms) scale 1 scale 2 scale 4 scale 8 scale (b) Circular steering Figure Steering time against ID Errors Besides the expected main effects on error rates of movement amplitude scale had a strong influence on error occurrence steering resulted in more errors than linear steering 95, p < :0001). A significant interaction between scale and tunnel type shows that the number of errors increases much faster for circular steering than for linear steering when the movement scale decreases (see Figure 8). Finally interactions between movement scale and width movement scale and movement amplitude that it is more likely to make errors in long or narrow tunnels when the tablet is very small: in the scale-16 con- ditions, natural tremor and biomechanical noise was greatly amplified such that subjects systematically made a few failed trials in a sequence, even though they did their best. scale IP (bits/s)515 linear tunnel circular tunnel Figure 7: Index of performance against scale Scale Average number of linear tunnel circular tunnel Figure 8: Average number of errors in each scale To conclude, the smaller the scale, the higher the error rates. Considering that the optimal scales were 2 and 4 for time performance, it appeared that the scale-2 condition had the best overall performance while considering both time performance and error rate. DISCUSSION AND CONCLUSION Movement scale, control gain, control-display ratio and motor joints performance differences, are a set of related concepts in input control literature without consistent conclu- sions. In terms of movement scale or control gain effects, some researchers found a U-shaped function [11, 3]; others straight linear function [9], and yet others did not believe gain or scale should matter much [12, 7]. Traditionally these issues have been studied in the framework of target acquisition tasks. We have conducted a systematic study on these issues in a new paradigm - the steering law. Furthermore, we focused on the most fundamental concept of them all - control movement scale. Our results supported the U-shaped performance-scale func- tion: scale does matter. The U-shaped function is easily plausible in the two extrema: too large a scale is beyond the arm's reach and too small a scale is beyond motor control precision. But even within the "moderate" range we tested, the U shape was still clearly demonstrated. The cause for the U-shaped function in this range is likely to be twofold: motor joints shift and the human motor precision limitation. The best performance appeared in the middle range (scale 2 and 4), when the movements were carried out by all parts of the upper limb (arm, hand, and finger), although the arm's role might be lesser than the hand and finger. In this range, the steering IP was (in bits=ms) 1=62 and 1=74 for linear steering; and 1=179 and 1=174 for circular steering. On the larger scale side, when the movements were primarily carried out by the arm movement, steering performance dropped to 1=81 for linear and 1=200 for circular steering. On the smaller scale side (scale 8 and 16), when the movements were carried out mostly by fingers, the performance also dropped (to 1=85 and 1=227 for scale 16). However, we can not make a conclusion that the fingers are inferior, because the other factor, motor control precision limitation, was increasingly more limiting as the movement scale de- creased. This was clearly demonstrated by the number of errors (failed attempts to complete the entire steering path) shown in Figure 6: participants increasingly "accidentally" moved out of the tunnel. Note that because we maintained the same visual display size for all scale conditions, the error had to be on the motor precision side. The more theoretical implication of the results pertains the validity of the steering law [1]. Similar to Fitts' law, the steering law states that the difficulty of movement lies in relative accuracy. The two steering tasks in Figure 1 are exactly the same in steering law terms. This study shows that while the fitness of the steering law held very well in all levels tested, the movement scale does have an impact on steering law's index of performance. A steering law model with strictly the same index of performance is only valid if the scale does not vary so widely that the motor joint combination shift fundamentally or the control precision becomes the primary limiting factor. It is interesting to realize that the impact of scale is much less significant than the steering law's index of difficulty. For example, the range of scale we tested varied by a factor of 16, but the largest steering time difference was only 17% - an impact equivalent to only 17% change of the steering ID (either 17% longer or 15% narrower tunnel). There are also many practical implications in our results. For example, the size of a computer input device (tablet or mouse and its pad) should be such that the fingers, wrists and to a lesser extent forearm are all allowed in the operation. Another practical implication lies in the design and use of zooming interface. In fact users often unconsciously make effort to stay at the bottom of the U-shaped scale function by zooming up when their motor precision limits their performance, and zooming down when too much of the movement had to be carried out by large arm movement. How we can deliberately apply the findings in the present study to assist zooming is an interesting future research issue. Based on the results of this study, we can begin to answer some of the questions we raised at the beginning of this paper on a more scientific ground. First, we found that device size and movement scale indeed affects input control quality: people do not accomplish the two steering tasks in Figure 1 in the same amount of time. Furthermore, small scale tends to be limited by motor precision, large scale limited by the arm dexterity, but scale in the medium range does not significantly influence performance. Consequently, substituting a large input device with a small one will not significantly impact user performance if there is no fundamental change in the muscle groups involved; but it will if the substituted input device is too small that human motor control precision becomes a limiting factor. The scale effects are also not very sensitive in comparison to the steering law index of difficulty effects (e.g. change the tunnel width while keeping the same length). Finally, the question with regard to control gain or control-display ration is ill-posed, because what matters is the control movement scale: for the same movement scale, the appropriate control gain depends on the display size. In summary, we have 1) introduced a new task paradigm to study scale effect; 2) contributed to the literature of scale ef- fect, confirming the U-shaped function and rejecting straight-line or no-effect hypotheses; improved the understanding of the steering law; guidelines to practical input design and selection issues. ACKNOWLEDGMENTS We would like to thank Fr-ed-eric Lepied of the XFree86 team for his advice on the Wacom tablet configuration. --R Beyond Fitts' law: models for trajectory Performance evaluation of input devices in trajectory-based tasks: an application of the steering law Is display/control gain a useful metric for optimizing an interface? Performance differences in the fingers Engineering data com- pendium: Human perception and performance Motor performance in relation to control-display gain and target width The information capacity of the human motor system in controlling the amplitude of move- ment Controller design: Interactions of controlling limbs Difficulty and scale as the basic dimensions of aimed movement. Nonadjectival rating scales in human response experiments. Powermice and user per- formance An investigation of Fitts' law using a wide range of movement amplitudes. Fitts' law as a research and design tool in human-computer interaction Human Factors Design Handbook. The influence of muscle groups on performance of multiple degree-of- freedom input --TR Powermice and user performance Is display/control gain a useful metric for optimizing an interface? The influence of muscle groups on performance of multiple degree-of-freedom input Beyond Fitts'' law Performance differences in the fingers, wrist, and forearm in computer input control Performance evaluation of input devices in trajectory-based tasks --CTR Mary Czerwinski, Humans in human-computer interaction, The human-computer interaction handbook: fundamentals, evolving technologies and emerging applications, Lawrence Erlbaum Associates, Inc., Mahwah, NJ, 2002 Robert Pastel, Measuring the difficulty of steering through corners, Proceedings of the SIGCHI conference on Human Factors in computing systems, April 22-27, 2006, Montral, Qubec, Canada Sergey Kulikov , I. Scott MacKenzie , Wolfgang Stuerzlinger, Measuring the effective parameters of steering motions, CHI '05 extended abstracts on Human factors in computing systems, April 02-07, 2005, Portland, OR, USA Yves Guiard , Michel Beaudouin-Lafon, Target acquisition in multiscale electronic worlds, International Journal of Human-Computer Studies, v.61 n.6, p.875-905, December 2004 Raghavendra S. Kattinakere , Tovi Grossman , Sriram Subramanian, Modeling steering within above-the-surface interaction layers, Proceedings of the SIGCHI conference on Human factors in computing systems, April 28-May 03, 2007, San Jose, California, USA Yves Guiard , Renaud Blanch , Michel Beaudouin-Lafon, Object pointing: a complement to bitmap pointing in GUIs, Proceedings of the 2004 conference on Graphics interface, p.9-16, May 17-19, 2004, London, Ontario, Canada Marcelo Mortensen Wanderley , Nicola Orio, Evaluation of Input Devices for Musical Expression: Borrowing Tools from HCI, Computer Music Journal, v.26 n.3, p.62-76, Fall 2002 Tue Haste Andersen, A simple movement time model for scrolling, CHI '05 extended abstracts on Human factors in computing systems, April 02-07, 2005, Portland, OR, USA Ken Hinckley, Input technologies and techniques, The human-computer interaction handbook: fundamentals, evolving technologies and emerging applications, Lawrence Erlbaum Associates, Inc., Mahwah, NJ, 2002 Carl Gutwin , Amy Skopik, Fisheyes are good for large steering tasks, Proceedings of the SIGCHI conference on Human factors in computing systems, April 05-10, 2003, Ft. Lauderdale, Florida, USA David Ahlstrm, Modeling and improving selection in cascading pull-down menus using Fitts' law, the steering law and force fields, Proceedings of the SIGCHI conference on Human factors in computing systems, April 02-07, 2005, Portland, Oregon, USA Taher Amer , Andy Cockburn , Richard Green , Grant Odgers, Evaluating swiftpoint as a mobile device for direct manipulation input, Proceedings of the eight Australasian conference on User interface, p.63-70, January 30-February 02, 2007, Ballarat, Victoria, Australia Shumin Zhai , Johnny Accot , Rogier Woltjer, Human action laws in electronic virtual worlds: an empirical study of path steering performance in VR, Presence: Teleoperators and Virtual Environments, v.13 n.2, p.113-127, April 2004 Xiang Cao , Shumin Zhai, Modeling human performance of pen stroke gestures, Proceedings of the SIGCHI conference on Human factors in computing systems, April 28-May 03, 2007, San Jose, California, USA

input device;movement scale;C-D ratio;joints;elbow;motor control;finger;control gain;steering law;wrist;device size

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Empirically validated web page design metrics.

A quantitative analysis of a large collection of expert-rated web sites reveals that page-level metrics can accurately predict if a site will be highly rated. The analysis also provides empirical evidence that important metrics, including page composition, page formatting, and overall page characteristics, differ among web site categories such as education, community, living, and finance. These results provide an empirical foundation for web site design guidelines and also suggest which metrics can be most important for evaluation via user studies.

INTRODUCTION There is currently much debate about what constitutes good web site design [19, 21]. Many detailed usability guidelines have been developed for both general user interfaces and for web page design [6, 16]. However, designers have historically experienced difficulties following design guidelines [2, 7, 15, 24]. Guidelines are often stated at such a high level that it is unclear how to operationalize them. A typical example can be found in Fleming's book [10] which suggests ten principles of successful navigation design in- cluding: be easily learned, remain consistent, provide feed- back, provide clear visual messages, and support users' goals and behaviors. Fleming also suggests differentiating design among sites intended for community, learning, information, shopping, identity, and entertainment. Although these goals align well with common sense, they are not justified with empirical evidence and are mute on actual implementation. Other web-based guidelines are more straightforward to im- plement. For example, Jakob Nielsen's alertbox column [18] of May 1996 (updated in 1999) claims that the top ten mistakes of web site design include using frames, long pages, non-standard link colors, and overly long download times. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy oth- erwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. SIGCHI'01, March 31-April 4, 2001, Seattle, WA, USA. These are based on anecdotal observational evidence. Another column (March 15, 1997) provides guidelines on how to write for the web, asserting that since users scan web pages rather than read them, web page design should aid scannabil- ity by using headlines, using colored text for emphasis, and using 50% less text (less than what is not stated) since it is more difficult to read on the screen than on paper. Although reasonable, guidelines like these are not usually supported with empirical evidence. Furthermore, there is no general agreement about which web design guidelines are correct. A recent survey of 21 web guidelines found little consistency among them [21]. We suspect this might result from the fact that there is a lack of empirical validation for such guidelines. Surprisingly, no studies have derived web design guidelines directly from web sites that have been assessed by human judges. In this paper we report the results of empirical analyses of the page-level elements on a large collection of expert- reviewed web sites. These metrics concern page composition (e.g., word count, link count, graphic count), page formatting (e.g., emphasized text, text positioning, and text clusters), and overall page characteristics (e.g., page size and down-load speed). The results of this analysis allows us to predict with 65% accuracy if a web page will be assigned a very high or a very low rating by human judges. Even more interest- ingly, if we constrain predictions to be among pages within categories such as education, community, living, and finance, the prediction accuracy increases to 80% on average. The remainder of this paper describes related work, our method- ology, including the judged web dataset, the metrics, and the data collection process; the results of the study in detail, and finally our conclusions. RELATED WORK Most quantitative methods for evaluating web sites focus on statistical analysis of usage patterns in server logs [5, 8, 11, 12, 26, 27]. Traffic-based analysis (e.g., pages-per-visitor or visitors-per-page) and time-based analysis (e.g., click paths and page-view durations) provide data that the evaluator must interpret in order to identify usability problems. This analysis is largely inconclusive since web server logs provide incomplete traces of user behavior, and because timing estimates may be skewed by network latencies. Other approaches assess static HTML according to a number of pre-determined guidelines, such as whether all graphics contain ALT attributes [4, 22]. Other techniques compare quantitative web page measures - such as the number of links or graphics - to thresholds [25, 27, 28]. However, concrete thresholds for a wider class of quantitative web page measures still remain to be established; our work is a first step towards this end. The Design Advisor [9] uses heuristics about the attentional effects of various elements, such as motion, size, images, and color, to determine and superimpose a scanning path on a web page. The author developed heuristics based on empirical results from eye tracking studies of multimedia presen- tations. However, the heuristics have not been validated for web pages. Simulation has also been used for web site evaluation. For example, WebCriteria's Site Profile [29] attempts to mimic a user's information-seeking behavior within a model of an implemented site. This tool uses an idealized user model that follows an explicit, pre-specified navigation path through the site and estimates several metrics, such as page load and optimal navigation times. As another example, Chi, Pirolli, and Pitkow [5] have developed a simulation approach for generating navigation paths for a site based on content similarity among pages, server log data, and linking structure. The simulation models hypothetical users traversing the site from specified start pages, making use of information "scent" (i.e., common keywords between the user's goal and content on linked pages) to make navigation decisions. Neither of these approaches account for the impact of various web page at- tributes, such as the amount of text or layout of links. Brajnik [3] surveyed 11 automated web site analysis meth- ods, including the previously mentioned static analysis tools and WebCriteria's Site Profile. The survey revealed that these tools address only a sparse set of usability features, such as download time, presence of alternative text for images, and validation of HTML and links. Other usability aspects, such as consistency and information organization are unaddressed by existing tools. Zhu and Gauch [30] gathered web site quality ratings criteria from a set of expert sites, including Internet Scout, Lycos Top 5%, Argus Clearinghouse, and the Internet Public Library. For each site they computed web page currency, availability, authority, popularity, cohesiveness, and information-to-noise ratio. This last metric is the only one related to the kind of metrics discussed below, and is computed as the number of bytes taken up by words divided by the total number of bytes in the page; in essense a word percentage measure. The authors assessed these metrics in terms of how well they aided in various information retrieval tasks, finding that weighted combinations of metrics improved search over text content Webby Overall Score9.007.005.003.001.00Number of Figure 1: Histogram of the the overall scores assigned to the sites considered for the 2000 Webby Awards. The x axis is the overall score and the y axis is the number of sites assigned this score. alone. They did not relate these metrics to web site usability or attempt to predict the judges ratings outside the context of search. The most closely related work is our earlier study [13] in which we reported a preliminary analysis of a collection of 428 web pages. Each page corresponded to a site that had either been highly rated by experts, or had no rating. The expertise ratings were derived from a variety of sources, such as PC Magazine Top 100, WiseCat's Top 100, and the final nominees for the Webby Awards. For each web page, we captured 12 quantitative measures having to do with page composition, layout, amount of information, and size (e.g., number of words, links, and colors). We found that 6 metrics cluster count, link count, page size, graphics count, color count and reading complexity - were significantly associated with rated sites. Additionally, we found 2 strong pair-wise correlations for rated sites, and 5 pairwise correlations for unrated sites. Our predictions about how the pairwise correlations were manifested in the layout of the rated and unrated sites' pages were supported by inspection of randomly selected pages. A linear discriminant classifier applied to the page types (rated versus unrated) achieved a predictive accuracy of 63%. The work reported in this paper expands on that preliminary analysis in several ways. First, rather than comparing highly rated sites to unrated sites, we are comparing sites that have been rated on a single scale, and according to several mea- sures, by one set of judges. Second, the sites within this dataset have been classified into topics (such as financial, ed- ucational, community), thus allowing us to see if preferred values for metrics vary according to type of category. Fi- nally, informed by the results of our preliminary study, we have improved our metrics and analyze a larger number of web pages. This work further validates our preliminary analysis This study computes quantitative web page attributes (e.g., number of fonts, images, and words) from web pages evaluated for the 2000 Webby Awards [20]. The Webby organizers place web sites into 27 categories, including news, per- sonal, finance, services, sports, fashion, and travel. A panel of over 100 judges from The International Academy of Digital Arts & Sciences use a rigorous evaluation process to select winning sites. 1 Webby organizers describe the judge selection criteria as follows: "Site Reviewers are Internet professionals who work with and on the Internet. They have clearly demonstrable familiarity with the category in which they review and have been individually required to produce evidence of such expertise. The site reviewers are given different sites in their category for review and they are all prohibited from reviewing any site with which they have any personal or professional affiliation. The Academy regularly inspects the work of each reviewer for fairness and accuracy." Judges rate web sites based on six criteria: content, structure & navigation, visual design, functionality, interactivity, and overall experience. Figure 1 shows the distribution of the overall criterion across all of the judged sites. We suspected that the six criteria were highly correlated, suggesting that there was one factor underlying them all. To test this hypoth- esis, we used a principles component analysis to examine the underlying factor structure. The first factor accounted for 91% of the variance in the six criteria. In the experiments reported below, we used both the overall Webby score and the extracted factor for doing discriminant classification. For our study, we selected sites from six topical categories - financial, educational, community, health, service, and living - because these categories contained at least 100 sites (in which the primary goal is to convey information about some topic). We used the overall score to define two groups of sites for analysis: good (top 33% of sites), and not-good (remaining 67% of sites). Specifically, we wanted to determine if there are significant differences between the groups - both overall and within each category. Furthermore, we wanted to construct models for predicting group membership. These models would enable us to establish concrete thresholds for each metric, evaluate them with user studies, and eventually provide guidance for design im- provement. We also used the composite rating to group sites into two categories: top 33% of sites, and bottom 33% of sites. The cutoffs for both sets, based on the overall criterion (ranging from 1 to 10) are: Webby Awards judging has three rounds. The data used in this study are derived from the first round of judging; only the list of nominees for the last round is available to the public. Throughout this paper, we assume a score assigned to a site applies uniformly to all the pages within that site. Community Education Finance Top 6.97 6.58 6.47 6.6 Bottom 5.47 5.66 5.38 5.8 Health Living Services Top 7.9 6.66 7.54 Bottom 6.4 5.66 5.9 The following section introduces the metrics and describes the data collected for this analysis. Web Page Metrics From a list of 42 web page attributes associated with effective design and usability [13], we developed an automated tool to compute the 11 metrics that we focus on in this study (see Table 1). (This subset was chosen primarily because it was the easiest to compute; we are in the process of extending the tool to compute a wider range of metrics.) The tool functions similarly to the Netscape Navigator browser in processing web pages and cascading stylesheets; it has limited support for inline frames, but does not support framesets, applets, scripts or other embedded objects. We analyzed the accuracy of the computed metrics using a set of 5 pages with widely different features, such as use of stylesheets, style tags, and forms. Overall, the metrics are about 85% accurate with text cluster and text positioning counts range from 38% to 74% accuracy. Data Collection We used the metrics tool to collect data for 1,898 pages from the six Webby Awards categories. These pages are from 163 sites and from 3 different levels in the site - the home page, pages directly accessible from the home page (level 1), and pages accessible from level 1 but not directly accessible from the home page (level 2). We attempted to capture 15 level 1 pages and 45 level 2 pages from each site. Because not every website has many pages at each level, our collection consists of an average of 11 pages per site. We employed several statistical techniques, including linear regression, and linear discriminant analysis, and t-test for equality of means, to examine differences between the good and not-good groups. The following sections discuss the findings in detail. Distinguishing Good Pages We used Linear Discriminant analysis to discriminate good from not-good pages, and top from bottom pages. This technique is suitable for cases where the predicted variable is dichotomous in nature. We built two predictive models for identifying good webpages using linear discriminant analysis . Model 1: A simple, conservative model that distinguishes "good" (top 33%) from "not good" (bottom 67%) websites, using the overall Webby criterion as the predictor. Metric Description Word Count Total words on a page Body Text % Percentage of words that are body vs. display text (i.e., headers) Emphasized Body Text % Portion of body text that is emphasized (e.g., bold, capitalized or near !'s) Text Positioning Count Changes in text position from flush left areas highlighted with color, bordered regions, rules or lists Link Count Total links on a page Page Size Total bytes for the page as well as elements graphics and stylesheets Graphic % Percentage of page bytes that are for graphics Graphics Count Total graphics on a page (not including graphics specified in scripts, applets and objects) Color Count Total colors employed Font Count Total fonts employed (i.e., face bold Table 1: Web page metrics computed for this study. . Model 2: A more complex model that uses the Webby factor and distinguishes "top" (top 33%) from "bottom" (bot- tom 33%) pages. Tables 2 and 3 summarize the accuracy of the predictions for both models for the entire sample as well as within each category. We report the Wilks Lambda along with the associated Chi-square for each of the models; all of the discriminant functions have significant Wilks Lambda. The squared canonical correlation indicates the percentage of variance in the metrics accounted for by the discriminant function. The final and most important test for the model is the classification accuracy. For Model 1, the overall accuracy is 67% (50.4% and 78.4% for good and not-good pages, respectively) if categories are not taken into account (see Table 2). Classification accuracy is higher on average when categories are assessed separately (70.7% for good pages and 77% for not-good pages). Our earlier results [13] achieved 63% overall accuracy but had a smaller sample size, did not have separation into category types, and had to distinguish between rated sites versus non- rated sites, meaning that good sites may have been included among the non-rated sites. Interestingly, the average percentage of variance explained within categories is more than double the variance explained across the dataset. The health category has the highest percentage of variance explained and also has the highest classification accuracy of 89% (80.9% and 94.6% for good and not-good pages, respectively). The accuracy for this model is indicative of the predictive power of this approach. In the future we plan to use more metrics and a larger dataset in our analysis. The model with the smallest percentage of variance explained (20% for the living category) is also the model with the lowest classification accuracy of 55% (47.4% and 62.3% for good and not-good pages, respectively). We partially attribute this lower accuracy to a smaller sample size; there are only 118 pages in this category. The results for Model 2 are shown in Table 3. The average category accuracy increases to 73.8% for predicting the top pages and 86.6% for predicting the bottom pages. (This prediction does not comment on intermediate pages, however.) The higher accuracy is caused both by the relatively larger differences between top and bottom pages (as opposed to top versus the rest) and by the use of the Webby factor. In related work [23] analyzing the Webby Award criteria in detail, we found that the content criterion was the best predictor of the overall score, while visual design was a weak predictor at best. Here we see that the metrics are able to better predict the Webby factor than the overall score. We think this happens because the overall criterion is an abstract judgement of site quality, while the Webby factor (consisting of contributions from content, structure & navigation, visual design, functionality, interactivity, and as well as overall rat- ings) reflects aspects of the specific criteria which are more easily captured by the metrics. The Role of Individual Metrics To gain insight about predictor metrics in these categories, we also employed multiple linear regression analysis to predict the overall Webby scores. We used a backward elimination method wherein all of the metrics are entered into the equation initially, and then one by one, the least predictive metric is eliminated. This process is repeated until the Adjusted R Square shows a significant reduction with the elimination of a predictor. Table 4 shows the details of the analysis. The adjusted R 2 for all of the regression analyses was significant at the .01 level, meaning that the metrics explained about 10% of the variance in the overall score for the whole dataset. This indicates that a linear combination of our metrics could significantly predict the overall score. We used standardized Beta coefficients from regression equations to determine the significance of the metrics in predicting good vs. not-good pages. Table 5 illustrates which of the metrics make significant contributions to predictions as well as the nature of their contributions (positive or negative). Significant metrics across the dataset are fairly consistent with Squared Classification Canonical Wilks Chi- Sample Accuracy Category Correlation Lambda square Sig. Size Good Not-Good Community Education 0.26 0.74 111.52 0.000 373 69.00% 90.20% Finance 0.37 0.63 85.61 0.000 190 63.20% 88.00% Health 0.60 0.4 104.8 0.000 121 94.60% 80.90% Living Services 0.34 0.66 100.6 0.000 311 82.50% 75.80% Cat. Avg. 70.70% 77.00% Table 2: Classification accuracy for predicting good and not-good pages. The overall accuracy ignores category labels. Discriminant analysis rejects some data items. Squared Classification Canonical Wilks Chi- Sample Accuracy Category Correlation Lambda square Sig. Size Top Bottom Community 0.60 0.40 275.78 0.000 305 83.2% 91.9% Education 0.28 0.72 118.93 0.000 368 75.7% 73.2% Finance 0.47 0.53 85.74 0.000 142 76.5% 93.4% Health Living 0.22 0.79 24.46 0.010 106 42.3% 75.9 % Services 0.36 0.64 90.51 0.000 208 85.7% 74.8% Cat. Avg. 76.07% 82.75% Table 3: Classification accuracy for predicting the top 33% versus the bottom 33% according to the Webby factor. The overall accuracy ignores category labels. profiles discussed in the next section; most of the metrics found to be individually significant play a major role in the overall quality of pages. Profiles of Good Pages Word count was significantly correlated with 9 other metrics (all but emphasized body text percentage), so we used it to subdivide the pages into three groups, depending on their size: low (avg. word count = 66.38), medium (avg. word (avg. word count = 827.15). Partitioning pages based on the word count metric created interesting profiles of good versus not-good pages. In addition, the regression score and discriminant analysis classification accuracy increases somewhat when the dataset is divided in this manner; Model 1 is most accurate for pages that fall into the medium-size group. To develop profiles of pages based on overall ratings, we compared the means and standard deviations of all metrics for good and not-good pages with low, medium, and high word counts (see Table 6). We employed t-tests for equality of means to determine their significance and also report 2-tailed significance values. Different metrics were significant among the different size groups, with the exception of graphic percentage, which is significant across all groups. The data suggests that good pages have relatively fewer graph- ics; this is consistent with our previously discussed finding that visual design was a weak predictor of overall rating [23]. Returning to Table 5, we see that in most cases, the positive Adj. R Std. F Sig. Sample Category Square Err. value Size Community 0.36 1.76 22.52 .000 430 Education 0.16 1.53 10.34 .000 536 Finance 0.24 1.90 7.78 .000 234 Health 0.56 0.79 27.98 .000 234 Living 0.11 1.62 2.70 .000 153 Services Table 4: Linear regression results for predicting overall rating for good and not-good pages. The F value and corresponding significance level shows the linear combination of the metrics to be related to the overall rating. or negative contribution of a metric aligns with differences in the means of good vs. bad pages depicted in Table 6, with the exception that page size and link count in the medium word count category appear to have opposite contribution than expected, since in general good pages are smaller and have more links on average than not-good pages. Looking in detail at Tables 5 and 6, we can create profiles of the good pages that fall within low, medium, and high word counts: Low Word Count. Good pages have slightly more content, smaller page sizes, less graphics and employ more font variations than not-good pages. The smaller page sizes and graphics count suggests faster download times for these Word Count Category Metric Low Med. High Com. Edu. Fin. Hlth. Lvng Serv. Freq. Word Count # 4 Body Text % # 3 Emp. Body Text % # 3 Text Pos. Count # 2 Link Count # 2 Page Size # 4 Graphic Graphics Count # 2 Color Count # 3 Font Count # 4 Table 5: Significant beta coefficients for all metrics in terms of whether they make a positive (#), negative (#), or no contribution in predicting good pages. The frequency column summarizes the number of times a metric made significant contributions within the categories. pages (this was corroborated by a download time metric, not discussed in detail here). Correlations between font count and body text suggest that good pages vary fonts used between header and body text. Medium Word Count. Good pages emphasize less of the body text; if too much text is emphasized, the unintended effect occurs of making the unemphasized text stands out more than emphasized text. Based on text positioning and text cluster count, medium-sized good pages appear to organize text into clusters (e.g., lists and shaded table areas). The negative correlations between body text and color count suggests that good medium-sized pages use colors to distinguish headers. High Word Count. Large good pages exhibit a number of differences from not-good pages. Although both groups have comparable word counts, good pages have less body text, suggesting pages have more headers and text links than not-good pages (we verified this with hand-inspection of some pages). As mentioned above, headers are thought to improve scannability, while generous numbers of links can facilitate information seeking provided they are meaningful and clearly marked. DISCUSSION It is quite remarkable that the simple, superficial metrics used in this study are capable of predicting expert's judgements with some degree of accuracy. It may be the case that these computer-accessible metrics reflect at some level the more complex psychological principles by which the Webby judges rate the sites. It turns out that similar results have been found in related problems. For example, one informal study of computer science grant proposals found that superficial features such as font size, inclusion of a summary section and section numbering distinguishes between proposals that are funded and those that are not [1]. As another example, programs can assign grades to student essays using only superficial metrics (such as average word length, essay length, number of commas, number of prepositions) and achieve correlations with teachers' scores that are close to those of between- teacher correlations [14]. There are two logical explanations for this effect; the first is that there is a causal relationship between these metrics and deeper aspects of information architecture. The second possibility is that high quality in superficial attributes is generally accompanied by high quality in all aspects of a work. In other words, those who do a good job do a good job overall. It may be the case that those site developers who have high-quality content are willing to pay for professional designers to develop the other aspects of their sites. Nevertheless, the fact that these metrics can predict a difference between good and not-good sites indicates that there are better and worse ways to arrange the superficial aspects of web pages. By reporting these, we hope to help web site developers who cannot afford to hire professional design firms. There is some question as to whether or not the Webby Awards judgements are good indicators of web site usability, or whether they assess other measures of quality. We have conducted a task-based user study on a small subset of the web sites within our sample, using the WAMMI Usability Questionnaire [17]. We plan to report the results of this study in future work. CONCLUSIONS AND FUTURE WORK The Webby Awards dataset is possibly the largest human- rated corpus of web sites available. Any site that is submitted is initially examined by three judges on six criteria. As such it is a statistically rigorous collection. However, since the criteria for judging are so broad, it is unclear or unknown what the specific web page components are that judges actually use for their assessments. As such it is not possible for those who would like to look to these expert-rated sites to learn Low Word Count Medium Word Count High Word Count Mean & (Std. Dev.) Mean & (Std. Dev.) Mean & (Std. Dev.) Metric G NG Sig. G NG Sig. G NG Sig. Word Count 74.6 62.7 0.002 231.77 228.1 0.430 803.0 844.7 0.426 Body Text % 62.4 60.0 0.337 68.18 68.5 0.793 73.5 80.3 0.000 Emp. Body Text % 12.1 14.5 0.180 9.28 18.1 0.000 11.2 17.1 0.001 Text Pos. Count 1.4 1.6 0.605 4.59 3.5 0.096 6.1 7.3 0.403 Text Clus. Count 1.3 1.1 0.477 5.17 3.4 0.007 7.8 7.8 0.973 Link Count 74.6 16.0 0.202 35.68 36.2 0.764 61.1 51.7 0.019 Page Size 23041.2 32617.6 0.004 53429.98 46753.0 0.163 77877.7 50905.0 0.000 Graphic % 28.8 48.9 0.000 40.64 56.0 0.000 37.8 45.4 0.004 Graphics Count 11.4 15.0 0.005 24.88 26.2 0.451 25.3 25.8 0.835 Color Count 6.1 5.9 0.224 7.47 7.1 0.045 8.1 7.2 0.000 Font Count 3.7 3.2 0.001 5.42 5.3 0.320 6.7 6.7 0.999 Table Means and standard deviations (in parenthesis) for the good (G) and not-good (NG) groups based on the low, medium, and high word count categories. The table also contains t-test results (2-tailed significance) for each profile; bold text denotes significant differences (i.e., p < 0.05). how to improve their own designs to derive value from these results. We hope that the type of analysis that we present here opens the way towards a new, bottom-up methodology for creating empirically justified, reproducible interface design recommendations, heuristics, and guidelines. We are developing a prototype analysis tool that will enable designers to compare their pages to profiles of good pages in each subject category. However, the lack of agreement over guidelines suggests there is no one path to good design; good web page design might be due to a combination of a number of metrics. For example, it is possible that some good pages use many text clusters, many links, and many colors. Another good design profile might make use of less text, proportionally fewer colors, and more graphics. Both might be equally valid paths to the same end: good web page design. Thus we do not plan to simply present a rating, nor do we plan on stating that a given metric exceeds a cutoff point. Rather, we plan to develop a set of profiles of good designs for each category, and show how the designer's pages differ from the various profiles. It is important to keep in mind that metrics of the type explored here are only one piece of the web site design puzzle; this work is part of a larger project whose goals are to develop techniques to empirically investigate all aspects of web site design, and to develop tools to help designers assess and improve the quality of their web sites. ACKNOWLEDGMENTS This research was supported by a Hellman Faculty Fund Award, a Gates Millennium Fellowship, a GAANN fellowship, and a Lucent Cooperative Research Fellowship Program grant. We thank Maya Draisin and Tiffany Shlain at the International Academy of Digital Arts & Sciences for making the WebbyAwards 2000 data available; Nigel Claridge and Jurek Kirakowski of WAMMI for agreeing to analyze our user study data; and Tom Phelps for his assistance with the extended metrics tool. --R Does typography affect proposal as- sessment? Communications of the ACM Automatic web usability evaluation: Where is the limit? Building usable web pages: An HCI per- spective The use of guidelines in menu interface design: Evaluation of a draft stan- dard Using web server logs to improve site design. Visually critiquing web pages. Web Navigation: Designing the User Experience. Measuring user motivation from server log files. Understanding patterns of user visits to web sites: Interactive starfield visualizations of WWW log data. Preliminary findings on quantitative measures for distinguishing highly rated information-centric web pages Beyond automated essay scoring. Web Style Guide: Basic Design Principles for Creating Web Sites. WAMMI web usability qusetion- naire The alertbox: Current issues in web usability. Designing Web Usability: The Practice of Simplicity. The International Academy of Arts and Sciences. Characterization and assessment of HTML style guides. Developing usability tools and techniques for designing and testing web sites. Content or graphics? Standards versus guidelines for designing user interface software. The rating game. Reading reader reaction: A proposal for inferential analysis of web server log files. Authoring tools: Towards continuous usability testing of web documents. A tool for systematic web authoring. Web Criteria. Incorporating quality metrics in centralized/distributed information retrieval on the world wide web. --TR Knowledge-based evaluation as design support for graphical user interfaces Characterization and assessment of HTML style guides Guidelines for designing usable World Wide Web pages Gentler Web navigation Using Web server logs to improve site design The scent of a site On Site: does typography affect proposal assessment? Incorporating quality metrics in centralized/distributed information retrieval on the World Wide Web Designing Web Usability Web Style Guide The use of guidelines in menu interface design --CTR Christopher C. Whitehead, Evaluating web page and web site usability, Proceedings of the 44th annual southeast regional conference, March 10-12, 2006, Melbourne, Florida Eleni Michailidou, ViCRAM: visual complexity rankings and accessibility metrics, ACM SIGACCESS Accessibility and Computing Hartmut Obendorf , Harald Weinreich , Torsten Hass, Automatic support for web user studies with SCONE and TEA, CHI '04 extended abstracts on Human factors in computing systems, April 24-29, 2004, Vienna, Austria Janice (Ginny) Redish , Randolph G. Bias , Robert Bailey , Rolf Molich , Joe Dumas , Jared M. Spool, Usability in practice: formative usability evaluations - evolution and revolution, CHI '02 extended abstracts on Human factors in computing systems, April 20-25, 2002, Minneapolis, Minnesota, USA Tetsuya Yoshida , Masato Watanabe , Shogo Nishida, An image based design support system for web page design, International Journal of Knowledge-based and Intelligent Engineering Systems, v.10 n.3, p.201-212, July 2006 Melody Y. Ivory , Marti A. Hearst, Statistical profiles of highly-rated web sites, Proceedings of the SIGCHI conference on Human factors in computing systems: Changing our world, changing ourselves, April 20-25, 2002, Minneapolis, Minnesota, USA Melody Y. Ivory , Marti A. Hearst, Improving Web Site Design, IEEE Internet Computing, v.6 n.2, p.56-63, March 2002 Tamara Sumner , Michael Khoo , Mimi Recker , Mary Marlino, Understanding educator perceptions of "quality" in digital libraries, Proceedings of the 3rd ACM/IEEE-CS joint conference on Digital libraries, May 27-31, 2003, Houston, Texas Ron Perkins, Remote usability evaluations using the internet, Proceedings of the 1st European UPA conference on European usability professionals association conference, p.79-86, September 02-06, 2002, London, UK Ed H. Chi , Adam Rosien , Gesara Supattanasiri , Amanda Williams , Christiaan Royer , Celia Chow , Erica Robles , Brinda Dalal , Julie Chen , Steve Cousins, The bloodhound project: automating discovery of web usability issues using the InfoScent simulator, Proceedings of the SIGCHI conference on Human factors in computing systems, April 05-10, 2003, Ft. Lauderdale, Florida, USA Sibylle Steinau , Oscar Daz , Juan J. Rodrguez , Felipe Ibez, A tool for assessing the consistency of websites, Enterprise information systems IV, Kluwer Academic Publishers, Hingham, MA, Anastasios Tombros , Ian Ruthven , Joemon M. Jose, How users assess web pages for information seeking, Journal of the American Society for Information Science and Technology, v.56 n.4, p.327-344, 15 February 2005 Elaine G. Toms , Adam R. Taves, Measuring user perceptions of web site reputation, Information Processing and Management: an International Journal, v.40 n.2, p.291-317, March 2004 Weiyin Hong , James Y. L. Thong , Kar Yan Tam, Designing product listing pages on e-commerce websites: an examination of presentation mode and information format, International Journal of Human-Computer Studies, v.61 n.4, p.481-503, October 2004 Surendra N. Singh , Nikunj Dalal , Nancy Spears, Understanding web home page perception, European Journal of Information Systems, v.14 n.3, p.288-302, September 2005 Aline Chevalier , Melody Y. Ivory, Web site designs: influences of designer's expertise and design constraints, International Journal of Human-Computer Studies, v.58 n.1, p.57-87, January Melody Y. Ivory , Rodrick Megraw, Evolution of web site design patterns, ACM Transactions on Information Systems (TOIS), v.23 n.4, p.463-497, October 2005 Ralf Gitzel , Axel Korthaus , Martin Schader, Using established Web Engineering knowledge in model-driven approaches, Science of Computer Programming, v.66 n.2, p.105-124, April, 2007 Melody Y. Ivory , Marti A Hearst, The state of the art in automating usability evaluation of user interfaces, ACM Computing Surveys (CSUR), v.33 n.4, p.470-516, December 2001

World Wide Web;automated usability evaluation;empirical studies;web site design

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Optimal Reward-Based Scheduling for Periodic Real-Time Tasks.

AbstractReward-based scheduling refers to the problem in which there is a reward associated with the execution of a task. In our framework, each real-time task comprises a mandatory and an optional part. The mandatory part must complete before the task's deadline, while a nondecreasing reward function is associated with the execution of the optional part, which can be interrupted at any time. Imprecise computation and Increased-Reward-with-Increased-Service models fall within the scope of this framework. In this paper, we address the reward-based scheduling problem for periodic tasks. An optimal schedule is one where mandatory parts complete in a timely manner and the weighted average reward is maximized. For linear and concave reward functions, which are most common, we 1) show the existence of an optimal schedule where the optional service time of a task is constant at every instance and 2) show how to efficiently compute this service time. We also prove the optimality of Rate Monotonic Scheduling (with harmonic periods), Earliest Deadline First, and Least Laxity First policies for the case of uniprocessors when used with the optimal service times we computed. Moreover, we extend our result by showing that any policy which can fully utilize all the processors is also optimal for the multiprocessor periodic reward-based scheduling. To show that our optimal solution is pushing the limits of reward-based scheduling, we further prove that, when the reward functions are convex, the problem becomes NP-Hard. Our static optimal solution, besides providing considerable reward improvements over the previous suboptimal strategies, also has a major practical benefit: Run-time overhead is eliminated and existing scheduling disciplines may be used without modification with the computed optimal service times.

Introduction In a real-time system each task must complete and produce correct output by the specified deadline. However, if the system is overloaded it is not possible to meet each deadline. In the past, several techniques have been introduced by the research community regarding the appropriate strategy to use in overloaded systems of periodic real-time tasks. One class of approaches focuses on providing somewhat less stringent guarantees for temporal con- straints. In [16], some instances of a task are allowed to be skipped entirely. The skip factor determines how often instances of a given task may be left unexecuted. A best effort strategy is introduced in [11], aiming at meeting k deadlines out of n instances of a given task. This framework is also known as (n,k)-firm deadlines scheme. Bernat and Burns present in [3] a hybrid and improved approach to provide hard real-time guarantees to k out of n consecutive instances of a task. The techniques mentioned above tacitly assume that a task's output is of no value if it is not executed completely. However, in many application areas such as multimedia applications [26], image and speech processing [5, 6, 9, 28], time-dependent planning [4], robot control/navigation systems [12, 30], medical decision making [13], information gathering [10], real-time heuristic search [17] and database query processing [29], a partial or approximate but timely result is usually acceptable. The imprecise computation [7, 19, 21] and IRIS (Increased Reward with Increased Service) [14, 15, 18] models were proposed to enhance the resource utilization and graceful degradation of real-time systems when compared with hard real-time environments where worst-case guarantees must be provided. In these models, every real-time task is composed of a mandatory part and an optional part. The former should be completed by the task's deadline to provide output of acceptable (minimal) quality. The optional part is to be executed after the mandatory part while still before the deadline, if there are enough resources in the system that are not committed to running mandatory parts for any task. The longer the optional part executes, the better the quality of the result (the higher the reward). The algorithms proposed for imprecise computation applications concentrate on a model that has an upper bound on the execution time that could be assigned to the optional part [7, 21, 27]. The aim is usually to minimize the (weighted) sum of errors. Several efficient algorithms are proposed to solve optimally the scheduling problem of aperiodic imprecise computation tasks [21, 27]. A common assumption in these studies is that the quality of the results produced is a linear function of the precision consequently, the possibility of having more general error functions is usually not addressed. An alternative model allows tasks to get increasing reward with increasing service (IRIS model) [14, 15, 18] without an upper bound on the execution times of the tasks (though the deadline of the task is an implicit upper bound) and without the separation between mandatory and optional parts [14]. A task executes for as long as the scheduler allows before its deadline. Typically, a nondecreasing concave reward function is associated with each task's execution time. In [14, 15] the problem of maximizing the total reward in a system of aperiodic independent tasks is addressed. The optimal solution with static task sets is presented, as well as two extensions that include mandatory parts and policies for dynamic task arrivals. Note that the imprecise computation and IRIS models are closely related, since the performance metrics can be defined as duals (maximizing the total reward vs. minimizing the total error). Similarly, a concave reward function corresponds to a convex error function, and vice versa. We use the term "Reward-based scheduling" to encompass scheduling frameworks, including Imprecise Computation and IRIS models, where each task can be logically decomposed into a mandatory and optional subtask. A nondecreasing reward function is associated with the execution of each optional part. An interesting question concerns the types of reward functions that represent realistic application areas. A linear reward function [19, 21] models the case where the benefit to the overall system increases uniformly during the optional execution. Similarly, a concave reward function [14, 15, 18, 26] addresses the case where the greatest increase/refinement in the output quality is obtained during the first portions of optional executions. Linear and general concave functions are considered as the most realistic and typical in the literature since they adequately capture the behavior of many application areas like image and speech processing [5, 6, 9, 28], multimedia applications [26], time-dependent planning [4], robot control/navigation systems [30], real-time heuristic search [17], information gathering [10] and database query processing [29]. In this paper, we show that the case of convex reward functions is an NP-Hard problem and thus focus on linear and concave reward functions. Reward functions with 0/1 constraints, where no reward is accrued unless the entire optional part is executed, or as step functions have also received some interest in the literature. Unfortunately, this problem has been shown to be NP-Complete in [27]. Periodic reward-based scheduling remains relatively unexplored, since the important work of Chung, Liu and Lin [7]. In that paper, the authors classified the possible application areas as "error non- cumulative" and "error cumulative". In the former, errors (or optional parts left unexecuted) have no effect on the future instances of the same task. Well-known examples of this category are tasks which receive, process and transmit periodically audio, video or compressed images [5, 6, 9, 26, 28] and information retrieval tasks [10, 29]. In "error cumulative" applications, such as radar tracking, an optional instance must be executed completely at every (predetermined) k invocations. The authors further proved that the case of error-cumulative jobs is an NP-Complete problem. In this paper, we restrict ourselves to error non-cumulative applications. Recently, a QoS-based resource allocation model (QRAM) has been proposed for periodic applications [26]. In that study, the problem is to optimally allocate several resources to the various applications such that they simultaneously meet their minimum requirements along multiple QoS dimensions and the total system utility is maximized. In one aspect, this can be viewed as a generalization of optimal CPU allocation problem to multiple resources and quality dimensions. Further, dependent and independent quality dimensions are separately addressed for the first time in this work. However, a fundamental assumption of that model is that the reward functions and resource allocations are in terms of utilization of resources. Our work falls rather along the lines of Imprecise Computation model, where the reward accrued has to be computed separately over all task instances and the problem is to find the optimal service times for each instance and the optimal schedule with these assignments. Aspects of the Periodic Reward-Based Scheduling Problem The difficulty of finding an optimal schedule for a periodic reward-based task set has its origin on two objectives that must be simultaneously achieved, namely: i. Meeting deadlines of mandatory parts at every periodic task invocation. ii. Scheduling optional parts to maximize the total (or average) reward. These two objectives are both important, yet often incompatible. In other words, hard deadlines of mandatory parts may require sacrificing optional parts with greatest value to the system. The analytical treatment of the problem is complicated by the fact that, in an optimal schedule, optional service times of a given task may vary from instance to instance which makes the framework of classical periodic scheduling theory inapplicable. Furthermore, this fact introduces a large number of variables in any analytical approach. Finally, by allowing nonlinear reward functions to better characterize the optional tasks' contribution to the overall system, the optimization problem becomes computationally harder. In [7], Chung, Liu and Lin proposed the strategy of assigning statically higher priorities to mandatory parts. This decision, as proven in that paper, effectively achieves the first objective mentioned above by securing mandatory parts from the potential interference of optional parts. Optional parts are scheduled whenever no mandatory part is ready in the system. In [7], the simulation results regarding the performance of several policies which assign static or dynamic priorities among optional parts are reported. We call the class of algorithms that statically assign higher priorities to mandatory parts Mandatory-First Algorithms. In our solution, we do not decouple the objectives of meeting the deadlines of mandatory parts and maximizing the total (or average) reward. We formulate the periodic reward-based scheduling problem as an optimization problem and derive an important and surprising property of the solution for the most common (i.e., linear and concave) reward functions. Namely, we prove that there is always an optimal schedule where optional service times of a given task do not vary from instance to instance. This important result immediately implies that the optimality (in terms of achievable utilization) of any policy which can fully use the processor in case of hard-real time periodic tasks also holds in the context of reward-based scheduling (in terms of total reward) when used with these optimal service times. Examples of such policies are RMS-h (Rate Monotonic Scheduling with harmonic periods) [20], EDF (Earliest Deadline First) [20] and LLF (Least Laxity First) [24]. We also extend the framework to homogeneous multiprocessor settings and prove that any policy which can fully utilize all the processors is also optimal for scheduling periodic reward-based tasks (in terms of total reward) on multiprocessors environments. Following these existence proofs, we address the problem of efficiently computing optimal service times and provide polynomial-time algorithms for linear and/or general concave reward functions. Note that using these optimal and constant optimal service times has also important practical advantages: (a) The runtime overhead due to the existence of mandatory/optional dichotomy and reward functions is removed, and (b) existing RMS with harmonic periods), EDF and LLF schedulers may be used without any modification with these optimal assignments. The remainder of this paper is organized as follows: In Section 2, the system model and basic definitions are given. The main result about the optimality of any periodic policy which can fully utilize the processor(s) is obtained in Section 3. In Section 4, we first analyze the worst-case performance of Mandatory-First approaches. We also provide the results of experiments on a synthetic task set to compare the performance of policies proposed in [7] against our optimal algorithm. In Section 6, we show that the concavity assumption is also necessary for computational efficiency by proving that allowing convex reward functions results in an NP-Hard problem. Then, we examine whether the optimality of identical service times still holds if the model is modified by dropping some fundamental assumptions (Section 5). We present details about the specific optimization problem that we use in Section 7. We conclude by summarizing our contribution and discussing future work. System Model We first develop and present our solution for uniprocessor systems, then we show how to extend it to the case of homogeneous multiprocessor systems. We consider a set T of n periodic real-time tasks . The period of T i is denoted by P i , which is also equal to the deadline of the current invocation. We refer to the j th invocation of task T i All tasks are assumed to be independent and ready at Each task T i consists of a mandatory part M i and an optional part O i . The length of the mandatory part is denoted by m i ; each task must receive at least m i units of service time before its deadline in order to provide output of acceptable quality. The optional part O i becomes ready for execution only when the mandatory part M i completes, it can execute as long as the scheduler allows before the deadline. Associated with each optional part of a task is a reward function R which indicates the reward accrued by task T ij when it receives t ij units of service beyond its mandatory portion. R is of the (1) where f i is a nondecreasing, concave and continuously differentiable function over nonnegative real numbers and is the length of the entire optional part O i . We underline that f the benefit of task T ij can not decrease by allowing it to run longer. Notice that the reward function R i (t) is not necessarily differentiable at Note also that in this formulation, by the time the task's optional execution time t reaches the threshold value the reward accrued ceases to increase. Clearly, the reward of executing an optional part O i for an amount of time will be the same as the reward for executing for Therefore, it is not beneficial to execute O i for more than time units. A function f(x) is concave if and only if for all x; y and 0 - ff - ff)f(y). Geometrically, this condition means that the line joining any two points of a concave curve may not be above the curve. Examples of concave functions are linear functions (kx functions (ln[kx decay functions (c th root functions (x 1=k ). Note that the first derivative of a nondecreasing concave function is nonincreasing. Having nondecreasing concave reward functions means that while a task T i receives service beyond its mandatory portion M i , its reward monotonically increases. However, its rate of increase decreases or remains constant with time. The concavity assumption implies that the early portions of an optional execution are not less important than the later ones, which adequately captures many application areas mentioned in the introduction. We mostly concentrate on linear and, in general, concave reward functions. A schedule of periodic tasks is feasible if mandatory parts meet their deadlines at every invocation. Given a feasible schedule of the task set T, the average reward of task T i is defined as: where P is the hyperperiod, that is, the least common multiple of P is the service time assigned to the j th instance of optional part of task T i . That is, the average reward of T i is computed over the number of its invocations during the hyperperiod P, in an analogous way to the definition of average error in [7] 1 . The average weighted reward of a feasible schedule is then given by: We note that the results we prove easily extend to the case in which one is interested in maximizing the total reward where w i is a constant in the interval (0,1] indicating the relative importance of optional part O i . Although this is the most general formulation, it is easy to see that the weight w i can always be incorporated into the reward function f i (), by replacing it by w Thus, we will assume that all weight (importance) information are already expressed in the reward function formulation and that REWW is simply equal to Finally, a schedule is optimal if it is feasible and it maximizes the average weighted reward. A Motivating Example: Before describing our solution to the problem, we present a simple example which shows the performance limitations of any Mandatory-First algorithm. Consider two tasks where 5. Assume that the reward functions associated with optional parts are linear Furthermore, suppose that k 2 associated with the reward accrued by T 2 is negligible when compared to k 1 , i.e. k 1 AE k 2 . In this case, the "best" algorithm among "Mandatory-First" approaches should produce the schedule shown in Figure 1.00000000000011111111111100000000000000000000001111111111111111111111 Figure 1: A schedule produced by Mandatory-First Algorithm Above, we assumed that the Rate Monotonic Priority Assignment is used whenever more than one mandatory task are simultaneously ready, as in [7]. Yet, following other (dynamic or static) priority schemes would not change the fact that the processor will be busy executing solely mandatory parts Mandatory-First approach. During the remaining idle interval [5,8], the best algorithm would have chosen to schedule O 1 completely (which brings most benefit to the system) for 1 time unit and O 2 for 2 time units. However, an optimal algorithm would produce the schedule depicted in Figure 2. As it can be seen, the optimal strategy in this case consisted of delaying the execution of M 2 in order to be able to execute 'valuable' O 1 and we would still meet the deadlines of all mandatory parts. By doing so, we would succeed in executing two instances of O 1 , in contrast to any Mandatory-First scheme which can execute only one instance of O 1 . Remembering that k 1 AE k 2 , one can conclude that the reward accrued by the 'best' Mandatory-First scheme may only be around half of that accrued by the optimal one, for this example. Also, observe that in the optimal schedule, the optional execution times of a given task did not vary from instance to instance. In the next section, we prove that this pattern O 2 Figure 2: An optimal schedule is not a mere coincidence. We further perform an analytical worst-case analysis of Mandatory-First algorithms in Section 4. 3 Optimality of Full-Utilization Policies for Periodic Reward-Based Scheduling We first formalize the Periodic Reward-Based Scheduling problem. The objective is clearly finding values to maximize the average reward. By substituting the average reward expression given by (2) in (3), we obtain our objective function: maximize The first constraint that we must enforce is that the total processor demand of mandatory and optional parts during the hyperperiod P may not exceed the available computing capacity, that is: Note that this constraint is necessary, but by no means sufficient for feasibility of the task set with values. Next, we observe that optimal t ij values may not be less than zero, since negative service times do not have any physical interpretation. In addition, the service time of an optional instance of T i does not need to exceed the upperbound of reward function R i (t), since the reward accrued by T i ceases to increase after Hence, we obtain our second constraint set: The constraint above allows us to readily substitute f i () for R i () in the objective function. Finally, we need to express the "full" feasibility constraint, including the requirement that mandatory parts complete in a timely manner at every invocation. Note that it is sufficient to have one feasible schedule with the involved fm i g and optimal values: There exists a feasible schedule with fm i g and ft ij g values We express this constraint in English and not through formulas since the policy or algorithm producing this schedule including optimal t ij assignments need not be specified at this point. To re-capture all the constraints, the periodic reward-based scheduling problem, which we denote by REW-PER, is to find values so as to: maximize subject to There exists a feasible schedule with fm i g and ft ij g values (7) Before stating our main result, we underline that if it is not possible to schedule mandatory parts in a timely manner and the optimization problem has no solution. Note that this condition is equivalent to ? 1, which indicates that the task set would be unschedulable, even if it consisted of only mandatory parts. Hence, thereafter, we suppose that there exists at least one feasible schedule. Theorem 1 Given an instance of Problem REW-PER, there exists an optimal solution where the optional parts of a task T i receive the same service time at every instance, i.e. Furthermore, any periodic hard-real time scheduling policy which can fully utilize the processor (EDF, LLF, RMS-h) can be used to obtain a feasible schedule with these assignments. Proof: Our strategy to prove the theorem will be as follows. We will drop the feasibility condition (7) and obtain a new optimization problem whose feasible region strictly contains that of REW-PER. Specifically, we consider a new optimization problem, denoted by MAX-REW, where the objective function is again given by (4), but only the constraint sets (5) and (6) have to be satisfied. Note that the new problem MAX-REW does not a priori correspond to any scheduling problem, since the feasibility issue is not addressed. We then show that there exists an optimal solution of MAX-REW . Then, we will return to REW-PER and demonstrate the existence of a feasible schedule (i.e. satisfiability of (7)) under these assignments. The reward associated with MAX-REW's optimal solution is always greater than or equal to that of REW-PER's optimal solution, for MAX-REW does not consider one of the REW-PER's constraints. This will imply that this specific optimal solution of the new problem MAX-REW is also an optimal solution of REW-PER. Now, we show that there exists an optimal solution of MAX-REW where be an optimal solution to also an optimal solution to MAX-REW. ffl We first show that ft 0 values satisfy the constraints (5) and (6), if already satisfy them. Since i the constraint (5) is not violated by the transformation. Also, by assumption, i , which is arithmetic mean of is necessarily less than or equal to max j the constraint set (6) is not violated either by the transformation. ffl Furthermore, the total reward does not decrease by this transformation, since ). The proof of this statement is presented in the Appendix. Using Claim 1, we can commit to finding an optimal solution of MAX-REW by setting t n. In this case, . Hence, this version of MAX-REW can be re-written as: maximize subject to Finally, we prove that the optimal solution t automatically satisfies the feasibility constraint (7) of our original problem REW-PER. Having equal optional service times for a given task greatly simplifies the verification of this constraint. Since t satisfy (9), we can write equivalently, - 1. This implies that any policy which can achieve 100% processor utilization in classical periodic scheduling theory (EDF, LLF, RMS-h) can be used to obtain a feasible schedule for tasks, which have now identical execution times at every instance. Hence, the "full feasibility" constraint (7) of REW-PER is satisfied. Furthermore, this schedule clearly maximizes the average reward since ft i g values maximize MAX-REW whose feasible region encompasses that of REW-PER. Corollary 1 Optimal t i values for the Problem REW-PER can be found by solving the optimization problem given by (8), (9) and (10). The details of the solution of this concave optimization problem are presented in Section 7. 3.1 Extension to Multiprocessors The existence proof of identical service times can be easily extended to homogeneous multiprocessors. The original formulation of REW-PER needs to be modified in order to reflect the multiprocessor environment. Note that the objective function (4), the lower and upper bound constraints (6) on optional service times and the full feasibility constraint (7) can be kept as they are. However, with k processors, the system can potentially have a task set whose total utilization is k instead of 1. Hence, we need to change the first constraint accordingly. By doing so, we obtain the formulation of periodic imprecise computation problem for k processors, denoted as MULTI-REW: maximize subject to There exists a feasible schedule on k processors with fm i g and ft ij g values (14) Following exactly the same line of reasoning depicted in Theorem 1, we can infer the following: Theorem 2 Given an instance of Problem MULTI-REW, there exists an optimal solution where the optional parts of a task T i receive the same service time at every instance, i.e. Furthermore, any scheduling policy which can achieve full utilization on k processors can be used to obtain a feasible schedule with these assignments. An example of such full-utilization policies for multiprocessors is provided by Mancini et al. in [23]. We note that the PFair scheduling policy [2] which can also achieve full-utilization, assumes that all the periods are multiples of the slot length and hence it can not be used in this context. Corollary 2 Optimal t i values for the Problem MULTI-REW can be found by solving the following optimization problem: maximize subject to Again, the details of the solution of this concave optimization problem are given in Section 7. 4 Evaluation and comparison with other approaches We showed through the example in Section 2 that the reward accrued by any Mandatory-First scheme may only be approximately half of that of the optimal algorithm. We now prove that, under worst-case scenario, the ratio of the reward accrued by a Mandatory-First approach to the reward of the optimal algorithm approaches zero. Theorem 3 There is an instance of periodic reward-based scheduling problem where the ratio Reward of the best mandatory\Gammafirst scheme Reward of the optimal scheme r for any integer r - 2. Proof: Consider two tasks T 1 and T 2 such that P 2 r which implies that r (r\Gamma1) This setting suggests that during any period of T 1 , a scheduler has the choice of executing (parts in addition to M 1 . Note that under any Mandatory-First policy, the processor will be continuously busy executing mandatory parts until Furthermore, the best algorithm among Mandatory-First policies should use the remaining idle times by scheduling O 1 entirely (since 2units of O 2 . The resulting schedule is shown in Figure 3.m 1 Figure 3: A schedule produced by Mandatory-First Algorithm The average reward that the best mandatory-first algorithm (MFA) can accrue is therefore: r However, an optimal algorithm (shown in Figure 4) would choose delaying the execution of M 2 for units of time, at every period of T 1 . By doing so, it would have the opportunity of accruing the reward of O 1 at every instance.m 1 om r r r r r Figure 4: An optimal schedule The total reward of the above schedule is: r The ratio of rewards for the two policies turns out to be (for any r - 2) RMFA ROPT =r r =r which can be made as close as possible to 0 by appropriately choosing r.Theorem 3 gives the worst-case performance ratio of Mandatory-First schemes. We also performed experiments with a synthetic task set to investigate the relative performance of Mandatory-First schemes proposed in [7] with different types of reward functions and different mandatory/optional workload ratios. The Mandatory-First schemes differ by the policy according to which optional parts are scheduled when there is no mandatory part ready to execute. Rate-Monotonic (RMSO) and Least-Utilization (LU) schemes assign statically higher priorities to optional parts with smaller periods and least utilizations respectively. Among dynamic priority schemes are Earliest-Deadline-First (EDFO) and Least-Laxity- First (LLFO) which consider the deadline and laxity of optional parts when assigning priorities. Least Attained Time (LAT) aims at balancing execution times of optional parts that are ready, by dispatching the one that executed least so far. Finally, Best Incremental Return (BIR) is an on-line policy which chooses the optional task contributing most to the total reward, at a given slot. In other words, at every slot BIR selects the optional part O ij such that the difference f is the largest (here t ij is the optional service time O ij has already received and \Delta is the minimum time slot that the scheduler assigns to any optional task). However, it is still a sub-optimal policy since it does not consider the laxity information. The authors indicate in [7] that BIR is too computationally complex to be actually implemented. However, since the total reward accrued by BIR is usually much higher than the other five policies, BIR is used as a yardstick for measuring the performance of other algorithms. We have used a synthetic task set comprising 11 tasks whose total (mandatory is 2.3. Individual task utilizations vary from 0.03 to 0.6. Considering exponential, logarithmic and linear reward functions as separate cases, we measured the reward ratio of six Mandatory-First schemes with respect to our optimal algorithm. The tasks' characteristics (including reward functions) are given in the Table below. In our experiments, we first set mandatory utilization to 0 (which corresponds to the case of all-optional workload), then increased it to 0.25, 0.4, 0.6, 0.8 and 0.91 subsequently. 7 90 9 Figures 5 and 6 show the reward ratio of six Mandatory-First schemes with respect to our optimal algorithm as a function of mandatory utilization, for different types of reward functions. A common pattern appears: the optimal algorithm improves more dramatically with the increase in mandatory utilization. The other schemes miss the opportunities of executing "valuable" optional parts while constantly favoring mandatory parts. The reward loss becomes striking as the mandatory workload increases. Figures 5.a and 5.b show the reward ratio for the case of exponential and logarithmic reward functions, respectively. The curves for these strictly concave reward functions are fairly similar: BIR performs best among Mandatory-First schemes, and its performance decreases as the mandatory utilization increases; for instance the ratio falls to 0.73 when mandatory utilization is 0.6. Other algorithms which are more amenable to practical implementations (in terms of runtime overhead) than BIR perform even worse. However, it is worth noting that the performance of LAT is close to that of BIR. This is to be expected, since task sets with strictly concave reward functions usually benefit from "balanced" optional service times. Utilization OPT Reward Ratio with Respect to Optimal RMSO LLFO EDFO (a) LLFO RMSO EDFO LU Reward Ratio with Respect to Optimal OPT Mandatory Figure 5: The Reward Ratio of Mandatory-First schemes: strictly concave reward functions (a) Exponential (b) Logarithmic functions Reward Ratio with Respect to Optimal Utilization OPT0.800.600.500.201.00 LLFO RMSO EDFO LU Figure The Reward Ratio of Mandatory-First schemes: linear reward functions Figure 6 shows the reward ratio for linear reward functions. Although the reward ratio of Mandatory- First schemes again decreases with the mandatory utilization, the decrease is less dramatic than in the case of concave functions (see above). However, note that the ratio is typically less than 0.5 for the five practical schemes. It is interesting to observe that the (impractical) BIR's reward now remains comparable to that of optimal, even in the higher mandatory utilizations: the difference is less than 15%. In our opinion, the main reason for this behavior change lies on the fact that, for a given task, the reward of optional execution slots in different instances does not make a difference in the linear case. In contrast, not executing the "valuable" first slot(s) of a given instance creates a tremendous effect for nonlinear concave functions. The improvement of the optimal algorithm would be larger for a larger range of k i values (where k i is the coefficient of the linear reward function). We note that even the worst-case performance of BIR may be arbitrarily bad with respect to the optimal one for linear functions, as Theorem 3 suggests. Further considerations on the optimality of identical service times We underline that Theorem 1 was the key to eliminate (potentially) an exponential number of unknowns thereby to obtain an optimization problem of n variables. One is naturally tempted to ask whether the optimality of identical service times is still preserved if some fundamental assumptions of the model are relaxed. Unfortunately, attempts to reach further optimality results for extended / different models remain inconclusive as the following propositions suggest. Proposition 1 The optimality of identical service times no longer holds if the Deadline = Period assumption is relaxed. Proof: We will prove the statement by providing a counter-example. Assume that we allow the deadline of a task to be less than its period. Consider the following two tasks: Further, assume that the deadline of T 2 is D while the deadline of T 1 coincides with its period, i.e. d 8. Note that the tight deadline of T 2 makes it impossible to schedule any optional after which we are able to schedule O 1 for 3 units. This optimal schedule is shown in Figure 7. On the other hand, if one commits to identical service times per instance, it is clear that we may not schedule any optional part, since we could not execute O 1 in the first instance of T 1 (Figure 8).Next, suppose that the deadlines are equal to the periods, but we have to adopt a static priority scheduling policy. It was already mentioned in Section 3, that if the periods are harmonic, then we can use RMS without compromising optimality. But, in the general case where the periods are not necessarily harmonic, this is not true even if we are investigating the 'best' schedule within the context of a given static priority assignment. Proposition 2 In the general case, the optimality of identical service times no longer holds if we commit to a static priority assignment. d O 110 Figure 7: The optimal schedule0 Figure 8: Best schedule with identical service times Proof: Again, consider the following task set: As we have only two tasks, we will consider the cases where T 1 or T 2 has higher priority and show that in every case, the reward of the optimal schedule differs from the one obtained with identical service times per instance assumption. Case higher priority: It is easy to see that we can construct a schedule which fully utilizes the timeline during the interval [0; lcm(6; This schedule is also immediately optimal since the reward function is linear (observe that hence we do not receive any reward for executing O 2 ). But, we remark that we can not execute O 1 for more than 1 unit on its first instance in any feasible schedule - without violating the deadline of T 2 . Therefore, we would have ended up with a lower reward after executing 1 unit of O 1 at each instance, if we had committed to identical service times (Figure 9)b. Case 2 - T 2 has higher priority: The optimality is still compromised if T 2 has higher priority. While the optimal schedule (Figure 10a) fully utilizes the timeline, the best schedule with t Figure remains suboptimal. We remark that the Proposition 2 has also implications for Q-RAM model [26], since it points to the impossibility of achieving optimality with identical service times by using a static priority assignment.0000000000001111111111110000000000001111111111110000000000001111111111110000000000001111111111110163 242 6 6 81 O 1 O 1 O O 1 O 1 O O 1(a) (b) Figure has the higher priority (a) The optimal schedule (b) The best schedule with identical service times O O MO 15OMM 1 O O MO 15OMM 1 O M5OMM 19 2118O O 111 (a) (b) Figure has the higher priority (a) The optimal schedule (b) The best schedule with identical service times The final proposition in this section illustrates that the optimality of identical service times is also sensitive to the concavity of reward functions. Proposition 3 The optimality of identical service times no longer holds if the concavity assumption about the reward functions is relaxed. Proof: Consider two harmonic tasks without mandatory parts whose parameters are given in the following table: Note that t 2 should be assigned its maximum possible value (i.e., the upper bound 6), since the marginal return of F 2 is larger than F 1 everywhere. An optimal schedule maximizing the average reward for these two tasks is depicted in Figure 11.00000000000000000011111111111111111111111100000000000000000000000000000000000000000000000000000000000000011111111111111111111111111111111111111111111111111111111111111111111111111111111111100000000000000000000000000000000000000000000000011111111111111111111111111111111111111111111111111111111111111111O O 2 O 12 8 Figure 11: An optimal schedule The sum of the average rewards in the optimal schedule is 2:6 However, if we commit ourselves to the equal service times per instances, we can find no better schedule than the one shown in Figure 12, whose reward is only It is not difficult to construct a similar example for the tasks with 0-1 constraints as well, which implies that even the number of variables to deal with (the t ij 's) may be prohibitively large in these problems. O 1 O 1O513 O 2 Figure service times 6 Periodic Reward-Based Scheduling Problem with Convex Reward Functions is NP-Hard As we mentioned before, maximizing the total (or average) reward with 0/1 constraints case had already been proven to be NP-Complete in [21]. Similarly, in section 5 we showed that, if the reward functions are convex, the optimality of identical service times is not preserved. In this section, we show that, in fact convex reward functions result in an NP-Hard problem, even with identical periods. We now show how to transform the SUBSET-SUM problem, which is known to be NP-Complete, to REW-PER with convex reward functions. SUBSET-SUM: Given a set of positive integers and the integer M, is there a set SA ' S such that We construct the corresponding REW-PER instance as follows. Let Now consider a set of n periodic tasks with the same period M and mandatory parts m 8i. The reward function associated with T i is given by: strictly convex and increasing function on nonnegative real numbers. Notice that f i (t i ) can be re-written as t i Also we underline that having the same periods for all tasks implies that REW-PER can be formulated as: maximize subject to Let us denote by MaxRew the total reward of the optimal schedule. Observe that for the quantity t i Otherwise, at either of the boundary values 0 or s i , MaxRew - WM . Now, consider the question: "Is MaxRew equal to WM ?". Clearly, this question can be answered quickly if there is a polynomial-time algorithm for REW-PER where reward functions are allowed to be convex. Furthermore, the answer can be positive only when and each t i is equal to either 0 or s i . Therefore, MaxRew equal to WM , if and only if there is a set SA ' S such that which implies that REW-PER with convex reward functions is NP-Hard. 7 Solution of Periodic Reward-Based Scheduling Problem with Concave Reward Functions Corollaries 1 and 2 reveal that the two optimization problems whose solutions provide optimal service times for uniprocessor and multiprocessor systems share a common form: maximize subject to where d (the 'slack' available for optional executions) and b 1 are positive rational numbers. In this section, we present polynomial-time solutions for this problem, where each f i is a nondecreasing, concave and differentiable 2 function. First note that, if the available slack is large enough to accommodate every optional part entirely (i.e., if then the choice t clearly maximizes the objective function due to the nondecreasing nature of reward functions. Otherwise, the slack d should be used in its entirety since the total reward never decreases by doing so (again due to the nondecreasing nature of the reward functions). In this case, we obtain a concave optimization problem with lower and upper bounds, denoted by OPT-LU. An instance of OPT-LU is specified by the set of nondecreasing concave functions g, the set of upper bounds and the available slack d. The aim is to: maximize subject to 2 In the auxiliary optimization problems which will be introduced shortly, the differentiability assumption holds as well. Special Case of Linear functions: We address separately the case when F comprises solely linear functions, since the time complexity can be considerably reduced by using this information. Note that for a function f i increase t i by \Delta then total reward increases by k i \Delta. However by doing so, we make use of b i \Delta units of slack (d is reduced by b i \Delta due to (22)). Hence, the "marginal return" of task T i per slack unit is w . Now consider another function f . If should be always favored with respect to T i since the marginal return of f j is strictly greater than f i everywhere. Repeating the argument for every pair of tasks, we can obtain the following optimal strategy. We first order the functions according to their marginal returns w the function with the largest marginal return, f 2 the second and so on (ties are broken arbitrarily). If d, then we set t and we are done (we are using the entire slack for T 1 , since transferring service time to any other task would not increase the total reward). If b 1 then we set t and d is reduced accordingly Next, we repeat the same for the next "most valuable" . After at most n iterations, the slack d is completely consumed. We note that this solution is analogous to the one presented in [26]. The dominant factor in the time complexity comes from the initial sorting procedure, hence in the special case of all-linear functions, OPT-LU can be solved in time O(n log n). When F contains nonlinear functions then the procedure becomes more involved. In the next two subsections, we introduce two auxiliary optimization problems, namely Problem OPT (which considers only the equality constraint) and Problem OPT-L (which considers only the equality and lower bound constraints), which will be used to solve OPT-LU. 7.1 Problem OPT: Case of the Equality Constraint An instance of the problem OPT is characterized by the set of nondecreasing concave functions and the slack d: maximize subject to As it can be seen, OPT does not take into account the lower and upper bound constraints of Problem OPT-LU. The algorithm which returns the solution of Problem OPT, is denoted by "Algorithm OPT". When F is composed solely of non-linear reward functions, the application of Lagrange multipliers technique [22] to the Problem OPT, yields:b i where - is the common Lagrange multiplier and f 0 is the derivative of the reward function f i . The quantity 1 actually represents the marginal return contributed by T i to the total reward, which we will denote as w i (t i ). Observe that since f i non-decreasing and concave by assumption, both non-increasing and positive valued. Equation (25) implies that the marginal returns should be equal for all reward functions in the optimal solution g. Considering that the equality should also hold, one can obtain closed formulas in most of the cases which occur in practice. The closed formulas presented below are obtained by this method. ffl For logarithmic reward functions of the form f and ffl For exponential reward functions of the form f and ffl For "k th root" reward functions of the form f and When it is not possible to find a closed formula, following exactly the approach presented in [14, 15, 18], we solve - in the equation is the inverse function of 1 (we assume the existence of the derivative's inverse function whenever f i is nonlinear, complying with [14, 15, 18]). Once - is determined, t is the optimal solution. We now examine the case where F is a mix of linear and nonlinear functions. Consider two linear functions f t. The marginal return of f and that of f j is w then the service time t i should be definitely zero, since marginal return of f i is strictly less than f j everywhere. After this elimination process, if there are linear functions with the same (largest) marginal return w max then we will consider them as a single linear function in the procedure below and evenly divide the returned service time t max among values corresponding to these p functions. Hence, without loss of generality, we assume that f n is the only linear function in F. Its marginal return is w We first compute the optimal distribution of slack d among tasks with nonlinear reward functions f . By the Lagrange multipliers technique, w and thus w 1 (t at the optimal solution t . Now we distinguish two cases: . In this case, t is the optimal solution to OPT. To see this, first remember that all the reward functions are concave and nondecreasing, hence w This implies that transferring some service time from another task T i to T n would mean favoring the task which has the smaller marginal reward rate and would not be optimal. . In this case, reserving the slack d solely to tasks with nonlinear reward functions means violating the best marginal rate principle and hence is not optimal. Therefore, we should increase drops to the level of w not beyond. Solving assigning any remaining slack d\Gamma bn to t n (the service time of unique task with linear reward function) clearly satisfies the best marginal rate principle and achieves optimality. 7.2 Problem OPT-L: Case of Lower Bounds In this section we present a solution for problem OPT-L and we improve on this solution in Section 7.2.1. An instance of Problem OPT-L is characterized by the set F=ff 1 of nondecreasing concave reward functions, and the available slack d: maximize subject to To solve OPT-L, we first evaluate the solution set SOPT to the corresponding problem OPT and check whether all inequality constraints are automatically satisfied. If this is the case, the solution set SOPT \GammaL of Problem OPT-L is clearly the solution SOPT . Otherwise, we will construct SOPT \GammaL iteratively as described below. A well-known result of nonlinear optimization theory states that the solution SOPT \GammaL of Problem OPT-L should satisfy so-called Kuhn-Tucker conditions [22, 25]. Furthermore, Kuhn-Tucker conditions are also sufficient in the case of concave reward functions [22, 25]. For Problem OPT-L, Kuhn-Tucker conditions comprise Equations (27), (28) and: are Lagrange multipliers. The necessary and sufficient character of Kuhn-Tucker conditions indicates that any 2n+ 1 tuple which satisfies conditions (27) through (31) provides optimal t i values for OPT-L. One method of solving the optimization problem OPT-L is to find a solution to the 2n+1 equations (27), (29) and (30) which satisfies constraint sets (28) and (31). Iteratively solving the 2n+ 1 nonlinear equations is a complex process which is not guaranteed to converge. In this paper, we follow a different approach. Namely, we use the Kuhn-Tucker conditions (29), (30) and (31) to prove some useful properties of the optimal solution. Our method is based on carefully using the properties that we derive in order to refine the solution of the optimization problem OPT. violates some inequality constraints given by (28) then 9i Proof: Assume to the contrary that 8i - In this case Kuhn-Tucker conditions reduce to the equality constraint (27), the set of inequality constraints (28) plus the Lagrangian condition given in (25). On the other hand, SOPT , which is the solution of OPT, should satisfy (27) and the Lagrangian condition (25). In other words, solving OPT is always equivalent to solving a set of non-linear equations which are identical to Kuhn-Tucker conditions of OPT-L except inequality constraints, by setting 8i - Hence, if there were a solution to OPT-L where 8i - then that solution would be returned by the algorithm solving OPT and would not violate inequality constraints. However, given that the solution SOPT failed to satisfy all the inequality constraints we reach a contradiction. Therefore, there exists at least one Lagrange multiplier - i which is strictly greater than 0.Claim 3 9j Proof: For the sake of contradiction, assume that 8j - j ? 0. In this case Equation (30) enforces that 8i t were true, will be equal to 0, leaving the slack d totally unutilized. In this case, this clearly would not be the optimal solution due to the nondecreasing nature of reward functions.In the rest of the paper, we use the expression "the set of functions" instead of "the set of indices of functions" unless confusion arises. Let: Remember that 1 is the marginal return associated with f x (t x ) and is denoted by w x (t x ). In- formally, \Pi contains the functions f x 2 F with the smallest marginal returns at the lower bound 0, Lemma 1 If SOPT violates some inequality constraints then, in SOPT \GammaL , t Proof: Assume that 9m 2 \Pi such that t m ? 0. In this case, Equation (30) implies that the corresponding we know that 9j such that - j ? 0. By Equation (30), t Using Equation (29), we can writeb m the concavity property of f m suggests that 1 (0). But in this case we contradicting the assumption that m 2 \Pi. Hence -m ? 0 and by Equation 0.In view of Lemma 1, we present the algorithm to solves Problem OPT-L in Figure 13 . Algorithm OPT-L(F,d) 1 Evaluate the solution SOPT of the optimization problem (without inequality constraints) 2 If all the inequality constraints are satisfied then SOPT \GammaL =SOPT ; exit 3 Compute \Pi from equation 6 goto Step 1 Figure 13: Algorithm to solve Problem OPT-L Complexity: The time complexity COPT (n) of Algorithm OPT is O(n) (If the mentioned closed apply, then the complexity is clearly linear. Otherwise the unique unknown - can be solved in linear time under concavity assumptions, as indicated in [14, 15, 18]). Lemma 1 immediately implies the existence of an algorithm which sets t re-invokes Algorithm OPT for the remaining tasks and slack (in case that some inequality constraints are violated by SOPT ). Since the number of invocations is bounded by n, the complexity of the algorithm which solves OPT-LU is O(n 2 ). 7.2.1 Faster Solution for Problem OPT-L In this section, we present a faster algorithm of time complexity O(n \Delta log n), to solve OPT-L. We will make use of the new (faster) algorithm in the final solution of OPT-LU. Consider Algorithm OPT-L depicted in Fig. 13. Let F k be the set of functions passed to OPT during the k th iteration of Algorithm OPT-L, and \Pi k be the set of functions with minimum marginal returns at the lower bounds (minimum w i (0) values) during the k th iteration (Formally, \Pi by f 0 y be the number of distinct w i (0) values among functions in F, and m - n be the iteration number where Algorithm OPT returns a solution set which satisfies the constraint set given by (28) for the remaining t i values. Note that the elements of \Pi be produced in O(n \Delta log n) time during the preprocessing phase. Clearly, Algorithm OPT-L sequentially sets returns a solution which does not violate the constraint set for the remaining unknowns at the (m ) th iteration. A tempting idea is to use binary search in the range [1; n ] to locate the critical index m in a faster way. However, to justify the correctness of such a procedure one needs to prove that if one had further set invoked subsequently the algorithm OPT, then SOPT obtained in this way would have still satisfied the constraint set given by (28). Notice that if this property does not hold, then it is not possible to determine the "direction" of the search by simply testing SOPT at a given index i, since we must be assured that there exists a unique index m such that: setting invoking OPT does not provide a solution SOPT which satisfies the inequality constraints whenever 1 - setting invoking OPT does provide a solurion SOPT which satisfies the inequality constraints whenever m The first of these properties follows directly from the correctness of Algorithm OPT-L. It turns out that the second property also holds for concave objective functions as proven below. Hence, the time complexity COPT \GammaL (n) may be reduced to O(n \Delta log n) by using a binary search like technique. Algorithm FAST-L which solves Problem OPT-L in time O(n \Delta log n) is shown in Figure 14. 7.2.2 Correctness proof of the Fast Algorithm We begin by introducing the following additional notation regarding the k th iteration of Algorithm OPT-L. assigned to the optional part of task T i by OPT during the k th iteration of Algorithm OPT-L solution produced by OPT during the k th iteration of Algorithm OPT-L 0g: the set of indices for which the solution SOPT;k violates inequality constraints. Clearly, Algorithm OPT-L successively sets t returns a solution which does not violate any constraints for functions in Fm at the (m ) th iteration. Algorithm FAST-L uses binary search to determine the critical index m efficiently. The correctness of Algprithm OPT-L assures that 8 invoking OPT Algorithm FAST-L(F,d) Evaluate SOPT of the corresponding Problem OPT 2 If all the constraints are satisfied then SOPT \GammaL =SOPT ; exit 3 Enumerate the functions in F according to the nondecreasing order of w j (0) values and construct the sets \Pi 6 Evaluate SOPT by invoking OPT(\Pi m+1 [ 7 if a constraint is violated 9 else f Evaluate SOPT by invoking 12 if a constraint is violated 14 else f Figure 14: Fast Algorithm for Problem OPT-L would yield a non-empty violating set V y for the remaining tasks. Finally, Proposition 4 establishes that 8 y - m \Gamma 1, V y will always remain empty after setting t and invoking OPT, since this would leave even more slack for the remaining tasks. In the algorithm FAST-L, a specific index m is tested at each iteration to check whether it satisfies the property and ;. If this is the case, then since there is only one index m satisfying this property. However, in case that Vm 6= ; then we can infer that and the next probe is determined in the range (m; n ). Finally, if both then we restrict the search in the range (0; Proposition 4 Suppose that, during the execution of Algorithm OPT-L, SOPT;k does not violate some inequality constraints (i.e., . Then the th invocation of Algorithm OPT for the remaining tasks yields SOPT;k+1 such that t i;k+1 - t i;k for all (which implies that V k+1 is still empty). Proof: Note that t i;k - 0 8 by assumption. Based on the optimality property of subproblems, if the k th invocation of Algorithm OPT yields an optimal solution, it will also generate the optimal distribution of d \Gamma among functions in F k \Gamma \Pi k . However, the th invocation provides optimal distribution of d among functions in F k \Gamma \Pi k as well (by setting t Thus, two successive invocations of Algorithm OPT can be written as: maximize subject to and maximize subject to Hence the proof will be complete if we show that 8i 2 F Any solution SOPT;k should satisfy first-order necessary conditions for Lagrangian [22]: The necessary conditions (34) giveb c b d For the sake of contradiction, assume that 9w 2 F there must be some y 2 F k \Gamma \Pi k such that t y;k+1 ? t y;k . We distinguish two cases: 1. f w is nonlinear, that is, its derivative is strictly decreasing. Since f y is also concave, we can by f 0 by f 0 are clearly inconsistent with Equations (35) and (36). This can be easily seen by substituting w for c and y for d in Equations (35) and (36), respectively. 2. f w is linear, which implies that f 0 j. In this case, to satisfy Equations (35) and y should be also linear of the form f y bw . Hence, two functions have the same marginal return by But remembering our assumption from Section 7.1 that Algorithm OPT treats all linear functions of the same marginal return "fairly" (that is, assigns them the same amount of service time), we reach a contradiction since t w;k was supposed to be greater than t y;k . Complexity: At most O(log n) probes are made during binary search and at each probe Algorithm OPT is called twice. Recall that Algorithm OPT has the time complexity O(n). The initial cost of sorting the derivative values is O(n log n). Hence the total complexity is COPT \GammaL log log n), which is O(n \Delta log n). 7.3 Combining All Constraints: Solution of Problem OPT-LU An instance of Problem OPT-LU is characterized by the set F= ff of nondecreasing, differentiable, and concave reward functions, the set O= of upper bounds on the length of optional execution parts, and available slack d: maximize subject to We recall that in the specification of OPT-LU. We first observe the close relationship between the problems OPT-LU and OPT-L. Indeed, OPT-LU has only an additional set of upper bound constraints. It is not difficult to see that if SOPT \GammaL satisfies the constraints given by Equation (39), then the solution SOPT \GammaLU of problem OPT-LU is the same as SOPT \GammaL . However, if an upper bound constraint is violated then we will construct the solution iteratively in a way analogous to that described in the solution of Problem OPT-L. contains the functions f x 2 F with the largest marginal returns at the upper bounds, w x (o x ). The algorithm ALG-OPT-LU (see Figure 15) which solves the problem OPT-LU is based on the successive invocations of FAST-L. First, we find the solution SOPT \GammaL of the corresponding problem OPT-L. We know that this solution is optimal for the simpler problem which does not take into account upper bounds. If the upper bound constraints are automatically satisfied, then we are done. However, if this is not the case, we set t \Gamma. Finally, we update the sets F, O and the slack d before going through the next iteration. Correctness: Most of the algorithm is self-explanatory in view of the results obtained in previous sections. However, Line 5 of ALG-OPT-LU requires further elaboration. In addition to constraints (38), (39) and (40), the necessary and sufficient Kuhn-Tucker conditions for Problem OPT-LU can be expressed as: Algorithm OPT-LU(F,O,d) 3 Evaluate SOPT \GammaL by invoking Algorithm FAST-L 4 if all upper bound constraints are satisfied then set F=F\Gamma\Gamma 9 set O=O\Gammafo x jx 2 \Gammag Figure 15: Algorithm to solve Problem OPT-LU are Lagrange multipliers. violates upper bound constraints given by Equation (39) then 9i - Proof: Assume that 8i - In this case, Kuhn-Tucker conditions (42) and (44) vanish. Also conditions (38), (40), (41), (43) and (45) become exactly identical to Kuhn-Tucker conditions of Problem OPT-L. Thus, SOPT \GammaL returned by Algorithm FAST-L is also equal to SOPT \GammaLU if and only if it satisfies the extra constraint set given by (39).Claim 5 8i Proof: Assume that 9i such that - In this case, (42) and (43) force us to choose which implies that this is contrary to our assumption that the specification of the problem. Now we are ready to justify line 5 of the algorithm: violates upper bound constraints given by (39) then , in SOPT \GammaLU , t \Gamma. Proof: We will prove that the Lagrange multipliers - are all non-zero, which will imply (by (42)) that t We know that 9j such that 5. Using (41) we can which gives (since t necessarily less than or equal to , we can deduce 1 contradicts our assumption that m 2 \Gamma.Complexity: Notice that the worst case time complexity of each iteration is equal to that of Algorithm FAST-L, which is O(n \Delta log n). Observe that the cardinality of F decreases by at least 1 after each iteration. Hence, the number of iterations is bounded by n. It follows that the total time complexity COPT \GammaLU (n) is O(n 2 \Delta log n). 8 Conclusion In this paper, we have addressed the periodic reward-based scheduling problem. We proved that when the reward functions are convex, the problem is NP-Hard. Thus, our focus was on linear and concave reward functions, which adequately represent realistic applications such as image and speech processing, time-dependent planning and multimedia presentations. We have shown that for this class of reward there exists always a schedule where the optional execution times of a given task do not change from instance to instance. This result, in turn, implied the optimality of any periodic real-time policy which can schedule a task set of utilization k on k processors. The existence of such policies is well-known in real-time systems community: RMS (with harmonic periods), EDF and LLF for uniprocessor systems, and in general, any scheduling policy which can fully utilize a multiprocessor system. We have also presented polynomial-time algorithms for computing the optimal service times. We believe that these efficient algorithms can be also used in other concave resource allocation/QoS problems such as the one addressed in [26]. We underline that besides clear and observable reward improvement over previously proposed sub-optimal policies, our approach has the advantage of not requiring any runtime overhead for maximizing the reward of the system and for constantly monitoring the timeliness of mandatory parts. Once optimal optional service times are determined statically by our algorithm, an existing (e.g., EDF) scheduler does not need to be modified or to be aware of mandatory/optional semantic distinction at all. In our opinion, this is another major benefit of having pre-computed and optimal equal service times for a given task's invocations in reward-based scheduling. In addition, Theorem 1 implies that as long as we are concerned with linear and concave reward functions, the resource allocation can be also made in terms of utilization of tasks without sacrificing optimality. In our opinion, this fact points to an interesting convergence of instance-based [7, 21] and utilization-based [26] models for the most common reward functions. About the tractability issues regarding the nature of reward functions, the case of step functions was already proven to be NP-Complete ([21]). By efficiently solving the case of concave and linear reward functions and proving that the case of convex reward functions is NP-Hard, efficient solvability boundaries in (periodic or aperiodic) reward-based scheduling have been reached by our work (assuming Finally, we have provided examples to show that the theorem about the optimality of identical service times per instance no longer holds, if we relax some fundamental assumptions such as the deadline/period equality and the availability of the dynamic priority scheduling policies. Considering dynamic aperiodic task arrivals and investigating good approximation algorithms for intractable cases such as step functions and error cumulative jobs can be major avenues for reward-based scheduling. --R A Polynomial-time Algorithm to solve Reward-Based Scheduling Prob- lem Fairness in periodic real-time scheduling Solving time-dependent planning problems Scalable Video Coding using 3-D Subband Velocity Coding and Multi-Rate Quan- tization Scalable Video Data Placement on Parallel Disk data arrays. Scheduling periodic jobs that allow imprecise results. Algorithms for Scheduling Real-Time Tasks with Input Error and End-to-End Deadlines An extended imprecise computation model for time-constrained speech processing and generation A dynamic priority assignment technique for streams with (m Architectural foundations for real-time performance in intelligent agents Reasoning under varying and uncertain resource constraints Efficient On-Line Processor Scheduling for a Class of IRIS (Increasing Reward with Increasing Service) Real-Time Tasks Algorithms and Complexity for Overloaded Systems that Allow Skips. Imprecise Results: Utilizing partial computations in real-time systems Scheduling Algorithms for Multiprogramming in Hard Real-time Environment Algorithms for scheduling imprecise computations. Linear and Nonlinear Programming Scheduling Algorithms for Fault-Tolerance in Hard-Real-Time Systems Fundamental Design Problems of Distributed systems for the Hard Real-Time Environment Classical Optimization: Foundations and Extensions A Resource Allocation Model for QoS Management. Algorithms for scheduling imprecise computations to minimize total error. Image Transfer: An end-to-end design Producing monotonically improving approximate answers to relational algebra queries. Anytime Sensing --TR --CTR R. M. Santos , J. Urriza , J. Santos , J. Orozco, New methods for redistributing slack time in real-time systems: applications and comparative evaluations, Journal of Systems and Software, v.69 n.1-2, p.115-128, 01 January 2004 Hakan Aydin , Rami Melhem , Daniel Moss , Pedro Meja-Alvarez, Power-Aware Scheduling for Periodic Real-Time Tasks, IEEE Transactions on Computers, v.53 n.5, p.584-600, May 2004 Shaoxiong Hua , Gang Qu , Shuvra S. Bhattacharyya, Energy reduction techniques for multimedia applications with tolerance to deadline misses, Proceedings of the 40th conference on Design automation, June 02-06, 2003, Anaheim, CA, USA Xiliang Zhong , Cheng-Zhong Xu, Frequency-aware energy optimization for real-time periodic and aperiodic tasks, ACM SIGPLAN Notices, v.42 n.7, July 2007 Melhem , Nevine AbouGhazaleh , Hakan Aydin , Daniel Moss, Power management points in power-aware real-time systems, Power aware computing, Kluwer Academic Publishers, Norwell, MA, 2002 Melhem , Daniel Moss, Maximizing rewards for real-time applications with energy constraints, ACM Transactions on Embedded Computing Systems (TECS), v.2 n.4, p.537-559, November Jeffrey A. Barnett, Dynamic Task-Level Voltage Scheduling Optimizations, IEEE Transactions on Computers, v.54 n.5, p.508-520, May 2005 Lui Sha , Tarek Abdelzaher , Karl-Erik rzn , Anton Cervin , Theodore Baker , Alan Burns , Giorgio Buttazzo , Marco Caccamo , John Lehoczky , Aloysius K. Mok, Real Time Scheduling Theory: A Historical Perspective, Real-Time Systems, v.28 n.2-3, p.101-155, November-December 2004

deadline scheduling;real-time systems;reward maximization;periodic task scheduling;imprecise computation

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Optimal covering tours with turn costs.

We give the first algorithmic study of a class of covering tour problems related to the geometric Traveling Salesman Problem: Find a polygonal tour for a cutter so that it sweeps out a specified region (pocket), in order to minimize a cost that depends not only on the length of the tour but also on the number of turns. These problems arise naturally in manufacturing applications of computational geometry to automatic tool path generation and automatic inspection systems, as well as arc routing (postman) problems with turn penalties. We prove lower bounds (NP-completeness of minimum-turn milling) and give efficient approximation algorithms for several natural versions of the problem, including a polynomial-time approximation scheme based on a novel adaptation of the m-guillotine method.

Introduction An important algorithmic problem in manufacturing is to compute effective paths and tours for covering ("milling") a given region ("pocket") with a cutting tool: Find a path or tour along which to move a prescribed cutter in order that the sweep of the cutter (exactly) covers the re- gion, removing all of the material from the pocket, while not "gouging" the material that lies outside of the pocket. This covering tour problem and its variants arise not only in NC machining applications but also in several other applications, including automatic inspection, spray paint- ing/coating operations, robotic exploration, arc routing, and even mathematical origami. While we will often speak of the problem as "milling" with a "cutter", many of its important applications arise in various contexts outside of machining. Department of Applied Mathematics and Statistics, State University of New York, Stony Brook, NY 11794-3600, festie, jsbmg @ams.sunysb.edu. y Department of Computer Science, State University of New York, Stony Brook, NY 11794-4400, fbender, saurabhg@cs.sunysb .edu. z Department of Computer Science, University of Waterloo, Water- loo, Ontario N2L 3G1, Canada, eddemaine@uwaterloo.ca. x Fachbereich Mathematik, Technische Universitat Berlin, 10623 Berlin, Germany, fekete@math.tu-berlin.de. The majority of research on these geometric covering tour problems as well as on the underlying arc routing problems in networks has focused on cost functions based on the lengths of edges. However, in many actual routing problems, this cost is dominated by the cost of switching paths or direction at a junction. A drastic example is given by fiber-optical networks, where the time to follow an edge is negligible compared to the cost of changing to a different frequency at a router. In the context of NC machining, turns represent an important component of the objective function, as the cutter may have to be slowed in anticipation of a turn. The number of turns (or the "link naturally as an objective function in robotic exploration (minimum-link watchman tours) and in various arc routing problems (snow plowing or street sweeping with turn penalties). In this paper, we address the problem of minimizing the cost of turns in a covering tour. This important aspect of the problem has been left unexplored so far in the algorithmic community; the arc routing community has examined only heuristics or exact algorithms that do not have performance guarantees. We provide several new results: (1) We prove that the covering tour problem with turn costs is NP-complete, even if the objective is purely to minimize the number of turns, the pocket is orthogonal (rectilinear), and the cutter must move axis- parallel. The hardness of the problem is not apparent, as our problem seemingly bears a close resemblance to the polynomially-solvable Chinese postman prob- lem; see the discussion below. (2) We provide a variety of constant-factor approximation algorithms that efficiently compute covering tours that are nearly optimal with respect to turn costs in various versions of the problem. While getting some O(1)-approximation is not difficult for most problems in this class, through a careful study of the structure of the problem, we have developed tools and techniques that enable significantly stronger approximation results. One of our main results is a 3.75-approximation for minimum-turn axis-parallel tours for a unit square cutter that covers an integral orthogonal polygon (with holes). Another main result gives a 4/3-approximation for minimum-turn tours in a "thin" pocket, as arises in the arc routing version of our problem. Table summarizes our various results. Cycle Cover Tour Simultaneous Maximum Milling problem APX APX Length APX Coverage Restricted-direction geometric 5d 5d Orthogonal 2.5 3.75 4 4 Integral orthogonal 4 6 4 4 Orthogonal thin 1 4=3 4 4 Table 1: Approximation factors achieved by our algorithms. (See Section 2 for the definitions of , , , d.) (3) We devise a polynomial-time approximation scheme (PTAS) for the covering tour problem in which the cost is given as a weighted combination of length and number of turns; e.g., the Euclidean length plus a constant C times the number of turns. For a polygon with h holes, the running time is O(2 h N O(C) ). The PTAS involves an extension of the m-guillotine method, which has previously been applied to obtain PTAS's in problems involving only length. Related Work. In the CAD community, there is a vast literature on the subject of automatic tool path genera- tion; we refer the reader to Held [21] for a survey and for applications of computational geometry to the prob- lem. The algorithmic study of the problem has focussed on the problem of minimizing the length of a milling tour: Arkin et al. [5, 6] show that the problem is NP-hard in general. Constant-factor approximation algorithms are given in [5, 6, 23], with the best current factor being a 2.5-approximation for min-length milling (11/5- approximation for orthogonal simple polygons). For the closely related lawn mowing problem (also known as the "traveling cameraman problem" [23]), in which the covering tour is not constrained to stay within P , the best current approximation factor is (3 (utilizing PTAS results for TSP). Also closely related is the watchman route problem with limited visibility (or "d-sweeper prob- lem"), as studied by Ntafos [31], who provides a 4/3- approximation, which is improved to a 6/5-approximation by [6]. The problem is also closely related to the Hamiltonicity problem in grid graphs; the recent results of [32] suggest that in simple polygons, minimum-length milling may in fact have a polynomial-time algorithm. Covering tour problems are related to watchman route problems in polygons, which have had considerable study in terms of both exact algorithms (for the simple polygon case) and approximation algorithms (in gen- see [29] for a recent survey. Most relevant to our problem is the prior work on minimum-link watchman tours: see [2, 3, 8] for hardness and approximation re- sults, and [14, 25] for combinatorial bounds. However, in these problems the watchman is assumed to see arbitrarily far, making them distinct from our tour cover problems. Other algorithmic results on milling include a recent study of multiple tool milling by Arya, Cheng, and Mount [9], who give an approximation algorithm for minimum-length tours that use different size cutters, and a recent paper of Arkin et al. [7], who examine the problem of minimizing the number of retractions for "zig-zag" machining without "re-milling", showing that the problem is NP-complete and giving an O(1)-approximation algorithm Geometric tour problems with turn costs have been studied by Aggarwal et al. [1], who prove NP-complete the angular-metric TSP, in which one is to compute a tour on a set of points in order to minimize the sum of the direction changes at each vertex. Fekete [17] and Fekete and Woeginger [18] have studied a variety of angle-restricted tour (ART) problems. In the operations research literature, there has been an extensive literature on arc routing problems, which arise in snow removal, street cleaning, road gritting, trash col- lection, meter reading, mail delivery, etc.; see the surveys of [10, 15, 16]. Arc routing with turn costs has had considerable attention recently, as it enables a more accurate modeling of the true routing costs in many sit- uations. Most recently, Clossey et al. [13] present six heuristic methods of attacking arc routing with turn penal- ties, without resorting to the usual transformation to a TSP problem; however, their results are purely based on experiments and provide no provable performance guarantees. The directed postman problem with turn penalties has been studied recently by Benavent and Soler [11], who prove the problem to be (strongly) NP-hard and provide heuristics (without performance guarantees) and computational results. (See also Soler's thesis [19] and [30] for computational experience with worst-case exponential-time exact methods.) Our covering tour problem is related to the Chinese postman problem, which is readily solved exactly in polynomial time. However, the turn-weighted Chinese postman problem is readily seen to be NP-complete: Hamiltonian cycle in line graphs is NP-complete (contrary to what is reported in [20]; see page 246, West [33]), implying that TSP in line graphs is also NP-complete. The Chinese postman problem on graph G with turn costs at nodes (and zero costs on edges) is equivalent to TSP on the corresponding line graph, L(G), where the cost of an edge in L(G) is given by the corresponding turn cost in G. Thus, the turn-weighted Chinese postman problem is also NP-complete. Summary of Results. As we show in Section 3, all of the variants of our problem mentioned so far are NP- thus, our main interest is in approximation al- gorithms. It turns out that all of these problems have essentially constant-factor approximations. Table 1 summarizes our best approximation factors for each problem. The term "coverage" indicates the number of times a point is visited, which is of interest in several practical applica- tions. This parameter also provides an upper bound on the total length. Preliminaries Problem Definitions. The general geometric milling problem is to find a closed curve (not necessarily sim- ple) whose Minkowski sum with a given tool (cutter) is precisely a given region (pocket), P . Subject to this con- straint, we may wish to optimize a variety of objective functions, such as the length of the tour, or the number of turns in the tour. We call these problems minimum-length and minimum-turn milling, respectively. While the latter problem is the main focus of this paper, we are also interested in bicriteria versions of the problem in which both length and number of turns must be small, or some linear combination of the two (see Section 5.8). In addition to choices in the objective function, the problem version depends on the constraints on the tour. In the orthogonal milling problem, the region P is an orthogonal polygonal domain (with holes) and the tool is an (axis-parallel) unit-square cutter constrained to axis-parallel motion, with links of the tour alternating between horizontal and vertical. All turns are 90 ; a "U-turn" has cost of 2. Instead of dealing directly with a geometric milling problem, we often find it helpful to consider a more combinatorial problem, and then adapt the solution back to the geometric problem. The integral orthogonal milling problem is a specialization of the orthogonal milling problem in which the region P has integral vertices. In this case, an optimal tour can be assumed to have its vertex coordinates of the form k milling in an integral orthogonal polygon (with holes) is equivalent to finding a tour of all the vertices ("pixels") of a grid graph; see Figure 1. A more general combinatorial model than integral orthogonal milling is the discrete milling problem, in which we discretize the set of possible links into a finite collection of "channels" which are connected together at Figure 1: An instance of the integral orthogonal milling problem (left), and the grid graph model (right). "vertices." More precisely, the channels have unit "width" so that there is only one way to traverse them with the given unit-size tool. At a vertex, the tour has a choice of (1) turning onto another channel connected at that end (costing one turn), (2) going straight if there is an incident channel collinear with the source channel (costing no turns), or (3) "U-turning" back onto the source edge (costing two turns). Hence, this problem can be modeled by a graph with certain pairs of incident edges marked as "collinear," in such a way that the set of collinear pairs at each vertex is a (not necessarily perfect) matching. The discrete milling problem is to find a tour in such a graph that visits every vertex. (The vertices represent the "pixels" to be covered.) Integral orthogonal milling is the special case of discrete milling in a grid graph. Let (resp., ) denote the average (resp., maximum) degree of a vertex and let denote the average number of distinct "directions" coming together at a vertex, that is, the average over each vertex of the cardinality of the matching plus the number of unmatched edges at that vertex. The thin milling problem is to find a tour in such a graph that traverses every edge (and visits every vertex). Thus, the minimum-length version of thin milling is exactly the Chinese postman problem. As we have already noted, the minimum-turn version is NP-complete. The orthogonal thin milling problem is the special case in which the graph comes from an instance of orthogonal milling. A generalization of orthogonal milling is the geometric milling problem with a constant number d of allowed directions, which we call restricted-direction geometric milling. In particular, the region P can only have edges with the d allowed directions. This problem is not a sub-problem of discrete milling, since it does not decompose into a collection of nonoverlapping "vertices;" however, it turns out that the same results apply. Other Issues. It should be stressed that using turn cost instead of (or in addition to) edge length changes several characteristics of distances. One fundamental problem is illustrated by the example in Figure 2: the triangle inequality does not have to hold when using turn cost. This implies that many classical algorithmic approaches for graphs with nonnegative edge weights (such as using optimal 2-factors or the Christofides method for the TSP) cannot be applied without developing additional tools. a c Figure 2: The triangle inequality may not hold when using turn cost as distance measure: d(a; c) In fact, in the presence of turn costs we distinguish between the terms 2-factor and cycle cover. While the terms are interchangeable when referring to the set of edges that they constitute, we make a distinction between their respective costs: a "2-factor" has a cost consisting of the sum of edge costs, while the cost of a "cycle cover" includes also the turn costs at vertices. It is often useful in designing approximation algorithms for optimal tours to begin with the problem of computing an optimal cycle cover, minimizing the total number of turns in a set of cycles that covers P . Specif- ically, we can decompose the problem of finding an optimal tour into two tasks: finding an optimal cycle cover, and merging the components. Of course, these two processes may influence each other: there may be several optimal cycle covers, some of which are easier to merge than others. (In particular, we say that a cycle cover is connected, if the graph induced by the set of cycles and their intersections is connected.) As we will show, even the problem of optimally merging a connected cycle cover is NP-complete. This is in contrast to minimum-length milling, where an optimal connected cycle cover can trivially be converted into an optimal tour that has the same cost. Another important issue is the encoding of the input and output. In integral orthogonal milling, one might think that it is most natural to encode the grid graph, since the tour will be embedded on this graph and will, in general, have complexity proportional to the number of pixels. But the input to any geometric milling problem has a natural encoding by specifying the vertices of the polygon P . In particular, long edges are encoded in binary (or with one real number, depending on the model) instead of unary. It is possible to get a running time depending only on this size, but of course we need to allow for the output to be encoded implicitly. That is, we cannot explicitly encode each vertex of the tour, because there are too many (it can be arbitrarily large even for a succinctly encodable rectangle). Instead, we encode an abstract description of the tour that is easily decoded. Algorithms whose running time is polynomial in the explicit encoding size (pixel count) are pseudo-poly- nomial. Algorithms whose running time is polynomial in the implicit encoding size are polynomial. For our purposes it will not matter whether lengths are encoded with a single real number or in binary. Finally, we mention that many of our results carry over from the tour (or cycle) version to the path version, in which the cutter need not return to its original position. We omit discussion here of the changes necessary to compute optimal paths. We also omit in this abstract discussions of how our results apply also to the case of lawn mowing, in which the sweep of the cutter is allowing to go outside P during its motion. With so many problems of interest, we specify in every lemma, theorem, and corollary to which class of problems it applies. The default subproblem is to find a tour; if this is not the case (e.g., it is to find a cycle cover), we state it explicitly. 3 NP-Completeness Arkin et al. [6] have proved that the problem of optimizing the length of a milling is NP-hard. This implies that it is NP-hard to find a tour of minimum total length that visits all vertices. If, on the other hand, we are given a connected cycle cover of a graph that has minimum total length, then it is trivial to convert it into a tour of the same length by merging the cycles into one tour. In this section we show that if the quality of a tour is measured by counting turns, then even this last step of turning an optimal connected cycle cover into an optimal tour is NP-complete. This implies NP-hardness of finding a milling tour that optimizes the number of turns for a polygon with holes. THEOREM 3.1. Minimum-turn milling is NP-complete, even when we are restricted to the orthogonal thin case, and assume that we know an optimal (minimum-turn) connected cycle cover. See the full version [4] of this paper for proofs of this and most other theorems and lemmas. Because orthogonal thin milling is a special case of thin milling as well as orthogonal milling, and it is easy to convert an instance of thin orthogonal milling into an instance of integral orthogonal milling, we have COROLLARY 3.1. Discrete milling, restricted-direction geometric milling, orthogonal milling, and integral orthogonal milling are NP-complete. Approximation Tools There are three main tools that we use to develop approximation algorithms: computing optimal cycle covers for milling the "boundary" of P (Section 4.1), converting cycle covers into tours (Section 4.2), and utilizing optimal (or nearly-optimal) "strip covers" (Section 4.3). 4.1 Boundary Cycle Covers We consider first the problem of finding a minimum-turn cycle cover for covering a certain subset, P , of P that is along its boundary. Specifically, in orthogonal milling we define the boundary links to be orthogonal offsets, towards the interior of P , by 0:5 of each boundary edge. (In the nonorthogonal case, the notion of boundary link can be generalized; we defer the details to the full paper.) The region P is defined, then, to be the Minkowski sum of the boundary links and the tool, and we say that a cycle cover or tour mills the boundary if it covers P . (P is the union of pixels having at least one edge against the boundary of P .) We exploit the property that a cycle cover that mills the boundary (or a cycle cover that mills the entire region) can be assumed to include the boundary links in their entirety (without turns): LEMMA 4.1. Any cycle cover that mills the boundary can be converted into one that includes each boundary link as a portion of a link, without changing the number of turns. (a) (b) Figure 3: By performing local modifications, an optimal cycle cover can be assumed to cover each piece of the boundary in one connected link. This property allows us to apply methods similar to those used in solving the Chinese postman problem: we know portions of links that must be in the cycle cover, and furthermore these links mill the boundary. What remains is to connect these links into cycles, while minimizing the number of additional turns. We can compute the "turn distance" (which is one less than the link distance) between an endpoint of one boundary link and an endpoint of another boundary link. The crucial knowledge that we are using is the orientation of the boundary links, so, for example, we correctly compute that the turn distance is zero when two boundary links are collinear. Now we can find a minimum-weight perfect matching in the complete graph on boundary- link endpoints, with each edge weighted according to the corresponding turn distance. This connects the boundary links optimally into a set of cycles. This proves THEOREM 4.1. A min-turn cycle cover of the boundary of a region can be computed in polynomial time. Remark. Note that the definition of the "boundary" region P used here does not include all pixels that touch the boundary of P ; in particular, it omits the "reflex pixels" that share a corner, but no edge, with the boundary of P . It seems difficult to require that the cycle cover mill reflex pixels, since Lemma 4.1 does not extend to this case, and an optimal cycle cover of the boundary (as defined above) may have fewer turns than an optimal cycle cover that mills the boundary P plus the reflex pixels; see Figure 4. Figure 4: Optimally covering pixels that have an edge against the boundary can leave reflex pixels uncovered. 4.2 Merging Cycles It is often easier to find a minimum-turn cycle cover (or constant-factor approximation thereof) than to find a minimum-turn tour. Here, we show that an exact or approximate minimum-turn cycle cover implies an approximation for minimum-turn tours. THEOREM 4.2. A cycle cover with t turns can be converted into a tour with at most t + 2c turns, where c is the number of cycles. COROLLARY 4.1. A cycle cover of a connected rectilinear polygon with t turns can be converted into a single milling tour with at most 3 turns. Proof: Follows immediately from Theorem 4.2 and the fact that each cycle has at least four turns. 2 COROLLARY 4.2. If we could find an optimal cycle cover in polynomial time, we would have a 3-approximation algorithm for the number of turns. Unfortunately, general merging is difficult (as illustrated by the NP-hardness proof), so we cannot hope to improve these general merging results by more than a constant factor. 4.3 Strip and Star Covers A key tool for approximation algorithms is a covering of the region by a collection of "strips." In general, a strip is a maximal link whose Minkowski sum with the tool is contained in the region. A strip cover is a collection of strips whose Minkowski sums with the tool cover the region. A minimum strip cover is a strip cover with the fewest strips. LEMMA 4.2. The size of a minimum strip cover is a lower bound on the number of turns in a cycle cover (or tour) of the region. In the discrete milling problem, a related notion is a "queen placement." A queen is just a vertex, which can attack every vertex to which it is connected via a single link. A queen placement is a collection of queens no two of which can attack each other. LEMMA 4.3. The size of a maximum queen placement is a lower bound on the number of turns in a cycle cover (or tour) for discrete milling. In the integral orthogonal milling problem, the notions of strip cover and queen placement are dual, and efficient to compute: LEMMA 4.4. For integral orthogonal milling, a minimum strip cover and a maximum queen (rook) placement have equal size, and furthermore can be computed in time O(n 2:5 ). For general discrete milling, it is possible to approximate an optimal strip cover as follows. Greedily place queens until you cannot place anymore, in other words, until there is no unattackable vertex. This means that every vertex is attackable by some queen, so by replacing each queen with all possible strips through that vertex, we obtain a strip cover of size times the number of queens. (We call this type of strip cover a star cover.) But each strip in a minimum strip cover can only cover a single queen, so this is a -approximation to the minimum strip cover. We have thus proved LEMMA 4.5. In discrete milling, the number of stars in a greedy star cover is a lower bound on the number of strips, and hence serves as an -approximation algorithm for minimum strip covers. LEMMA 4.6. A greedy star cover can be found in linear time. 5 Approximation Algorithms 5.1 Discrete Milling Our most general approximation algorithm for the discrete milling problem has the additional feature of running in linear time. First we take a star cover according to Lemma 4.5 which approximates an optimal strip cover within a factor of . Then we tour the stars using an efficient method described below. Finally we merge these tours using Theorem 4.2. THEOREM 5.1. For min-turn cycle covers in discrete milling, there is a linear-time ( Furthermore, the maximum coverage of a vertex (i.e., the maximum number of times a vertex is swept) is , and the cycle cover is an -approximation on length. COROLLARY 5.1. For min-turn discrete milling, there is a linear-time the maximum coverage of a vertex is , and the tour is an -approximation on length. Note that, in particular, these algorithms give a 6- approximation on length if the discrete milling problem comes from a planar graph, since the average degree of a planar graph is bounded by 6. COROLLARY 5.2. For minimum-turn integral orthogonal milling, there is a linear-time 12-approximation that covers each pixel at most 4 times and hence is also a 4- approximation on length. Proof: In this case, 5.2 Restricted-Direction Geometric Milling As we already mentioned above, restricted-direction geometric milling is not a special case of discrete milling, but the same method and result applies: THEOREM 5.2. In restricted-direction geometric milling, there is a 5d-approximation on minimum-turn cycle covers that is linear in the number N of pixels, and a (5d+2)- approximation on minimum-turn tours of same complex- ity. In both cases, the maximum coverage of a point is at most 2d, so the algorithms are also 2d-approximations on length. Proof: Lemma 4.6 still holds, and each strip in a star (as described in the previous section) will be a full strip. The claim follows. 2 Note that this approximation algorithm applies to geometric milling problems in arbitrary dimensions provided that the number of directions is bounded, e.g., or- thohedral milling. As mentioned in the preliminaries, just the pixel count N may not be a satisfactory measure for the complexity of an algorithm, as the original region may be encoded more efficiently by its boundary, and a tour may be encoded by structuring it into a small number of pieces that have a short description. It is possible to use the above ideas for approximation algorithms in this extended framework. For simplicity, we describe how this can be done for the integral orthogonal case, where the set of pixels is bounded by n boundary edges. THEOREM 5.3. There is a 10-approximation of (strongly polynomial) complexity O(n log n) on minimum-turn cycle cover for a region of pixels bounded by n integral axis-parallel segments, and a 12-approximation on minimum- turn tours of same complexity. In both cases, the maximum coverage of a point is at most 4, so the algorithms are also 4-approximations on length. For the special case where the boundary is connected (i.e., the region does not have any holes), the complexities drop to O(n). 5.3 Integral Orthogonal We have already shown a 12-approximation for min-turn integral orthogonal milling, using a star cover, with a running time of O(N) or O(n log n). If we are willing to invest more time for computation, we can find an optimal rook cover (instead of a greedy one). As discussed in the proof of Lemma 4.4, this yields an optimal strip cover. This can be used to get a 6-approximation, with a running time of O(n 1:5 ). THEOREM 5.4. There is an O(n 2:5 )-time algorithm that computes a milling tour with number of turns within 6 times the optimal, and with length within 4 times the optimal. By more sophisticated merging procedures, it may be possible to reduce this approximation factor to something between 4 and 6. However, our best approximation algorithm uses a different strategy. THEOREM 5.5. There is an O(n 2:5 ) 2:5-approximation algorithm for minimum-turn cycle covers, and hence a polynomial-time 3:75-approximation for minimum-turn tours, for integral orthogonal milling. Proof: As described in Lemma 4.4, find an optimal strip cover S. Let s be its cardinality, then OPT s. Now consider the end vertices of the strip cover. By construction, they are part of the boundary. Any end point of a strip is either crossed orthogonally, or the tour turns at the boundary segment. In any case, a tour must have a link that crosses an end vertex orthogonally to the strip. (Note that this link has zero length in case of a u-turn.) Next consider the following distance function between end points of strips: For any pair of end points u and v (possibly of different strips s u and s v ), let w(u; v) be the smallest number of links from u to v when leaving u in a direction orthogonal to s u , and arriving at v in a direction orthogonal to s v . By a standard argument, an optimal matching M satisfies w(M) OPT=2. By construction, the edges of M and the strips of S induce a 2-factor of the end points. Since any matching edges leaves a strip orthogonally, we get at most 2 additional turns at each strip for turning each 2-factor into a cycle. The total number of turns is 2s+w(M) 2:5OPT. Since the strips cover the whole region, we get a feasible cycle cover. Finally, we can use Corollary 4.1 to turn the cycle cover into a tour. By the corollary, this tour does not have more than 3.75OPT turns. 2 A simple class of examples in [4] shows that the cycle cover algorithm may use 2OPT turns, and the tour algorithm may use 3OPT turns, assuming that no special algorithms are used for matching and merging. Moreover, the same example shows that this 3:75-approximation algorithm does not give an immediate length bound on the resulting tour. However, we can use a local modification argument to show the following: THEOREM 5.6. For any given feasible tour of an integral orthogonal region, there is a feasible tour of equal length that covers each pixel at most four times. This implies a performance ratio of 4 on the total length. 5.4 Nonintegral Orthogonal Polygons Nonintegral orthogonal polygons present a difficulty in that no polynomial-time algorithm is known to compute a minimum strip cover for such polygons. Fortunately, however, we can use the boundary tours from Section 4.1 to get a better approximation algorithm than the 12 from Corollary 5.2. THEOREM 5.7. In nonintegral orthogonal milling, there is a polynomial-time 6:25-approximation for minimum- turn cycle covers and 6-approximation for minimum-turn tours. The running time is O(n 2:5 ). Milling Thin Pockets In this section we consider the special case of milling thin pockets. Intuitively, a pocket is thin if it is composed of a network of width-1 corridors, where each pixel is adjacent to some part of the boundary of the region. A width-1 polygon is defined more formally as follows: DEFINITION 1. An orthogonal polygon has width-1 if no axis-aligned 2x2 square fits into the feasible region. Equivalently, a width-1 polygon is such that each pixel has all four of its corners on the boundary of the polygon. A width-1 pocket has a natural graph representation, described as follows. We associate vertices with some of the squares that comprise the pocket. Specifically, a vertex is associated with each square that is adjacent to more than two squares or only one square. Squares that have exactly two neighbors are converted into edges as follows: Vertices u and v are connected by an edge iff there is path in the pocket from u to v for which all other pixels visited have two neighbors. The weight of edge (u; v) is the number of turns in this path from u to v. In other words: Pixels with one, three, or four neighbors can be considered vertices of degree one, three, or four. Chains (possibly of length zero) of adjacent pixels of degree two can be considered edges between other vertices, if there are any. (Clearly, the problem is trivial if there are no pixels of degree three or higher; moreover, it is not hard to see that all pixels adjacent to a pixel of degree four can have at most degree two, and each pixel of degree three must have at least one neighbor of degree not exceeding two.) In the following, we will refer to this interpretation whenever we speak of the "induced graph" of a region. This interpretation illustrates that the problem is closely related to the Chinese Postman Problem (CPP), where the objective is to find a cheapest round trip in a graph with nonnegative edge weights, such that all edges are traversed. It is well-known that the CPP can be solved optimally in polynomial time by finding a minimum-cost matching of the odd-degree vertices in the graph. For thin milling, however, this reduction does not work: While it is possible to find a minimum-cost matching of the odd- degree vertices, the cost of traversing the resulting Eulerian graph is more than just the sum of edge weights, since we may have to add a cost for turning at the vertices where the matching is merged with the other edges in the graph. As we noted in the introduction, this implies that triangle inequality does not hold in general, and standard combinatorial algorithms based on edge weights only may fail to achieve a reasonable guaranteed performance. The difficulty of having turn costs at vertices is also illustrated by the proof in Section 3: Even in an Eulerian graph, where all pixels have degree 2 or 4, it is NP-complete to find a minimum cost roundtrip. In this section we describe how refined combinatorial arguments can achieve efficient constant-factor approxi- mations. Without loss of generality, we assume that there are no pixels of degree one: They force a doubled path to and from them, which can be merged with any tour of the remaining region at no extra cost. We already know from Theorem 4.1 that a minimum- turn cycle cover of a thin region can be found in polynomial time, because any cycle cover of the boundary can be turned into a cycle cover of the entire thin region. By also applying Theorem 4.2, we immediately obtain COROLLARY 5.3. In thin milling, there is a polynomial-time algorithm for computing a min-turn cycle cover, and a polynomial-time 1.5-approximation for min-turn tours. More interesting is that we can do much better than general merging in the case of thin milling. The idea is to decompose the induced graph into a number of cheap cycles, and a number of paths. 5.6 Milling Thin Eulerian Pockets We first solve the special case of milling Eulerian pockets, that is, pocket that can be milled without retractions, so that each edge in the corresponding graph is traversed by the cutting tool exactly once. In an Eulerian pocket, all nodes have either two or four neighbors, and so all vertices in the graph have exactly four neighbors. Although one might expect that the optimal milling is one of the possible Eulerian tours of the graph, in fact, this is not always true; see the full version [4] for a class of examples with this property. However, we can achieve the following approximation THEOREM 5.8. There is a linear-time algorithm that finds a tour of length at most 6=5OPT. 5.7 Milling Arbitrary Thin Pockets Now we consider the case of general width-1 pockets, where vertices may have degree one or three. (As above, we can continue to disregard vertices of degree one.) For any odd-degree vertex, and any feasible solution, some edges must be traversed multiple times. As motivation for this more general milling problem, recall the solution to the Chinese Postman Problem. Find a minimum-cost perfect matching in the complete graph on the odd-degree vertices, where the weight of an edge is the length of the shortest path between the two endpoints. Double the edges along the paths chosen in the perfect matching. (No edge will be doubled more than once by a simple local-modification argument.) Now take an Euler tour of this graph with doubled edges. This idea can be applied to our turn-minimization problem. Consider a degree three vertex, and for naming convenience give it a canonical orientation of the letter "T". We will need to visit and leave this vertex twice in order to cover it; that is, two paths through the T are required in order to mill it. There are several possibilities, which depend on which edge of the T is covered twice by the cutting tool. (See Figure 5.) (a) (b) (c) Figure 5: Ways of covering a degree three node, which we call a "T". (a) If one of the top edges is milled twice then there is only a single turn at the T. Said more concisely, if the stalk is milled i times then exactly i turns are required to mill the T. If the stalk of the T is milled twice then there are two ways to mill the T: (b) One path mills the top of the T, and the other enters on the stalk makes a "U" turn and exits the T on the same path that it entered. (c) Both paths mill the stalk and one of the top edges of the T. In both cases, if the stalk of the T is milled twice, then there are 2 turns at the T. We find a minimum weight cycle cover by having as few turns at vertices as possible. Specifically, the tool starts traveling along any edge and proceeds straight through each intersection without making any additional turns. Whenever the tool would retraverse an edge, a cycle is obtained, and the tool continues the process setting out from another edge. Whenever the tool enters a T from the stalk it stops. Thus, we obtain a disjoint collection of cycles and paths. Now we connect the odd degree vertices together as in a Chinese postman problem. Thus, we define the length of a path between two degree three vertices to be the number of turns along the path, plus a penalty of 1 for each time the path ends at the bottom leg of a T. Next, we find a minimum weight perfect matching on paths that connect odd degree vertices. When we combine the paths of the matching with the disjoint collection of paths and cycles we obtain an Eulerian graph. We connect the ends of the paths with the ends of the matching paths to form cycles. CLAIM 1. The above strategy yields a minimum cost cycle cover. Observe that this cycle cover is different from the cycle cover in the Eulerian case because the cycles are not disjoint sets of edges of the graphs. Some edges may appear multiple times. We now describe a 4=3-approximation algorithm for connecting the cycles together. 1. Find an optimal cycle cover as described in Claim 1. 2. Repeat until there is only one cycle in the cycle cover: If there are any two cycles that can be merged without any extra cost, perform the merge. Find a vertex at which two cycles cross each other. Modify the vertex to incorporate at most two additional turns, thereby connecting the two cycles. THEOREM 5.9. The algorithm described above finds a tour of length at most 4=3OPT. An example (shown in the full version [4] of this paper) proves that the estimate for the performance ratio is tight. 5.8 PTAS Here we outline a PTAS for the problem of minimizing a weighted average of the two cost criteria: length and number of turns. Our technique is based on using the theory of m-guillotine subdivisions [28], properly extended to handle turn costs. We prove the following result: THEOREM 5.10. For any fixed " > 0, there is a (1 ")-approximation algorithm with running time O(2 h that computes a milling tour for an integral orthogonal polygon P with h holes, where the cost of the tour is its length plus C times the number of (90-degree) turns, and N is the number of pixels in P . Proof: (sketch) Let T be a minimum-cost tour. Following the notation of [28], we first use the main structure theorem to show that there exists an m-guillotine subdivi- sion, obtained from T , with length at most (1 times the length of T . (Note that part of TG may lie outside the pocket P , since we added m-spans to make it m-guillotine.) We then convert TG into a new graph, G , which has the added properties (since we keep only those portions of the m-span that lie inside P ), (2) the number of edges of T incident on each component of the m-span is even (since T is a tour), and (3) the total cost of T 0 G is at most (1 times the cost of T . We then use dynamic programming to obtain a min-cost (modified) m-guillotine subdivision, T G , which has certain specified properties, including connect- edness, coverage, "bridge-doubling", and an even number of edges incident on each connected component of each m-span (this is the source of the 2 h term in the running time, as we need to be able to require a given parity at each sub-bridge, in order to have a connected Eulerian graph in the end, from which we can extract a tour). The techniques of [27] can be applied to reduce the exponent on N to a term independent of ". 2 We expect to be able to use the same methods to obtain a PTAS that is polynomial in n (versus N ), with a careful consideration of implicit encodings of tours. We have not yet been able to give a PTAS for minimizing only the number of turns in a covering tour; this remains an intriguing open problem. 6 Conclusion Many open problems remain: 1. Can we find a minimum-turn cycle cover in polynomial time? This would immediately lead to a 1.5- approximation for the orthogonal case, and a (1+ 2 approximation for the general case. Note that finding a minimum-cost cycle cover for a planar set of points was shown to be NP-complete by Aggarwal et al. [1]. 2. Is there a polynomial-time algorithm for exactly computing a minimum-turn covering tour for simple orthogonal polygons? The related problem for cost corresponding to distances is still open, but there is some evidence that it is indeed polynomial. 3. Is the analysis of the 3.75-approximation algorithm There may be some redundancy in the combination of all the estimates; however, our example shows that even one of the basic simplifications (considering only maximal strips with endpoints on the boundary) may lead to a factor 2 or 3, depending on how the merging is done. 4. What is the complexity of computing a minimum strip cover in nonintegral orthogonal polygons? 5. What is the complexity of computing minimum strip covers in nonorthogonal polygons? 6. Is there a strip cover approximation algorithm for d directions whose performance is independent of d? 7. Can one obtain approximation algorithms for unrestricted directions in an arbitrary polygonal domain? Acknowledgments We thank Regina Estkowski for helpful discussions. This research is partially supported by NSF grant CCR- 9732221, and by several grants from Bridgeport Ma- chines, Hughes Research Labs, ISX Corporation, Sandia National Labs, Seagull Technology, and Sun Microsystems --R The angular-metric traveling salesman problem Minimal link visibility paths inside a simple polygon. Finding an approximate minimum-link visibility path inside a simple polygon Optimal covering tours with turn costs. The lawnmower problem. Approximation algorithms for lawn mowing and milling. Optimization problems related to zigzag pocket machining. Approximation algorithms for multiple-tool milling Arc routing methods and applications. The directed rural postman problem with turn penalties. Triangulating a simple polygon in linear time. Solving arc routing problems with turn penalties. Improved lower bounds for the link length of rectilinear spanning paths in grids. Arc routing problems Arc routing problems Geometry and the Travelling Salesman Problem. Computers and In- tractability: A Guide to the Theory of NP-Completeness On the Computational Geometry of Pocket Machining Hamilton paths in grid graphs. The traveling cameraman problem Computational complexity and the traveling salesman problem. Link length of rectilinear Hamiltonian tours on grids. Combinatorial Optimization: Networks and Matroids. Guillotine subdivisions approximate polygonal subdivisions: Part III - Faster polynomial-time approximation schemes for geometric network optimiza- tion Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP Geometric shortest paths and network optimization. Watchman routes under limited visibility. Hamiltonian cycles in solid grid graphs. An Introduction to Graph Theory. --TR On the computational geometry of pocket machining Triangulating a simple polygon in linear time Watchman routes under limited visibility Minimal link visibility paths inside a simple polygon Finding an approximate minimum-link visibility path inside a simple polygon Angle-restricted tours in the plane Approximation algorithms for multiple-tool miling Improved lower bounds for the link length of rectilinear spanning paths in grids Guillotine Subdivisions Approximate Polygonal Subdivisions The angular-metric traveling salesman problem Approximation algorithms for lawn mowing and milling Computers and Intractability The Traveling Cameraman Problem, with Applications to Automatic Optical Inspection The Directed Rural Postman Problem with Turn Penalties Hamiltonian Cycles in Solid Grid Graphs --CTR D. Demaine , Sndor P. Fekete , Shmuel Gal, Online searching with turn cost, Theoretical Computer Science, v.361 n.2, p.342-355, 1 September 2006 Sndor P. Fekete , Marco E. Lbbecke , Henk Meijer, Minimizing the stabbing number of matchings, trees, and triangulations, Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms, January 11-14, 2004, New Orleans, Louisiana

lawn mowing;approximation algorithms;manufacturing;traveling salesman problem TSP;NP-completeness;NC machining;turn costs;milling;covering;m-guillotine subdivisions;polynomial-time approximation scheme PTAS

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Resolving Motion Correspondence for Densely Moving Points.

AbstractThis paper studies the motion correspondence problem for which a diversity of qualitative and statistical solutions exist. We concentrate on qualitative modeling, especially in situations where assignment conflicts arise either because multiple features compete for one detected point or because multiple detected points fit a single feature point. We leave out the possibility of point track initiation and termination because that principally conflicts with allowing for temporary point occlusion. We introduce individual, combined, and global motion models and fit existing qualitative solutions in this framework. Additionally, we present a new efficient tracking algorithm that satisfies thesepossibly constrainedmodels in a greedy matching sense, including an effective way to handle detection errors and occlusion. The performance evaluation shows that the proposed algorithm outperforms existing greedy matching algorithms. Finally, we describe an extension to the tracker that enables automatic initialization of the point tracks. Several experiments show that the extended algorithm is efficient, hardly sensitive its few parameters, and qualitatively better than other algorithms, including the presumed optimal statistical multiple hypothesis tracker.

Introduction Motion correspondence has a number of applications in computer vision, ranging from motion analysis, object tracking and surveillance to optical flow and structure from motion [11], [24], [25], [26]. Motion # The authors are with the Department of Mediamatics, Faculty of Information Technology and Systems, Delft University of Technology, P.O.Box 5031, 2600 GA, Delft, The Netherlands. E-mail: fC.J.Veenman, M.J.T.Reinders, E.Backerg@its.tudelft.nl correspondence must be solved when features are to be tracked that appear identical or that are retrieved with a simple feature detection scheme which loses essential information about its appearance. Hence, the motion correspondence problem deals with finding corresponding points from one frame to the next in the absence of significant appearance identification (see Fig. 1a). The goal is to determine a path or track of the moving feature points from entry to exit from the scene, or from the start to the end of the sequence. During presence in the scene, a point may be temporarily occluded by some object. Additionally, a point may be missed and other points may be falsely detected because of a failing detection scheme, as in Fig. 1b and Fig. 1c 1 . (a) Figure 1: Three moving points are measured at three time instances. The lines represent the point correspondences in time. In (a) all points are measured at every time instance. In (b) there is an extra or false measurement at t k+1 , and in (c) there is a missing measurement at t k+1 A candidate solution to the correspondence problem is a set of tracks that describes the motion of each point from scene entry to exit. We adopt a uniqueness constraint, stating that one detected point uniquely matches one feature point. When 2-D projections from a 3-D scene are analyzed, this is not trivial, because one feature point may obscure another. If we further assume that all M points are detected in all n frames, the number of possible track sets is (M!) n-1 . Among these solutions, there is a unique track set that describes the true motion of the M points. In order to identify the true motion track set, we need prior knowledge about the point motion, because otherwise all track sets are equally plausible. This knowledge 1 In the remainder of this paper we display the measurements from different time instances in one box and use t k labels to indicate the time the point was detected. can range from general physical properties like inertia and rigidity to explicit knowledge about the observed objects, like for instance the possible movements of a robot arm in the case that points on a robot arm are to be tracked. Clearly, generic motion correspondence algorithms cannot incorporate scene information. Moreover, they do not differentiate between the points in the scene, i.e. all points are considered to have similar motion characteristics. When many similar points are moving through a scene, ambiguities may arise, because a detected point may well fit correctly to the motion model of multiple features points. Additional ambiguities are caused by multiple detected points that fit correctly to the model of a single feature. These correspondence ambiguities can be resolved if combined motion characteristics are modeled, like for instance least average deviation from all individual motion models. Besides resolving these ambiguities we also have to incorporate track continuation in order to cope with point occlusion and missing detections. Other events that we may need to model are track initiation and track termination, so that features can enter and leave the scene respectively. The available motion knowledge is usually accumulated in an appropriate model. Then, a specific strategy is needed to find the optimal solution among the huge amount of candidate solutions defined by the model. When the nearest neighbor motion criterion is used, (see also Section 3) and there are neither point occlusions nor detection errors, track set optimality only depends on the point distances between any two consecutive frames. It is then legitimate to restrict the scope of the correspondence decision to one frame ahead, which we call a greedy matching solution to the correspondence problem. In other cases in which velocity state information is involved, correspondence decisions for one frame influence the optimal correspondence for the next frames and the problem becomes increasingly more complex. In such cases only a global matching over all frames can give the optimal result. In this paper, we consider the more difficult cases, i.e. dense and fast moving points, which makes the use of velocity state information essential. Because there are no efficient algorithms to find the optimal track set by global matching, only approximation techniques apply. Several statistical [2] and qualitative approximation techniques have been developed both in the field of target tracking and computer vision. Statistical Methods The two best known statistical approaches are the Joint Probabilistic Data-Association Filter (JPDAF) [9] and the Multiple Hypothesis Tracker (MHT) [20]. The JPDAF matches a fixed number of features in a greedy way and is especially suitable for situations with clutter. It does not necessarily select point measurements as exact feature point locations, but, given the measurements and a number of corresponding probability density functions, it estimates these positions. The MHT attempts to match a variable number of feature points globally, while allowing for missing and false detections. Quite a few attempts have been made to restrain the consequent combinatorial explosion, such as [3], [4], [5], [6], [15], [16]. More recently, the equivalent sliding window algorithms have been developed, which match points using a limited temporal scope. Then, these solve a multidimensional assignment problem, which is again NP-hard, but real-time approximations using Lagrangian relaxation techniques are available [7], [8], [17], [18], [23]. A number of reasons make the statistical approaches less suitable as solution to the motion correspondence problem. First, the assumptions that the points move independently and, more strongly, that the measurements are distributed normally around their predicted position may not hold. Second, since statistical techniques model all events as probabilities these techniques typically have quite a number of param- eters, such as the Kalman filter parameters, and a priori probabilities for false measurements, and missed detections. In general, it is certainly not trivial to determine optimal settings for these parameters. In the experiments section we show that the best known statistical method (MHT) is indeed quite sensitive to its parameter setting. Moreover, the a priori knowledge used in the statistical models is not differentiated between the different points. As a consequence, the initialization may be severely hampered if the initial point speeds are widely divergent, because the state of the motion models only gradually adapts to the measure- ments. Finally, the statistical methods that optimize over several frames, are despite their approximations computationally demanding, since the complexity grows exponentially with the number of points. Heuristic Methods Alternatively, a number of attempts has been made to solve the motion correspondence problem with deterministic algorithms [1], [12], [14], [19], [22]. These algorithms are usually conceptually simpler and have less parameters. Instead of probability density functions, qualitative motion heuristics are used to constrain possible tracks and to identify the optimal track set. By converting qualitative descriptions like smoothness of motion and rigidity into quantitative measures, a distance from the optimal motion can be expressed (where a zero distance makes a correspondence optimal). The most commonly known algorithm is the conceptually simple greedy exchange algorithm [22], which iteratively optimizes a local smoothness of motion criterion averaged over all points in a sequence of frames. The advantage of such deterministic algorithms is that it is quite easy to incorporate additional constraints, like (adaptive) maximum speed, and a maximum deviation from smooth motion, while this a priori knowledge can restrain the computational cost and improve the qualitative performance, e.g. [1], [10]. The main contributions of this paper are the presentation of a 1) qualitative motion modeling framework for the motion correspondence problem. We introduce the notion of individual motion models, combined motion models, and a global motion model, and we differentiate between strategies to satisfy these mod- els. Further, we propose a 2) new efficient algorithm that brings together the motion models, an optimal strategy, and an effective way to handle detection errors and occlusion. Finally, we present an extensive comparative performance evaluation of a number of different qualitative methods. The outline of the paper is as follows. We start by giving a formulation of the motion correspondence problem in the next section. Then in Section 4, we present our qualitative motion model and show how existing deterministic motion correspondence algorithms can be fit into it. Additionally, we present a new algorithm that effectively resolves motion correspondence using the presented model in Section 5. In Section 6, we compare the qualitative performance, the efficiency, and the parameter sensitivity of the described algorithms. Further, we show how the proposed algorithm can be extended with self-initialization and evaluate it with synthetic data experiments in Section 7. We broaden this evaluation in Section 8, with real-data experiments. We finish the paper with a discussion on possible extensions and some conclusions. Problem statement In this section we describe the motion correspondence problem as treated in this paper. In motion corre- spondence, the goal is tracking points that are moving in a 2-D space that is essentially a projection of a 3-D world. The positions of the points are measured at regular times, resulting in a number of point locations for a sequence of frames. For the moment, we assume that we have initial motion information of all points, which is given by point correspondences between the first two frames. From Section 7 onwards this restriction is lifted. Since the measured points are projections, points may become occluded and thus miss- ing. Moreover, the point detection may be imperfect, resulting in missing and false point measurements. Because long occlusion on the one hand and scene entrance and exit on the other hand are conflicting re- quirements, we leave out the possibility of track initiation and track termination, so the number of features to be tracked is constant. Applications using this problem definition range from object tracking in general, like animal tracking to perform behavior analysis, particle tracking, and cloud system tracking, to feature tracking for motion analysis. In the remainder of this paper, we abbreviate the moving points to 'points' and their measured 2-D projections to 'measurements'. More formally: There are M points, p i , moving around in a 3-D world. Given is a sequence of n time instances for which at each time instance t k there is a set X k of m k measurements x k . The measurements x k are vectors representing 2-D coordinates in a 2-D space, with dimensions S w (width) and S h (height). The number of measurements, m k , at t k , can be either smaller (occlusion) or larger (false measurements) than M . At t 1 , the M points are identified among the Moreover, the corresponding M measurements at t 2 are given. The task is to return a set of M tracks that represent the (projected) motion of the M points through the 2-D space from t 1 to t n using the movements between t 1 and t 2 as initial motion characteristics. A track T i , with is an ordered n-tuple of corresponding measurements: #x 1 . It is assumed that points do not enter or leave the scene and that the movement can be modeled independently. A track that has been formed up to t k is called a track head and is denoted as T k Qualitative Motion Modeling The assumption underlying the qualitative model that we advocate is that points move smoothly from time instance to time instance. That is, besides that individual points move smoothly, also the total set of points moves smoothly between time instances as well as over the whole sequence. Hereto, we define a qualitative model in which these qualitative statements are explicitly represented by a composition of motion models, that we have called the global motion model, the combined motion model and the individual motion model. The individual motion model represents the motion of individual points. To embed the motion smoothness constraints, we can make use of well-known general physical properties like rigidity and inertia. Without loss of generality, we only consider first-order motions, and thus leave out acceleration-state information. Consequently, the motion vector of a feature point can be estimated from only two consecutive measure- ments. On the basis of the motion vector and the adopted individual motion model, the position of the point at the next time instance can be predicted. The measurement that is closest to this prediction can then be selected as corresponding measurement. In reality, however, the points do not move exactly according to their predictions, because of shortcomings of the adopted individual motion model. These are among others caused by the limited order of the motion model, the fact that measurements are 2-D projections of 3-D movements, and by noise in the system. To express the misfit between a measurement and the predicted position, the candidate motion vector between the candidate measurement and the last measurement in the track is calculated. Using the inertia argument the cost representing the misfit is expressed in terms of the candidate motion and the previous true motion vector. These cost can be used to select the appropriate candidate measurement to make a correspondence. When points are moving far apart from each other or when they move reasonably according to their models, their measurements can easily be assigned to the corresponding feature point. With densely moving point sets, however, assignment conflicts can easily occur. That is, one measurement fits correctly multiple individual motion models or multiple measurements correctly fit one motion model. To resolve these ambiguities, the motion smoothness constraint is also imposed on the complete set of points. To this end, we introduce the combined motion model, that expresses the deviation from this motion constraint. As an example, we could enforce that the average deviation from the individual motion models is minimal. Even with the use of the combined motion model it is not always possible to decide on point correspon- dences. For that reason the motion smoothness constraint is additionally extended over the whole sequence in the global motion model. In the remainder of this section, we present some individual motion models, combined motion models and a global motion model and we give quantitative expressions for each of them. To simplify expressing the criteria that lead to the point tracks T i , we introduce the assignment matrix A where the entries a k have the following meaning: a k only if measurement x k+1 j is assigned to track head i and otherwise zero. Because some measurements are false and others are missing, there can be some measurements that are not assigned to a track head (all zeros in a column in A k ), and some track heads that have no measurement assigned to them (all zeros in a row in A k ). Or, more formally: a k a k We use two alternative notations for a correspondence between a measurement and a track head. First, we define # k as: Second, we use ordered pairs (i, j ) to indicate that measurement x k+1 j has been assigned to track head i . Z k then contains all assignment pairs from t k to t k+1 according to: Tracks T i can now be derived from A, which is the concatenation of the assignment matrices A k . We introduce a deviation matrix D k to denote all individual assignment costs c k i and measurements x . The assignment matrix identifies all correspondences from frame to frame, while the deviation matrix quantifies the deviation from the individual motion track per correspondence. The matrices A k and D k both have M rows and m k+1 columns. The rows represent the M track heads, T k , and the columns represent the j , that have been detected at t k+1 . Individual Motion Models We now formulate three individual motion models, together with an expression to compute a deviation from the optimal track. The first model uses only one previous measurement to predict the new position. We have indicated the dependence of only one previous measurement by the order of the individual model: O The other two individual models depend on two measurements and consequently have order O 2. The following motion criteria coefficients c k are all defined from track head T k i to a measurement x . im1 The nearest neighbor model does not incorporate velocity information. It only states that a point moves as little as possible from t k to t k+1 . h . (4) im2 The smooth motion model as introduced by Sethi and Jain [22] assumes that the velocity magnitude and direction change gradually. The smooth motion is formulated quantitatively in the following criterion: x r x im3 The proximal uniformity model by Rangarajan and Shah [19] assumes little motion in addition to constant speed. The deviation is quantified in the following criterion: Combined Motion Models Combined motion models serve to resolve correspondence conflicts between two successive frames in case of dense moving point sets, making the individual model errors dependent on each other. Next, we give two combined model criterions C k as a function of A k and D k , that are defined at t k over all established track heads. cm1 The average deviation model. This is a typical combined model which usually is realistic. It accounts for the average deviation from the optimal track according to the individual model [21], [22], [26]. Quantitatively, we use the generalized mean, which has a z parameter to differentiate between emphasis on large and small deviations from the optimal individual track (see Fig. 2). a k z (a) 04080120 (b) 04080120 Figure 2: Three moving points that are matched with im1 and cm1 using either As a consequence larger deviations are penalized more in (b). cm2 The average deviation conditioned by competition and alternatives model is derived from [1], [19]. In this combined model measurements are assigned to that track head that gives low deviation from the optimal track, while both the other tracks are less attractive for this measurement and the other measurements are less attractive for this track. a k where: R a (i ) =m R a (i ) represents the average cost of alternatives for T k i and R c ( j ) the average cost for competitors of . Global Motion Model To find the optimal track set (over all frames) according to a certain combined model, we need to compute the accumulated global motion deviation S(D) as in the following expression: A#U k=O im where U is the set of matrices A that satisfy Eq.1. That is, the overall minimum of averaged combined criterions defines the optimal track set. Because finding this minimum is computationally expensive a greedy matching is considered in this paper. This means that instead of finding correspondences over all frames, we establish optimal correspondences between two successive frames, given the state of the individual motion models and the combined model up to that moment. After these sub-optimal correspondences have been established the states of the individual models are adjusted and the next frame is considered. In other words, Eq.10 is approximated by minimizing separately, i.e.: k=O im min (D k ), where C k min (D k This approximation approach reduces the complexity of the problem considerably, although at the cost of greedy, possibly less plausible, correspondence decisions (see for example Fig. 3). In the remainder, we leave out the D and D k parameter for S, " S, and C k min respectively. (a) 10203010 20 (b) 10203010 20 Figure 3: Two moving points at four time instances. When the smooth motion model (im2) with the average deviation model (cm1) are assumed, (b) gives a two times lower deviation from the optimal path than (a). However, (a) is decided for when greedy matching is used. Model Constraints The motion models we have described so far allow for any point speed and for any deviation from smooth- ness. The models only state that those assignments are preferred that have little deviation from the individual model. There are, however, situations in which there is more knowledge available about the point motions, like the minimum speed (d min ) and the maximum speed (d max ) [1], [12], [14], [21], maximum violation of smoothness spatial or temporal adaptive speed and smoothness violation constraints [10]. When imposed on the individual motion models, these constraints enable the recognition of impossible assignments, which can be both qualitatively and computationally beneficial. These constraints can for instance be implemented by setting the individual criterion to a very high value, when some constraint is violated. The strategy (see next sub-section) that satisfies the models can exploit these constraints more adequately by leaving out of consideration those correspondences that violate the motion constraints. To find the optimal track set, we compute the global motion deviation. However, we are not interested in the actual value of S, but in the assignment matrix A that results in the minimal global motion deviation. In the next section, we first show how existing algorithms approximate the minimization of C k and consequently deliver a sub-optimal solution A k . In Section 5, we present an optimal as well as efficient algorithm to find that A k that minimizes C k . Algorithms Having modeled the feature point motion and having described quantitative expressions that can be used to identify the optimal track set, we now review a number of existing algorithms and fit them in our motion framework using our concept of individual and combined motion models. Further, we describe the strategy they use to find the optimal correspondences. Because all algorithms perform greedy matching, their task is to find C k min . S&S algorithm (im2/cm1/z=1) The first algorithm we looked at was originally developed by Sethi and Jain [22]. The original algorithm assumes a fixed number of feature points to be tracked and does not allow for occlusion and detection errors. Here, we describe the adjusted algorithm by Salari and Sethi [21] that partially fixes these shortcomings. The algorithm adopts a smooth motion model for individual motion (im2). The combined motion model is an average deviation model (cm1/z=1). To find an optimum of the global motion, the algorithm iteratively exchanges measurements between tracks to minimize the criterion on average. Initially, the tracks are led through the nearest neighboring measurements in the sequence. In this stage conflicts are 'resolved' on a first come first served basis. That is, at t k+1 measurements are assigned to the closest track parts T k i that have been formed up to t k to which no point was assigned yet. Consequently, the initialization procedure is a greedy im1/cm1 approximation. Then, each iteration step modifies at most two assignment pairs somewhere in the sequence, by exchanging the second entry of the pair. The algorithm considers all possible exchanges within the d max range of two track heads in the whole sequence and the exchange that gives the highest gain by decreasing the average criterion deviation is executed. The iteration phase stops when gain can no longer be obtained. The exchange gain between assignment pairs (i, p) and ( j, q) (see Eq.3) is defined in the following way: To achieve even better tracking results, the algorithm first optimizes correspondences over all frames in forward direction and then (after this iteration phase stops) it optimizes correspondences in backward direction. Only when the optimization process did not change anything in either direction, the algorithm stops. This bi-directional optimization process can indeed increase the tracking quality, but, unfortunately, this process is not guaranteed to converge, especially with densely moving points [19]. In contrast with what we said before, this algorithm seems to optimize over the whole sequence. How- ever, when we look carefully at the optimization process within one iteration phase, we see that this is only partially true. As long as the tracks are wrong at the start, exchanges in the remainder of the track will mostly be useless. This is due to the fact that the tracks were initialized using another criterion than the one that is considered in the iteration phase. Consequently, the optimization is only effective at the initial measurements of the tracks. This problem is most severe when the sequence is long and when the difference between the initialization criterion (nearest neighbor) and the optimization criterion (smooth motion) is large, i.e. with high speeds and high densities. We tested this statement by feeding to the S&S algorithm the example shown in Fig.3. If we do not optimize in both directions, the S&S algorithm indeed makes greedy correspondences as in Fig.3a, which supports the statement that S&S is a greedy matching algorithm. The Salari and Sethi version of this, so-called greedy exchange algorithm, additionally proposes a way to resolve track continuation, initiation and termination. They introduce a number of phantom points to the set of measurements in each frame. These phantom points serve as replacements of missing measurements, while satisfying local constraints. By imposing the maximum allowed local smoothness criterion and a maximum speed, missed measurements are recognized and filled in with phantom points. Moreover, the constraints also allow the detection of false measurements. Effectively, false measurements are replaced by phantom points if the introduction of a phantom point results in a lower criterion value. This approach generally works fine except that missing measurements (represented by a phantom point) always have the maximum criterion and displacement. For instance, if point p i has not been measured at t k , the algorithm can easily associate a measurement of p i at t k+1 to another point which is within the criterion range # max . It is important to remark that the phantom points only enforce that the local movement constraints are satisfied, but when a phantom point is put in a track, the track is in fact divided up into two tracks. In other words, this maximum criterion approach solves the correspondence problem up to the maximum criterion. Choosing a low maximum criterion leads to many undecided track parts and a higher maximum criterion leads to possibly wrong correspondences. This is where the track initiation/termination and occlusion events become conflicting requirements as already mentioned in Section 2. R&S algorithm (im3/cm2) A different approach to the correspondence problem is chosen by Rangarajan and Shah [19]. They have a different combined motion model and do not use an iterative optimization procedure. The R&S algorithm assumes a fixed number of feature points and it allows for temporary occlusion or missing point detections, but not for false detections. It uses the proximal uniformity model (im3) as individual motion model and cm2 as the combined motion model. This algorithm does not constrain the individual point motion, i.e. it does not have a d max or # max parameter. To find the minimum of the combined model (Eq.8), the authors use a greedy non-iterative algorithm. In each step of the algorithm, that particular point x k+1 j is assigned to track head T k i that has a low deviation from the optimal motion (low individual deviation) while on average all alternative track heads have a larger deviation with respect to x k+1 j and on average all other measurements have a worse criterion with respect to We continue the description of the algorithm in terms that fit the proposed motion framework as established in Section 3. The algorithm selects that assignment pair (i, j ) that maximizes R # a (i ) all minimal track head extensions, where R # a (i ) and R # c are derived from Eq.9 according to: a (i ) =m Then, an optimal assignment pair g(X t , repeatedly selected in the following way: a (p) q#Xm is the set of track head indices that have not yet been assigned a measurement, and X m is the set of measurement indices that have not yet been assigned to a track head. After an assignment has been found, the track head and measurement are removed from the respective index sets X t and X m . The algorithm accumulates the assignment costs, and eventually stops when X t is empty. The criterion computation can be summarized in the recurrence relation as follows: The matching assignment pairs are collected similarly: #, if X Consequently, this strategy results in the following approximation of C k and the set of assignment pairs as defined in Eq.3: Additionally, the algorithm differentiates between two cases: 1) all measurements are present and 2) some measurements are missing, by occlusion or otherwise. In the first case, the algorithm works as described above. Otherwise, because there is a lack of measurements at t k+1 , the problem is not which measurement should be assigned to which track head, but which track head should be assigned to which measurement. Then, the assignment strategy is similar to the above. When all track head assignments T k to measurements x k are found, it is clear for which tracks a measurement is missing. The R&S algorithm directly fills in these points with extrapolated points. The disadvantage of this track continuation scheme becomes apparent when the point occlusion lasts for a number of frames. Direct extrapolation results in a straight extension of the last recognized motion vector, which on the long term can deviate much from the true motion track so that recovering becomes increasingly difficult (see the experiments in Section 6.2.4). C&V algorithm (im2/cm2) The third and last scheme we describe, has been developed by Chetverikov and Verestoy [1]. Their method allows for track initiation, track termination, and occlusion only during two time instances. C&V assume the smooth motion model (im2) and cm2 as combined motion model. The algorithm extends track heads T k by first collecting all candidate measurements x k+1 j in the circle with radius d max around x k whose criterion does not exceed # max . The candidate measurements are considered in optimal criterion order with respect to the track head. Then, for each measurement all competing track heads are collected. The candidate measurement will be rejected if it is the best alternative for any of the competing track heads. When there are no candidates left, the track head will not be connected. Remaining unconnected track parts, caused by occlusion or otherwise, are handled in a post-processing step, which we leave out of the discussion. This scheme does not maximize the cost of the alternatives (i.e. w in Eq.8) and track heads are only then considered as competitors if they are within the d max as well as # max range. Moreover, their cost is not averaged as in Eq.8: any competitor that fits a measurement best, prevents that the measurement is assigned to T k The basics of this algorithm can be summarized as follows 2 . Let X a (i ) be the set of alternative track head extensions for track head T k i as defined below: X a (i q#Xm where each measurement x j has a set of competing track heads X c ( j ) according to: The algorithm selects a measurement from X a (i ) for a track head from X t according to: q#X a (i) Substituted into Eq.15-18, this leads to the minimal combined criterion approximation and corresponding set of assignment pairs Z k . The advantage of this scheme is that the d max parameter is exploited very efficiently. With low point densities, there is usually just one candidate point and there are no competing track heads for that point. However, higher point densities or large d max values can reveal the inadequacy of the strategy to find the minimal combined motion model deviation. Because the deviation is not averaged over competitors and alternatives, greedy assignment decisions are the result. 5 Optimal algorithm to minimize C k In the previous section we saw that known algorithms adopt a sub-optimal search strategy to minimize C k . In this section, we propose an algorithm that finds the minimum of the combined motion models efficiently. We describe the algorithm in our own terms assuming a fixed number of points and verification depth = 2, see [1] for details. To this end, we use the Hungarian algorithm, which efficiently finds the solution of the classical assignment problem [13]. Danchick and Newman [6] first used it in a similar context; to find hypotheses for the Multiple Hypothesis Tracker. In general, the algorithm minimizes the following expression: a subject to: a a It typically finds the minimal cost assignment, which can be represented in a weighted bipartite graph consisting of two sets of vertices, X and Y . The m vertices from X are connected to all m vertices of Y with weighted edges w i j . The algorithm then assigns every vertex from X to a separate vertex in Y in such a way that the overall cost is minimized. In order to be able to apply the Hungarian algorithm and to handle detection errors and occlusion, we prepare the measurement data such that the problem becomes squared. We propose to handle the false detection problem by introducing false tracks as proposed earlier in [26]. False tracks do not have to adhere to any motion criterion, so that measurements that do not fit the motion model of any true track will be moved to these false tracks. By associating a maximum cost deviation (# max ) to assignments to false tracks, we even recognize false measurements if other measurements are missing. We propose to implement track continuation by introducing the concept of slave measurements (Fig. 4a), similar to the interpolation scheme in [26]. Slave measurements have two states: free and bound. A free slave is not willing to be assigned to a track. Consequently, it has a maximum deviation cost from the optimal motion track. Free slave measurements serve similar goals as the phantom points in [21]. A slave measurement is bound when it has been assigned to a track, despite its high deviation. Bound slaves imitate the movements of their neighboring measurements. Their position is calculated by interpolating the positions of preceding and succeeding measurements in the track established so far (Fig. 4b). The interpolated positions enable more accurate calculation of the motion criterion. In this way, we retain as much motion information as possible and we are therefore able to make plausible correspondences. Additionally, we assign high cost (> # max ) to correspondences that have d max exceeded. This ensures that in such cases, a slave measurement is preferred over a measurement that does not fit the model constraints. (a) false measurement true measurement optional bound slave positions optional track head extensions Figure 4: (a) shows a true measurement, a false measurement and a free slave measurement at t k+1 . The slave measurement is on the border of the dotted circle. (b) shows possible bound slave measurement positions related to possible track head extensions. Greedy Optimal Assignment (GOA) Tracker: Formal description To properly handle missing and false measurements, we extend the assignment matrices A k . That is, we want to be able to assign false measurements to false tracks and slave measurements to true track heads that have no measurement at t k+1 . Since all measurements can be false and all track heads may miss their measurement, we add m k+1 rows to allow for m k+1 false tracks, and we add M columns to allow for M slave measurements, resulting in the definition of the square matrix A k (resembling the dummy rows and columns in the validation matrix as proposed in [9]). The size of the individual criterion matrix is adjusted similarly. The entries in the m k+1 extra rows and in the M extra columns all equal the maximum cost resulting in cost matrix D k Having defined these square matrices the linear assignment problem can be solved for one frame after the other, assuming that the correspondences between the first two frames are given (in case O i m > 1) to be able to compute the initial velocity vector. In order to calculate the motion criterion, the individual motion models with O need the vector need either of x # k or x k i is a slave measurement, we estimate these vectors by scanning back in T k i to collect two true measurements in the nearest past being x and x respectively, with 1 # p < q # k and # k#q means k - q times recursive application of # k Consequently, the vector estimates are defined as follows: Having obtained these velocity vector estimates, we can now compute the individual motion criteria c k We transform the criterion matrix to a bipartite graph and prune all edges with weights that exceed # max . Then, to satisfy the combined motion model, we adjust the edge weights w i j as defined below. cm1 average deviation model: cm2 average deviation conditioned by competition and alternatives, using Eq. 13; a (i As mentioned before, the actual value of the minimized C k is not important. Therefore in cm1, the 1/z power can be ignored, because the 1/z power function is monotonic increasing. Algorithm 1. Starting with in the cost matrix D k as follows: (a) true tracks to true measurements, i.e. 1 # If the maximum speed (d max ) constraint is violated then c k Otherwise c k is according to the individual motion model. (b) all other entries: c k 2. Construct a bipartite graph based on the criterion matrix D k 3. Prune all edges that have weights exceeding # max . 4. Adjust the edge weights according to the combined motion model in Eq.24 and 25. 5. Apply the Hungarian algorithm to this graph, which results in the minimal cost assignment. The resulting edges (assignment pairs) correspond to an output A k , from which the first M rows and m k+1 columns represent the assignment matrix A k . 6. Increase k; if k < n go to 1, otherwise done. 6 Performance evaluation To evaluate the performance of the different algorithms, we compared them qualitatively and quantitatively. In Section 6.1, we start by looking at their correspondence quality by using a specially constructed example that (also) tests the algorithm's track continuation capabilities. Then, in Section 6.2, we explore the sensitivity of the algorithms to some problem parameters like point density and the total number of points, and algorithm parameters like d max . In all experiments in this section, the correspondences between the first two frames are known and passed to all algorithms (even to those that are capable of self-initialization avoiding favoring one of the methods). 6.1 Constructed example The carefully constructed example shows two crossing feature points with a missing measurement at t 4 for the first and at t 5 for the second point (see Fig. 5a). The difficulty of this data set is that in two consecutive frames a measurement is missing, but for different points. With all algorithms we used the smooth motion model (im2). For algorithms that have a # max parameter, we varied its value from 0.05 to 1 (lower values do not allow the initial motion of p 2 ). Further, we fixed the d max value to 20. 6.1.1 S&S results The S&S algorithm either leads to wrong correspondences or to disconnected track parts. We used two different settings of # max to show the shortcomings of S&S. First, with a high # max (0.1 # max # 1), the algorithm makes wrong correspondences (Fig. 5b). When assigning measurements to track heads T k i , the algorithm prefers track heads that have a true measurement at t k over track heads that have a phantom point at t k . Of course the motion criterion for that true measurement assignment may not exceed the maximum criterion. On the other hand, if # max is lower (e.g 0.05), the algorithm separates four track parts, while correspondences between the track parts have to be made afterwards (see Fig. 5c). 6.1.2 R&S results The R&S algorithm, which has no parameters, chooses the right correspondence when one measurement lacks at t 4 . Then, it estimates the missing measurement by extrapolation and continues with the next frame. With point extrapolation for one frame only, the deviation is limited. In the next frame (t 5 ) the situation is similar to the previous frame. The algorithm connects the single present measurement to the right track head and extrapolates the missing measurement. The processing of the last 3 frames is straightforward (see Fig. 5d). 6.1.3 C&V results At t 4 , C&V assigns the single measurement to the right track head (T 3 ). Then at t 5 only one track head T 4remains to which the measurement can be assigned. If it wouldn't fit because the distance was too great this measurement could start a new track. Since it is not too far away, the only point at t 5 is also assigned to T 2 . The two track parts that belong to p 1 are not connected in the post-processing step (Fig. 5e). 6.1.4 GOA tracker results When the algorithm proposed in this paper is applied to this data set with the smooth motion and average deviation model, all correspondences are made correctly. Moreover, the algorithm interpolates the missing measurements better than R&S and, hence, forms the most plausible tracks (see Fig. 5f). 6.2 Performance with generated data In this section we describe the tests we did to evaluate various aspects of the described algorithms. To this end we used a data set generator that is able of creating data sets of uncorrelated random point tracks of various densities and speeds. Among the described algorithms only the R&S algorithm does not exploit the d max parameter to improve quality and efficiency. For the experiments, we added the d max parameter to R&S (now called similar to the GOA tracker. Then, we tuned all algorithms to find the optimal setting for each of them and used that setting in all experiments. For C&V, R&S*, and the GOA tracker the true maximum is optimal and for S&S a very high value, d In Section 6.2.6, we consider the sensitivity of the algorithms for the d max parameter setting. We did not test the # max sensitivity, because it constrains the motion similarly. Other experiments evaluate the performance for increasingly difficult data sets, an increasing number of missing point detections, and the efficiency of the algorithms. For the generation of the uncorrelated tracks, we used the data set generator called Point Set Motion Generator (PSMG) according to [27] (see example in Fig. 6). Because this data generator model allows feature points to enter and leave the 2-D scene, which we do not consider in this paper, we modified the model to prevent this by replacing invalid tracks until all tracks are valid. The PSMG has the following 3 When dmax is very high then R&S* behaves like the original R&S, i.e. unconstrained speed. (a) 1030507090 missing missing (b) 1030507090 (c) 1030507090 (d) Figure 5: (a) Two input measurements at 8 time instances. At t 4 a measurement for point p 1 is missing and at t 5 a measurement of point p 2 is missing. The figures show the results of (b) S&S using (c) S&S using # and (f) the GOA tracker respectively. In the figures the estimated points are shown as non-filled boxes and crosses indicate the true positions of the missing points. parameters (defaults in brackets): 1. Number of feature point tracks 2. Number of frames per point track 3. Size (S of the square space 4. Uniform distributions for both dimensions of initial point positions between 0 and S. 5. Normal distribution for the magnitude of the initial point velocity vector: 6. Uniform distribution for the angle of the initial velocity vector, between 0 and 2# . 7. Normal distribution for the update of the velocity vector magnitude v k i , from t k to t 8. Normal distribution for the update of the velocity vector angle # k i from t k to t 9. Probability of occlusion ( Figure Example PSMG data set with 15 points during 8 time steps. A number of different measures has been proposed to quantify the quality of performance, like the distortion measure [19] and the link-based error and track-based error [27]. We use the track-based error as in [27], which is defined as follows: correct otal is the total number of true tracks and T correct is the number of completely correct tracks. Some remarks about the experiments. First, in all cases the shown results are an average of 100 runs. We did not incorporate significance levels because the minimal possible track error depends on the actual presented data, hence, the appropriateness of the individual motion model. Nevertheless, the ranking and relative quality were for each experiment the same as illustrated in the figures. Second, in this section we ran the S&S algorithm only with a forward optimization loop, because otherwise the algorithm would not converge (see also Section 4). 6.2.1 Tuning individual and combined models To find an optimal combination of individual and combined motion models, we assume that the individual models and combined models are independent. In order to find the best individual model for the PSMG generated data, we ran experiments with the individual models im1, im2 and im3, together with the combined model implemented in the GOA tracker. In Fig. 7a, we show the results of this experiment. Clearly, the model im2 fits this generated data set best. In order to identify the best combined model for this data set, we ran tests with shown in Fig. 7b. We chose w 1 equal to w 2 , because we want to express that the lack of alternatives is equally important as the absence of competing track heads. cm2 with even lower w 1 and w 2 values becomes better until it finally equals From these tests we conclude that the smooth motion model deviation model is the best combined modeling for PSMG data. Hence, we used these models in the remaining experiments, if possible. That is, only the GOA tracker allows for combined model settings and can be adjusted in that sense. track error track number of points (M) nearest neighbor proximal uniformity smooth motion (b)0.050.150.25 track error track number of points (M) Figure 7: (a) Track error of the GOA tracker with the average deviation model (cm1), in combination with the nearest neighbor, smooth motion or proximal uniformity model. (b) Track error of the GOA tracker with the smooth motion model in combination with cm1 and cm2. 6.2.2 Variable density performance To show how the algorithms perform with an increasing number of conflicts, we applied them to several data sets with an increasing point density. To this end, we generated the data in a fixed sized 2-D space and vary the number of point tracks. In Fig. 8a, we display the results of all algorithms. The figure clearly shows that the GOA tracker performs best. 6.2.3 Variable velocity performance Another experiment to test the tracking performance of the algorithms is varying the mean velocity and keeping the number of points constant. In order to have reasonable speed variances with all mean velocities, we scaled both # v 0 and # v u with the mean values according to # v and # v . In addition, we enlarged the space in which the point tracks are generated to to prevent that mainly diagonal tracks are allowed. The ranking of the algorithms is similar to the variable density experiment and again the GOA tracker performs better than all other schemes (see Fig. 8b). (a)0.020.060.10.14 track error track number of points (M) RS* SS GOA track error track mean velocity RS* SS GOA Figure 8: (a) Results of the algorithms applied to increasingly dense point sets. (b) Track error as a function of the mean velocity. 6.2.4 Track continuation performance In this experiment, we compared the track continuation performance of the R&S extrapolation scheme and the slave measurements interpolation, as proposed in this paper. We left out the other two algorithms because S&S does not really handle track continuation and C&V only allows very limited occlusion. In order to properly compare the track extrapolation and the slave measurements interpolation, we implemented them both in the GOA tracker. We tested the track continuation performance in a variable occlusion exper- iment, with In Fig. 9a, we display the track error results of the GOA tracker with both track continuation schemes with either 50 or 100 points. As illustrated in this figure, the slave measurements approach proposed in this paper clearly achieves better track continuation results than the track extrapolation scheme as proposed by Rangarajan and Shah [19]. The difference is larger with higher probability of occlusion ( p because then there will be more often occlusion during a number of consecutive frames, in which case the difference between interpolation and extrapolation becomes apparent. 6.2.5 Variable volume performance This test is directed towards measuring the computational efficiency of the different algorithms. Hereto, we keep the point density constant while increasing the number of point tracks (and thus enlarging the size of the 2-D space proportionally). Consequently, the correspondence problem remains equally difficult, but the problem size grows. In Fig. 9b, we show the results with logarithmically scaled axes. The figure shows that, with optimal d max , C&V is the fastest. Further, the computation time of the algorithms is widely divergent but all algorithms have polynomial complexity. We list the polynomial orders in the summary of the experiments in Section 6.3. (a)0.050.150.250.350.45 track error track probability of occlusion (p (b)0.1101000 time number of points (M) SS RS* GOA Figure 9: (a) Track error of the GOA tracker with either slave interpolation (Inter) or the R&S extrapolation scheme (Extra) in a variable occlusion experiment with 50 or 100 points. (b) Illustration of the efficiency of the algorithms in a variable volume experiment. 6.2.6 Sensitivity for d max parameter setting As mentioned, so far all algorithms used the tuned and optimal settings of the d max parameter. In this sensitivity experiment, we show the importance of the a priori knowledge about a reasonable value for this parameter. To this end, we varied the d max parameter from the known true value up to a high upper limit, values than the true maximum speed are clearly not sensible). Fig. 10a clearly shows that both S&S and R&S* are most sensitive to variations in this parameter. Remarkably, S&S performs better when d max is set far too high. We expect that the ill initialization, together with the exchange optimization causes this effect because every point exchange must obey the d max constraint. Both C&V and the GOA tracker are hardly sensitive to d max variations (which implies that they do not take advantage of it either). Computationally, especially the C&V algorithm is hampered by an incorrect or ignorant d max value as Fig. 10b illustrates. Consequently, the GOA tracker is the fastest when d max is over 5 times the true maximum speed. track error track RS* SS GOA time SS RS* GOA Figure 10: Illustrates the sensitivity of the algorithms to d max variations. (a) shows the track error performance and (b) shows the computational performance. 6.3 Summary of experiments In conclusion, for tracking a fixed number of points the GOA tracker is qualitatively the best algorithm among the algorithms we presented, according to its track continuation handling in the first test and its performance in all PSMG experiments. Moreover, it is hardly sensitive to the d max parameter setting. S&S performs only slightly worse, when we used the optimal d max setting (d but it is an order of magnitude slower than the GOA tracker. Moreover S&S did not perform well on the specially constructed example, nor does it give interpolated positions of the missed points. The version of R&S*, with added d max parameter and modified individual model, is efficient and qualitatively good as long as it has an accurate estimate of d max . The sensitivity experiment shows that R&S* performs worst of all if this value is not known (or not used as in the original R&S implementation). With (near) optimal maximum velocity setting, C&V is the fastest. If this optimal value is not known (which is usually the case), then the efficiency of C&V degrades rapidly. We should also note that, in our experiments, S&S performed consistently better than C&V, which does not agree with the results reported in [27]. This is probably because in [27] a different d max setting for S&S is used, for which we showed in Section 6.2.6 the S&S algorithm is quite sensitive. This implies that S&S can not exploit the d max parameter effectively to handle missing and spurious measure- ments. Finally, the variable occlusion experiment clearly showed that the slave measurements implement track continuation better than the point extrapolation scheme [19]. In Table 1, we summarize the PSMG experiments. The last column shows the polynomial order of complexity of the algorithms as derived from the variable volume experiment. Table 1: Summary of the PSMG experiments. Algorithm variable density track variable velocity track variable volume time polynomial order a in O(M a ) 7 Algorithm extension with self-initialization In the problem statement in Section 2, the correspondences between the first two frames were assumed to be known. In this section, we generalize the problem, by lifting this restriction and elaborate on how self-initialization is incorporated in the GOA tracker. Two algorithms we discussed have an integrated way of automatically initializing the point tracks. That is, both S&S and C&V only use the measurement positions, for the initialization. R&S on the other hand use additional information, i.e. the optical flow field, which is computed between the first two frames. We advocate the integrated approach, because it is more generally applicable and it allows for optimizing the initial correspondences using a number of frames as we proposed in the global motion model in Section 3. Here, we propose to extend the GOA tracker with features of the S&S algorithm. After that, we demonstrate the appropriateness of this extension and again analyze the parameter sensitivity of the algorithms that support self-initialization. 7.1 Up-Down Greedy Optimal Assignment Tracker (GOA/up-down) The S&S algorithm has a number of shortcomings, of which its computational performance has been shown to be the most apparent. Also, as mentioned, we deliberately left out the bi-directional optimization which quite often does not converge. However, for self-initialization the bi-directional optimization is essential. We propose to modify the GOA tracker in the spirit of [21] and [22] by initializing the correspondences between the first two frames using the described optimal algorithm to minimize C k with im1/cm1. After these correspondences are made, we continue the optimization of the remaining frames (up) in the normal way and additionally optimize the same frames backwards (down). Further forward and backward optimization proved to be useless, because the optimization process already converged. The reason for this fast convergence is that both the initial correspondences and the optimization scheme have been improved considerably compared to S&S. 7.2 Self-initialization experiments To test the performance of the algorithms that are capable of self-initializing the tracks, together with the just described extended GOA/up-down tracker, we did another variable density experiment, and a sensitivity experiment using the PSMG track generator. The individual models need not be tuned again because the parameter settings of the PSMG are the same as in Section 6.2. This time, we left out S&S because of serious convergence problems with their bi-directional optimization scheme, which is essential for self- initialization. R&S does not implement self-initialization using only point measurements, so it can not be applied within these experiments. Although we did not discuss statistical motion correspondence techniques in detail in this paper, we included the multiple hypothesis tracker (MHT) as described and implemented by Cox and Hingorani [3] in this experiment in order to see how it relates to non-statistical greedy matching algorithms. We should note that this MHT implementation is not the most efficient (for improvements see e.g. [15]), though qualitatively equivalent to the state of the art of the statistical motion correspondence algorithms. 7.2.1 Variable density experiment For this experiment we tuned the algorithms optimally for the given data sets. That is, both C&V and GOA/up-down use the true d max . The (eight) parameters of the MHT (like the Kalman filter and Mahalanobis distances), were tuned with a genetic algorithm, for which we used the track error as fitness function. Actually the only difference with the variable density experiment in Section 6.2.2 is that here the initial correspondences are not given. Fig. 11a shows the performance of the algorithms. Clearly, GOA/up- performs best and, remarkably, almost as good as when the initial correspondences were given. The performance of the MHT is similar to the GOA tracker until it seriously degrades, when the number of points exceeds 50, see Fig. 11a. This can be explained from the fact that the parameters for the MHT were trained for (only) 50 points. We did not include more points, because the training was already very time consuming (> 2 days on a Silicon Graphics Onyx II). It is, however, striking to see that the GOA tracker also performs consistently better than the MHT even with less than 50 points, although the latter optimizes over several frames. Among others this may be caused by the effective self-initialization scheme of the GOA tracker. The up-down scheme can be said to optimize the initial correspondences over the whole sequence when optimizing up. Then the remaining correspondences are established in the down phase. (a)0.050.150.250.350.45 track error track number of points (M) GOA/up-down MHT track error track GOA/up-down Figure 11: (a) shows the track error as a function of the number of points in a variable density experiment with self-initialization, and (b) shows the track error as a function of the d max setting. 7.2.2 Sensitivity experiment When the correspondences for the initial frames are not given, we expect the algorithms to be more sensitive to the d max setting. Namely, when the initial velocity is unconstrained, the greedy matching algorithms easily make implausible initial choices, from which they can not recover. To study this behavior, we did another sensitivity experiment for C&V and the GOA tracker and additionally an experiment to test the sensitivity of the MHT. We studied the MHT separately, because is has different parameters (and no d First, Fig. 11b indeed shows that for C&V a good estimate of d max is essential. GOA/up-down, however, hardly suffers from lack of a priori knowledge concerning d max , which is partially because the global cost was optimized for the initial frames. Moreover, the up/down optimization scheme can no longer be considered purely greedy, because correspondences are reconsidered in backward direction. Since both algorithms were computationally influenced similarly as when the initial correspondences were given, we did not include the figure here. We have to mention, however, that the computation time of C&V increased even 10 times faster (11 sec. when d because in this experiment the number of alternatives becomes much higher in the first frame. As a consequence, the GOA tracker was already the fastest when d max was set over 3 times the true maximum speed. In order to fairly test the sensitivity of the MHT and to show the results for all parameters in the same figure, we tested the performance in the range from 1/5 of the optimal setting to 10 times the optimal setting of all essential parameters (10 runs per setting). Consequently, the results in Fig. 12a can easily be compared in relation to Fig. 11a, in which 5 (d max ) is also optimal. The figure clearly shows that there is a small parameter range, in which the performance is (sub)optimal. Especially, increasing or decreasing the mahalanobis distance or the initial state variance parameter with 1/5th results in a performance penalty of roughly a factor two. Also the computation time increases dramatically if the parameters are not properly set, as Fig. 12b shows. We plotted the names of the essential parameters in the figures, but refer to [3], [20] for a complete description. track error track value / (5optimalvalue) position variance x position variance y process variance max.mah. velocity model initial state variance value (5optimalvalue) position variance x position variance y process variance max.mah. velocity model initial state variance Figure 12: Parameter sensitivity experiment for the MHT. (a) shows the track error as function of parameter variations and (b) shows the computation time. 8 Real data experiment: tracking seeds on a rotating dish Our final experiment is based on real image data. In this experiment we put 80 black seeds on a white dish and rotated the dish with more or less constant angular velocity, which implies the use of the smooth motion model (im2/# 0.1). The scene was recorded with a 25 Hz progressive scan camera using 4 ms shutter speed, resulting in a 10 image video sequence with very little motion blur 4 . The segmentation of the images was consequently rather straightforward, i.e. in all 10 images all 80 seeds were detected and there were no false measurements. There was a large difference in speed of the seeds ranging from 1 pixel/s in the center to 42 pixel/s at the outer dish positions. Like in Section 7, we tested only those algorithms that have self-initialization capability and again we included the MHT. Clearly, in contrast with Section 7, in this experiment the point motion is strongly dependent. Since all algorithms are hampered equally, this experiment actually tests the general applicability of the algorithms 5 . To be able to run the MHT properly we tuned its main parameters by applying a genetic algorithm (The ground truth was established by manual inspection.) For this experiment we also added the S&S algorithm, because this time it converged consistently, that is with different d max settings. Fig. 13 shows the resulting tracks overlaid on the first image of the sequence. Only the GOA/ up-down tracker was able to find all the true seed tracks, while the d max setting did not influence the results. Even GOA/up (not down!) was able to track the 80 seeds correctly over all 10 frames, regardless the d max value. Not surprisingly, the C&V algorithm that already proved to be sensitive to d max suffers severely from the divergent seed speeds. The S&S algorithm, which is also sensitive to d max , again makes less errors when d max is relaxed. In general the behavior of S&S turned out to depend greatly on the d max and # max settings. Although the MHT is extensively tuned and it optimizes over several frames simultaneously, the MHT still makes a few errors. Besides, the MHT is substantially slower than the other algorithms. 9 Discussion Throughout this paper, we introduced a framework for motion modeling, and we presented the greedy optimal assignment (GOA) tracker that we extended with self-initialization. In this section we discuss some potential other extensions and improvements. Although the tracking of a variable number of points conflicts with occlusion handling, it is certainly a feature that should be considered as an extension to the GOA tracker. Among the described algorithms we have seen two ways to approach this conflict of requirements, either by actually not implementing track continuation (S&S) or by only allowing occlusion during a very limited number of frames (C&V). First, 4 The rotating dish sequence is available for download at http://www-ict.its.tudelft.nl/tracking/datasets/sequences/rotdish80.tgz. 5 One could, of course, argue that for this data set a rotational individual model or polar coordinates for the measurement positions would fit better. (a) S&S: d p/s: 25 errors, 7.4 sec. (b) S&S: d (c) C&V: d Figure 13: Results of applying the self-initializing algorithms to the rotating dish sequence, consisting of frames with each 80 seeds; true d the GOA tracker can support track initiation and termination by replacing the slave measurements with the phantom points as in S&S. Alternatively, the GOA scheme can be incorporated in the C&V algorithm. The idea is that at each time instance, the GOA scheme is applied first to find corresponding measurements for all point tracks that have been established so far. Then, the original C&V scheme links the remaining measurements if possible. As a result the tracking features of C&V still apply and its performance increases 6 . Further, to deal more effectively with the underlying physical motion, the order of the individual motion models could be increased, e.g by modeling point acceleration. Clearly, extending the scope of the individual models implies difficulties for the model initialization and the track continuation capabilities. Finally, the scope over which the global matching is approximated can be extended. In this paper, we approximated S(D) in a greedy sense, i.e. we only minimized the combined model over two successive frames. We already illustrated in Fig. 3 that extending the scope for this minimization would yield more plausible tracking results. Extending the scope, however, implies that we need to cope with an increasingly complex problem, to which the efficient Hungarian algorithm as such can not be applied anymore. In this paper we showed an adequate way to model the motion correspondence problem of tracking a fixed number of feature points in a non-statistical way. By fitting existing algorithms in this motion framework, we showed which approximations these algorithms make. An approximation that all described algorithms have in common is that they greedily match measurements to tracks. For this approximation, we proposed an optimal algorithm, the Greedy Optimal Assignment (GOA) tracker, which obviously qualitatively outperforms all other algorithms. The way the proposed algorithm handles detection errors and occlusion turned out to be effective and more accurate than the other described algorithms. Moreover, the experiments show clearly that its computational performance is among the fastest. Also the self-initializing version of the GOA tracker turned out to be adequate and hardly sensitive to the maximum speed constraint (d max ) setting. Briefly, for the tracking of a fixed number of feature points the proposed tracker has proven to be efficient and qualitatively best. Among the described algorithms the R&S algorithm is completely surpassed because it operates under the same conditions, while the GOA tracker outperforms R&S both qualitatively and computationally. 6 We have already implemented this idea, but we did not include it in the experiments for the sake of clarity. With a fixed number of points, its performance indeed rated in between GOA/up-down and the original C&V algorithm. The S&S algorithm, which does not support track continuation, is computationally very demanding. The major drawbacks of the C&V algorithm are its relatively poor performance, especially with respect to the initialization, its restricted track continuation capability, and its sensitivity to the d max setting. Still, S&S and C&V may be considered because both support the tracking of a variable number of points and C&V can be very fast. In the previous section we indicated how their performance can be improved by incorporating GOA features in these algorithms. In a number of experiments we included the statistical multiple hypothesis tracker. Even though the MHT optimizes over several frames, which makes it computationally demanding, it turned out that it does not perform better than the GOA tracker. Possible causes are the effective initialization of the GOA tracker and the fact that the MHT models the tracking of a varying number of points, although we set the respective probabilities as to inform that the number of points is fixed. Most importantly, the MHT has quite a few parameters for which the tuning proved to be far from trivial. In conclusion, the proposed qualitative motion framework has proven to be an adequate modeling of the motion correspondence problem. As such, it reveals a number of possibilities to achieve qualitative improvements, ranging from more specialized individual models to S(D) approximations with an extended temporal scope. Acknowledgments This work was supported by the foundation for Applied Sciences (STW). The authors would like to thank Dr. Dmitry Chetverikov for the discussions on the details of his tracking algorithm and the anonymous reviewers for their comments and suggestions. --R A review of statistical data association techniques for motion correspondence. An efficient implementation of Reid's multiple hypothesis tracking algorithm and its evaluation for the purpose of visual tracking. On finding ranked assignments with applications to multi-target tracking and motion correspondence A comparison of two algorithms for determining ranked assignments with application to multi-target tracking and motion correspondence A fast method for finding the exact N-best hypotheses for multitarget tracking A new algorithm for the generalized multidimensional assignment problem. A generalized S-D assignment algorithm for multisensor-multitarget state estimation Sonar tracking of multiple targets using joint probabilistic data association. Adaptive constraints for feature tracking. Determining optical flow. Tracking feature points in time-varying images using an opportunistic selection ap- proach The hungarian method for solving the assignment problem. Establishing motion-based feature point correspondence Optimizing Murty's ranked assignment method. Combinatorial problems in multitarget tracking - a comprehensive solution Multidimensional assignments and multitarget tracking. Data association in multi-frame processing Establishing motion correspondence. An algorithm for tracking multiple targets. Feature point correspondence in the presence of occlusion. Finding trajectories of feature points in a monocular image sequence. Computational experiences with hot starts for a moving window implementation of track maintanance. Uniqueness and estimation of three-dimensional motion parameters of rigid objects with curved surface The Interpretation of Visual Motion. A fast and robust point tracking algorithm. --TR --CTR Meghna Singh , Mrinal K. Mandal , Anup Basu, Gaussian and Laplacian of Gaussian weighting functions for robust feature based tracking, Pattern Recognition Letters, v.26 n.13, p.1995-2005, 1 October 2005 C. J. Veenman , M. J. T. Reinders , E. Backer, Establishing motion correspondence using extended temporal scope, Artificial Intelligence, v.145 n.1-2, p.227-243, April Khurram Shafique , Mubarak Shah, A Noniterative Greedy Algorithm for Multiframe Point Correspondence, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.27 n.1, p.51-65, January 2005 Alper Yilmaz , Omar Javed , Mubarak Shah, Object tracking: A survey, ACM Computing Surveys (CSUR), v.38 n.4, p.13-es, 2006

target tracking;algorithms;motion correspondence;feature point tracking

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A framework for symmetric band reduction.

We develop an algorithmic framework for reducing the bandwidth of symmetric matrices via orthogonal similarity transformations. This framework includes the reduction of full matrices to banded or tridiagonal form and the reduction of banded matrices to narrower banded or tridiagonal form, possibly in multiple steps. Our framework leads to algorithms that require fewer floating-point operations than do standard algorithms, if only the eigenvalues are required. In addition, it allows for space-time tradeoffs and enables or increases the use of blocked transformations.

INTRODUCTION Reduction to tridiagonal form is a major step in eigenvalue computations for symmetric matrices. If the matrix is full, the conventional Householder tridiagonal- ization approach (e.g., [Golub and Van Loan 1989]) or a block variant thereof [Dongarra et al. 1989] is usually considered the method of choice. Hovever, for banded matrices this approach is not optimal if the semibandwidth b (the number of the outmost nonzero off-diagonal) is very small compared with the matrix dimension n, since the matrix being reduced has completely filled in after steps. It is well known that the algorithms of Rutishauser [1963] and Schwarz This work was supported by the Advanced Research Projects Agency, under contracts DM28E04120 and P-95006. Bischof also received support from the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under contract W-31-109-Eng-38. Lang also received support from the Deutsche Forschungsgemeinschaft, Gesch-aftszeichen Fr 755/6-1 and La 734/2-1. This work was partly performed while X. Sun was a postdoctoral associate with the Mathematics and Computer Science Division at Argonne National Laboratory. Argonne Preprint ANL/MCS-P586-0496, submitted to ACM Trans. Math. Software x Figure 1. Rutishauser's tridiagonalization with Householder transformations. [1968] (called RS-algorithms in the rest of the paper) are more economical than the standard approach when b - n. In these algorithms, elements are annihilated one at a time by Givens rotations; Rutishauser's algorithm annihilates the elements by diagonals, whereas Schwarz's algorithm proceeds by columns. Each Givens rotation generates a fill-in element outside of the current band, and the fill-in is chased out by a sequence of Givens rotations before more fill-in is introduced. Rutishauser [1963] also suggested another band reduction scheme based on Householder transformations that annihilates all elements of the current column instead of only one. Rutishauser used an analogous scheme to chase the triangular bulge generated by the reduction with a sequence of QR factorizations, as shown in Figure 1. However, because of the significant work involved in chasing the triangular bulges, this algorithm is not competitive with the rotation-based RS-algorithms. The Schwarz algorithm is the basis of the band reduction implementations in EISPACK [Smith et al. 1976; Garbow et al. 1977]. It can be vectorized along the diagonal [Kaufman 1984], and this variant is the basis of the band reduction algorithm in LAPACK [Anderson et al. 1995]. The RS-algorithms require storage for one extra subdiagonal, and Rutishauser's Householder approach requires storage for subdiagonals. To assess the storage requirements of various algorithms, we introduce the concept of working semibandwidth. The working semibandwidth of an algorithm for a symmetric band matrix is the number of sub(super)diagonals accessed during the reduction. For instance, the working semibandwidth is for the RS-algorithms and Rutishauser's second algorithm with Householder transformations. In both algorithmic approaches, each reduction step has two parts: -annihilation of one or several elements, and -bulge chasing to restore the banded form. Either way, the bulk of computation is spent in bulge chasing. The tridiagonaliza- tion algorithm described in [Murata and Horikoshi 1975] and, for parallel comput- ers, in [Lang 1993] (called MHL-algorithm in the following) improves on the bulge chasing strategy. It employs Householder transformations to eliminate all subdiagonal entries in the current column, but instead of chasing out the whole triangular bulge (only to have it reappear the next step) it chases only the first column of the bulges. These are the columns that (if not removed) would increase the working semibandwidth in the next step. The working semibandwidth for this algorithm is also 2b \Gamma 1. By leaving the rest of the bulges in place, the algorithm requires roughly the same number of floatingpoint operations (flops) as does the RS-algorithms if the latter are implemented with Givens rotations, and 50% more flops, if the RS-algorithms are based on fast Givens rotations. Using Householder transformations considerably improves data locality, as compared with the rota- tionbased algorithms. On the other hand, the RS-algorithms require less storage and may be still preferable if storage is tight. In this paper, we generalize the ideas behind the RS-algorithms and the MHL- algorithm. We develop a band reduction algorithm that eliminates d subdiagonals of a symmetric banded matrix with semibandwidth b (d ! b), in a fashion akin to the MHL tridiagonalization algorithm. Then, like the Rutishauser algorithm, the band reduction algorithm is repeatedly used until the reduced matrix is tridiagonal. If it is the MHL-algorithm; and if used for each reduction step, it results in the Rutishauser algorithm. However, d need not be chosen this way; in- deed, exploiting the freedom we have in choosing d leads to a class of algorithms for banded reduction and tridiagonalization with favorable computational properties. In particular, we can derive algorithms with (1) minimum algorithmic complexity, (2) minimum algorithmic complexity subject to limited storage, and (3) enhanced scope for employing Level 3 BLAS kernels through blocked orthogonal reductions. Setting results in the (nonblocked) Householder tridiago- nalization for full matrices. Alternatively, we can first reduce the matrix to banded form and then tridiagonalize the resulting band matrix. This latter approach significantly improves the data locality because almost all the computations can be done with the Level 3 BLAS, in contrast to the blocked tridiagonalization [Don- garra et al. 1989], where half of the computation is spent in matrix-vector products. This paper extends the work of [Bischof and Sun 1992]. The paper is organized as follows. In the next section, we introduce our frame-work for band reduction of symmetric matrices. In Section 3 we show that several known tridiagonalization algorithms can be interpreted as instances of this frame- work. Then we derive new algorithms that are optimal with respect to either computational cost or space complexity. In Section 4 we discuss several techniques for blocking the update of an orthogonal matrix U ; this is required for eigenvector computations. Then, in Section 5 we present some experimental results. Section 6 sums up our findings. 2. A FRAMEWORK FOR BAND REDUCTION In this section we describe a framework for band reduction of symmetric matrices. The basic idea is to repeatedly remove sets of off-diagonals. Therefore, we first present an algorithm for peeling off some diagonals from a banded matrix. Suppose an n-by-n symmetric band matrix A with semibandwidth b ! n is to be reduced to a band matrix with semibandwidth ~ b. That is, we want to eliminate the outermost d of the b nonzero sub(super)diagonals. Pre Sym Post Post Pre Sym Pre Post Sym Figure 2. Annihilation and chasing in the first sweep of the band reduction algorithm. "QR" stands for performing a QR decomposition, "Pre" and "Post" denote pre- and post-multiplication with Q T and Q, resp., and "Sym" indicates a symmetric update (Pre and Post). The last picture shows the block partitioning before the second sweep. Because of symmetry, it suffices to access either the upper or the lower triangle of the matrix A; we will focus on the latter case. Our algorithm is based on an "annihilate and chase" strategy, similar to the RS- and MHL-algorithms for tridiagonalization. As in the MHL-algorithm, Householder transformations are used to annihilate unwanted elements, but in the case ~ b ? 1 we are able to aggregate n b - ~ b of the transformations into the WY or compact WY representation [Bischof and Van Loan 1987; Schreiber and Van Loan 1989]. Thus, data locality is further improved. First, the d outmost subdiagonals are annihilated from the first n b columns of A. This step can be done with a QR decomposition of an h \Theta n b , upper trapezoidal block of A, as shown in the first picture of Figure 2. Then the WY representation of the transformation matrix is generated. To complete the similarity transformation, we must apply this block transform from the left and from the right to A. This requires applying Q from the left to an h \Theta (d \Gamma n b ) block of A ("Pre"), from both sides to an h \Theta h lower triangular block ("Sym"), and from the right to a b \Theta h block ("Post"). The "Post" transformation generates fill-ins in d diagonals below the band. The first n b columns of the fill-in are removed by another QR decomposition (second picture in Figure 2), with "Pre", "Sym", and "Post" to complete the similarity transformation. The process is then repeated on the newly generated fill-in, and so on. Each step amounts to chasing n b columns of the fill-in down along the diagonal by b diagonal elements until they are pushed off the matrix. Then we can start the second "annihilate and chase" sweep. Each sweep starts with a matrix that has the following properties: -The remainder of the matrix, (i.e., the current trailing matrix) is block tridiag- onal, with all diagonal blocks but the last one being of order b and the last one being of order - b; see the last picture in Figure 2. -Every subdiagonal block is upper triangular in its first ~ -The matrix is banded with semibandwidth b + d. Every similarity transformation within the sweep -maintains the working semibandwidth b -involves the same number h of rows and columns, and -restores the form described above in one subdiagonal block, while destroying it in the next subdiagonal block. We arrive at the following algorithm. Algorithm 1. One-step band reduction bandr1(n; A; b; d; n b ) Input. An n \Theta n symmetric matrix with semibandwidth b, n. The number of subdiagonals to be eliminated, and a block size n b , 1 Output. An n \Theta n symmetric matrix with semibandwidth b \Gamma d. while QR: Perform a QR decomposition of the block B and replace B by Pre: Replace the block B Sym: Replace the block B Post: Replace the block B while end for The working semibandwidth of the algorithm is b+d. Given the one-step band reduction algorithm, we can now derive a framework for band reduction in a straight-forward fashion by "peeling off" subdiagonals in chunks. Algorithm 2. Multistep band reduction bandr(n; A; b; fd (i) Input. An n \Theta n symmetric matrix with semibandwidth b, sequence of positive integers fd (i) g k b, and a sequence fn (i) of block sizes, where n (i) d (i) . Output. An n \Theta n symmetric matrix with semibandwidth b \Gamma d. call bandr1( n, end for The working semibandwidth for the multistep algorithm is (b d (j) ); which is b + d (1) if the d (i) satisfy There is also a class of systolic array algorithms that use Givens rotations to remove several outer diagonals at a time [Bojanczyk and Brent 1987; Ipsen 1984; Schreiber 1990]. Here, d (i) is chosen as large as b (i) =2 [Bojanczyk and Brent 1987] in order to increase the parallel scope per systolic operation and hence minimize the total number of systolic operations. 3. INSTANCES OF THE FRAMEWORK In this section we discuss several instances of Algorithm 2, including the (non- blocked) Householder tridiagonalization, Rutishauser's algorithm, and the MHL- algorithm for tridiagonalizing symmetric band matrices. In addition to these, the multistep framework allows for new tridiagonalization methods featuring lower flops count and/or better data locality. 3.1 One-step Tridiagonalization of Full Symmetric Matrices A full symmetric matrix can be tridiagonalized with bandr(n; A; 1g). Note that ~ 1. Then, the QR decomposition and the Sym step of Algorithm 1 reduce to determining and applying a suitable Householder transformation, while the Pre and Post steps vanish. Thus, we arrive at the nonblocked standard Householder tridiagonalization. 3.2 Two-step Tridiagonalization of Full Symmetric Matrices Another way to tridiagonalize a full symmetric matrix is the two-step sequence is some intermediate semibandwidth and n (1) - ~ b. That is, we first reduce A to banded form and then tridiagonalize the resulting banded matrix. In contrast to the blocked Householder tridiagonalization [Dongarra et al. 1989], where one half of the approximately 4=3n 3 flops for the reduction of A is confined to matrix-vector products, almost all the operations in the reduction to banded form can be done within the Level 3 BLAS. For b - n this first reduction constitutes the vast majority of flops. Tridiagonalizing the banded matrix requires roughly 6bn 2 flops, which can be done with Level 2 BLAS. Therefore, the two-step tridiagonalization may be superior on machines with a distinct memory hierarchy (see Table I in Section 5.1). 3.3 One-step Tridiagonalization of Symmetric Band Matrices As for full matrices, the simplest way to tridiagonalize a matrix with semibandwidth b is the one-step sequence bandr(n; A; b; fd 1g). Again, the QR decomposition and the Sym steps reduce to determining and applying single Householder transformations, and the Pre and Post steps vanish. This one-step sequence is equivalent to the MHL-algorithm. 3.4 Tridiagonalization by Peeling off Single Diagonals The sequence bandr(n; A; b; fd tridiagonalizes a banded matrix by repeatedly peeling off single diagonals. This corresponds to Rutishauser's algorithm, except that the rotations are replaced with length-2 Householder transformations. Figure 3. Nonzero structure of the matrices W and Y for 3.5 Optimal Tridiagonalization Algorithms for Band Matrices In the following, we derive tridiagonalization algorithms that have a minimum flops count for a given working semibandwidth, that is, minimize number of flops to tridiagonalize an n \Theta n banded matrix with semibandwidth b subject to working semibandwidth - s, where s (2) For the reduction of banded matrices, blocking the transformations always significantly increases the flops count. Figure 3 shows the nonzero pattern of the that result from aggregating n b length-(d+1) Householder transformations into a blocked Householder transformation. The nonzeros in Y form a parallelogram, while W is upper trapezoidal. Multiplying W to the columns of some m \Theta (d of A takes approximately n b (2d flops and multiplication with Y T costs another 2n b (d even if the multiplication routine GEMM is able to take full advantage of zeros. Thus, in the banded context, using the WY representation increases the overall flops count in two ways. First, the W and Y factors must be generated. Second, applying the blocked Householder transform requires more than the 4n b (d+1)m operations that would be needed for the n b single length-(d transformations in W and Y . The same argument applies also to the compact WY representation [Schreiber and Van Loan 1989]. In addition, this blocking technique requires one more matrix multiplication, . The cost for this additional multiplication is not negligible in the banded case, in contrast to the reduction of full matrices, where the average length of the Householder vectors significantly exceeds n b . Therefore, we prefer the "standard" WY representation for blocking the Householder transformations. Because this section focuses on minimizing the flops count, we consider only nonblocked reduction algorithms (i.e., n (i) On machines with a distinct memory hierarchy, however, the higher performance of the Level 3 BLAS may more than compensate for the overhead introduced by blocking the transforma- tions. Thus, both flops count and BLAS performance should be taken into account in order to minimize the execution time on such machines. For clarity in our analy- sis, we omitted the resulting weighting of the various algorithmic components, but the ideas presented here easily can be extended to this more general analysis. In the nonblocked case, the arithmetic cost of Algorithm 1 in flops is cost(n; b; d) - (4(d For any band difference sequence fd (i) g, Algorithm 2 therefore requires flops. Given a limit on s, we can use dynamic programming [Aho et al. 1983] to determine an optimal sequence fd (i) g from the cost function given in (3). By taking storage requirements into account, we allow for space tradeoffs to best use the available memory. 3.5.1 No Storage Constraints. We first consider Problem (2) with s - 2b \Gamma 1. Here are no storage constraints, since the maximum working semibandwidth of Algorithm 2 is 2b \Gamma 1. However, the optimal sequence fd (i) g is quite different from the one-step sequence of the MHL-algorithm. For example, for a 50; 000-by-50; 000 symmetric matrix with semibandwidth 300, the optimal sequence is the reduction requires 3:48\Delta10 12 floatingpoint operations, and the working semiband- width is 310. In contrast, the MHL-algorithm requires its working semibandwidth is 599. We also note that the constant sequence requires flops, with a working semibandwidth of 316. Another constant sequence requires working semibandwidth of 332. Hence a constant- stride sequence seems to be just as good a choice as the optimal one from a practical point of view, and saves the dynamic programming overhead. 3.5.2 Minimum Storage. Now consider another extreme case of the constraint in Problem 1. That is, we have space for at most one other subdiagonal. Even for small b, the MHL-algorithm is not among the candidates, since its working semibandwidth is candidate d-sequence for this case should satisfy the condition (1) with d 1. The sequence d satisfies this condition. Instead, we suggest We call it the doubling-stride sequence, since the bandwidth reduction size doubles in each round. For the example in the preceding subsection, the RS-algorithms require the doubling-stride algorithm requires 3:90 flops. row update rank-1 column update symmetric rank-2 update x Figure 4. Data area visited by the MHL-algorithm. 3.5.3 Understanding Optimality. Rutishauser's algorithm peels off subdiagonal one by one and requires minimum storage, but it is not optimal among the algorithms with minimum storage. On the other hand, although the MHL-algorithm eliminates all subdiagonals in one step, it is also not optimal when b is large. In the following, we give an intuitive explanation why the complexity of our scheme is superior to both these approaches. First, let us compare the MHL-algorithm and Algorithm 2 with a sequence fd; b \Gamma 1g. During the bulge chasing following the reduction in column 1, the data area accessed by the MHL-algorithm with rank-1 row and/or column updating and symmetric rank-2 updating is shown in Figure 4. The data area accessed by Algorithm 2 is shown in Figure 5. In comparing different bandreduction sequences, we take the rank-1 updating as basic unit of computation and examine the total data area visited by each algorithm for the purpose of rank-1 updating. Since a update applied to an m \Theta n matrix requires approximately 4mn flops, the area involved in a rank-1 update is a good measure of complexity. The symmetric rank-2 update of a triangular n \Theta n matrix is as expensive as a rank-1 update of a full matrix of that size; both require about 4n 2 flops. For the MHL-algorithm, the total area visited with rank-1 updates is For the two successive band reductions, it is a Denoting 1)=b, we obtain as the ratio aMHL which takes its minimum ae - 11=12 at ffi Therefore, the two-step reduction can save some 8% of the flops, as compared with the MHL-algorithm. Of course, the same idea can then be applied recursively on the tridiagonal reduction for the matrix with reduced semibandwidth b \Gamma d. b-d b-d Figure 5. Data area visited in the two steps bandr1(n, A, b, d) and bandr1(n, A, of Algorithm 2. x x Figure 6. Data area visited by Rutishauser's algorithm in the ``elimination and bulge chasing'' rounds for the three outmost elements a 91 , a 81 , and a 71 of the first column (light, medium, and dark grey shading, resp. Let us consider now how often a row or column is repeatedly involved in a reduction or chasing step. The data area visited by Rutishauser's algorithm for annihilating 3 elements in the first column in 3 rounds is shown in Figure 6. We see in particular that the total area visited in the first b \Theta b block is almost twice as big as the one by Algorithm 1 with as a result of the revisiting of the first row and column in the last visited area. Thus, in comparison with Rutishauser's and the MHL-algorithm, our algorithm can be interpreted as balancing the counteracting goals of -decreasing the number of times a column is revisited (by the use of Householder transformations), and -decreasing the area involved in updates (by peeling off subdiagonals in several chunks). 3.6 Bandwidth Reduction In some contexts it is not necessary to fully reduce the banded matrix to tridiagonal form. For example, in the invariant subspace decomposition approach (ISDA) for eigensystem computations [Lederman et al. 1991], the spectrum of a matrix A is condensed into two narrow clusters by repeatedly applying a function f to the matrix. If f is a polynomial of degree 3, each application of f roughly triples A's bandwidth. Therefore, A will be full after a moderate number of iterations if no countermeasures are taken. To prevent this situation, the bandwidth of A can be periodically reduced to a "reasonable" value after a few applications of f . This bandwidth reduction can also be done by using Algorithm 2, either in one or multiple reduction steps. 4. AGGREGATING THE TRANSFORMATIONS If the eigenvectors of the matrix A are required, too, then all the transformations from Algorithm 1 must also be applied to another matrix, say, an n \Theta n matrix U . For banded matrices A, the update of U dominates the floating-point complexity (2n 3 for updating U versus 6bn 2 for the reduction of A). Therefore, we should strive to maximize the use of BLAS 3 kernels in the update of U to decrease the total time. In this section we discuss several techniques for updating U with blocked Householder transformations. Some of these methods are tailored to the cases ~ where the transformations of A cannot be blocked. 4.1 On-the-Fly Update The update can easily be incorporated in the reduction by inserting the lines if the aggregate transformation matrix U is required Replace U (:; between the QR and Pre steps of Algorithm 1. That is, each transformation is applied to U as soon as it is generated and applied to A (on-the-fly update). If the matrix is not reduced to tridiagonal form, i.e., ~ then the work on A and U can be done with blocked Householder transformations, thus enabling the use of the Level 3 BLAS. 4.2 Backward Accumulation For tridiagonalization, however, ~ which means that the work on A cannot be done in a blocked fashion. At first glance, this fact seems also to preclude the use of Level 3 BLAS in the update of U . Fortunately, however, the obstacle can be circumvented by decoupling the work on U from the reduction of A. Let us first consider the tridiagonalization of a full symmetric matrix A. As in the LAPACK routine SYTRD, the update of U can be delayed until the reduction of A is completed. Then, the Householder transformations are aggregated into blocked Householder transforms with arbitrary n U b , and these are applied to U . In addition to enabling the use of the Level 3 BLAS, the decoupling of the update from the reduction allows reducing the flop count by reverting the order of the transformations (backward accumulation). For complexity reasons, the backward accumulation technique should also be used in the reduction from full to banded form, even if the on-the-fly update can also be done with blocked Householder transforms. 4.3 Update in the Tridiagonalization of Banded Matrices When a full matrix is reduced to either banded or tridiagonal form, the Householder vectors can be stored conveniently in the zeroed-out portions of A and an additional vector - . In the tridiagonalization of banded matrices, this strategy is no longer possible: eliminating one length-(b \Gamma 1) column of the band requires n=b length- b Householder vectors, because of the bulge chasing. Therefore, it is impractical to delay the update of U until A is completely tridiagonalized. We use another technique [Bischof et al. 1994] in this case. denote the kth Householder transformation that is generated in the jth sweep of Algorithm 1. That is, H j 1 eliminates the first column of the remaining band, and H j 3 , . are generated during the bulge chasing. During the reduction, the transformations of each sweep must be determined and applied to A in the canonical order H j 3 , . , because each H j depends on data modified in the Post step of H j . Once the transformations are known, this dependence no longer exists. Since the transformations from one sweep involve disjoint sets of U 's columns, they may be applied to U in any order (see Figure 7). We are, however, not entirely free to mix transformations from different sweeps. must be preceded by and , since it affects columns that are modified by these two transformations in sweep sweep \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi sweep oe oe oe oe oe oe oe oe oe oe oe oe oe oe oe \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi \Gamma\Psi Figure 7. Interdependence of the Householder transformations H j for the work on A (left picture) and on U (right picture). "H / ~ H cannot be determined and applied to A (cannot be applied to U) until H has been applied to A (to U ). To make use of the additional freedom, we delay the work on U until a certain number n U b of reduction sweeps are completed. Then the update of U is done bottom up by applying the transformations in the order Figure 8. Columns of U affected by each transformation H j of the first four reduction sweeps. In this example, 6. The transformations with the same hatching pattern can be aggregated into a blocked Householder transformation. , . , H J 1 . That is, the transformations in Figure 7 would be applied in the order H 1 6 , 5 , . , H 1 1 . This order preserves the inter-sweep dependence mentioned above. In addition, the n b transformations H k with the same index k can be aggregated into a blocked Householder transformation. In Figure 8, the transformations contributing to the same block transformation are hatched identically. A similar technique can also be used, for example, in the QR algorithm for computing the eigensystem of a symmetric tridiagonal matrix [Lang 1995]. It allows updating the eigenvector matrix with matrix-matrix products instead of single rotations 5. EXPERIMENTAL RESULTS The numerical experiments were performed on single nodes of the IBM SP parallel computer located at the High-Performance Computing Research Facility, Mathematics and Computer Science Division, Argonne National Laboratory, and on single nodes of the Intel Paragon located at the Zentralinstitut f?r Angewandte Mathe- matik, Forschungszentrum J-ulich GmbH. All timings are for computations in double precision. The matrices had random entries chosen from [0; 1]; since none of the algorithms is sensitive to the actual matrix entries, the following timings can be considered as representative. Five codes were used in the experiments: -DSYTRD: LAPACK routine for blocked tridiagonalization of full symmetric matrices [Dongarra et al. 1989], Table I. Timings (in seconds) on one node of the IBM SP for the one-step (LAPACK routine DSYTRD) and two-step reduction (routines DSYRDB and DSBRDT) of full symmetric matrices of order to tridiagonal form. The timings do not include the update of U . The intermediate semibandwidth in the two-step reduction was always b One-Step Reduction Two-Step Reduction Total Time full \Gamma! banded banded \Gamma! tridiagonal -DSBTRD: LAPACK routine for tridiagonalizing banded matrices using Kaufman's modification of Schwarz's algorithm [Schwarz 1968; Kaufman 1984] (called SK- algorithm in the following) to improve vectorization, -DSYRDB: blocked reduction of full symmetric matrices to banded form (Algo- rithm 1), -DSBRDB: blocked reduction of banded matrices to narrower banded form (Algo- rithm 1), and -DSBRDT: tridiagonalization of banded matrices (MHL-algorithm with the technique from Section 4.3 for updating U ). The latter three codes are described in more detail in [Bischof et al. 1996]. All programs are written in Fortran 77. For the IBM SP node, which for our purposes can be viewed as a 66 MHz IBM RS/6000 workstation, the codes were compiled with xlf -O3 -qstrict and linked with -lessl for the vendor-supplied BLAS. For the Intel Paragon node, the compilation was done with if77 -Mvect -O4 -nx, and the BLAS were linked in with -lkmath. The data presented in this section demonstrate that it may be advantageous to consider multistep reductions instead of conventional direct methods. In particular, on machines with memory hierarchies, it is not clear a priori which approach is superior for a given problem. 5.1 Tridiagonalization of Full Matrices We first compared the one-step tridiagonalization routine DSYTRD from LAPACK with the two-step reduction (routines DSYRDB and DSBRDT) from Section 3.2. Table I shows that for large matrices, the two-step reduction can make better use of the Level 3 BLAS than can the direct tridiagonalization, where onehalf of the operations is confined to matrix-vector products. The reduction to banded form runs at up to MFlops, which is close to the peak performance of the IBM SP node. Note that the tridiagonalization of the banded matrix cannot be blocked; therefore, the block size n b affects only the reduction to banded form. The bad performance of this reduction with due to the fact that the routine DSYRDB does not provide optimized code for the nonblocked case. If the transformation matrix U is required, too, then direct tridiagonalization is always superior because it has a significantly lower flops count. Matrix size SP1, no update Matrix size SP1, with update Matrix size Paragon, no update 200 400 600 800 1000 1200 Matrix size Paragon, with update Figure 9. Speedup of the MHL-algorithm (routine DSBRDT) for tridiagonalizing banded matrices over the SK-algorithm (LAPACK routine DSBTRD) with and without updating the matrix U (left and right pictures, resp. In the update, n U transformations were aggregated into a block transform. 5.2 One-step Tridiagonalization of Banded Matrices Next, we compared the SK-algorithm (LAPACK routine DSBTRD) with the MHL- algorithm DSBRDT. Both are one-step tridiagonalization algorithms for band matri- ces. In the MHL-algorithm, the update of U was done by using blocked Householder transformations with the default block size n U Figure 9 shows the results for various matrix dimensions and semibandwidths on both machines. Lang [1993] already noted that the LAPACK implementation is not optimal except for very small semibandwidths. The reason is that the bulk of computations must be done in explicit Fortran loops, since no appropriate BLAS routines cover them. Therefore, the routine runs at Fortran speed, whereas the MHL-algorithm can rely on the Level 2 BLAS for the reduction and Level 3 BLAS for the update of U . On the SP node, the MHL-algorithm runs at up to 45 MFlops in the reduction and 53 MFlops in the update, while DSBTRD only reaches 20 and 30 Mflops, respectively. (In addition, the SK-algorithm requires 50% more flops when U is updated.) This situation may be different on some vector machines because DSBTRD features vector operations with a higher average vector length, albeit with nonunit stride. Table II. Timings (in seconds) on one node of the IBM SP for tridiagonalizing banded symmetric matrices of order 1200. The intermediate semibandwidth in the two-step reduction was One-step reduction with DSBRDT 8.3 15.0 25.0 Two-step reduction with DSBRDB and DSBRDT 11.3 13.6 22.9 5.3 Two-step Tridiagonalization of Banded Matrices In Section 3.5.3 we showed that peeling off the diagonals in two equal chunks requires fewer flops than direct tridiagonalization with the MHL-algorithm. Table II shows that the time for the two-step approach can be lower, too, if the semiband- width is large enough. In contrast to the theoretical results, however, the intermediate semibandwidth b=2 is not always optimal. For example, it took only 20.03 seconds to first reduce the semibandwidth from then tridiagonalize that matrix. The explanation is that in the case b more of the work is done in the bandwidth reduction, which can rely on blocked Householder transformations, whereas the final tridiagonalization cannot. Therefore, the higher performance of the Level 3 BLAS more than compensates for the slighly higher flops count as compared with b For small semibandwidths b, the lower flops count of the two-step scheme was outweighed by the lower overhead (e.g., fewer calls to the BLAS with larger subma- trices) of the one-step reduction. As in the reduction of full matrices, the one-step tridiagonalization is always superior when U is required, too. 5.4 Doubling-stride Tridiagonalization of Banded Matrices While the MHL-algorithm is clearly superior on machines where the BLAS performance significantly exceeds that of pure Fortran code, it may not be applicable if storage is tight. Therefore, we also compared two algorithms that need only one additional subdiagonal as working space: -the SK-algorithm (routine DSBTRD from LAPACK), and -the doubling-stride sequence from Section 3.5.2: multiple calls to DSBRDB (with for the reduction of A and the update of U ) and one call to DSBRDT (with The results given in Figure 10 show that the doubling-sequence tridiagonalization, too, can well outperform the SK-algorithm on both machines. 6. CONCLUSIONS We introduced a framework for band reduction that generalizes the ideas underlying the Householder tridiagonalization for full matrices and Rutishauser's algorithm and the MHL-algorithm for banded matrices. By "peeling off" subdiagonals in chunks, we arrived at algorithms that require fewer floating-point operations and less storage. We also provided an intuitive explanation of why our approach, which eliminates subdiagonals in groups, has a lower computational complexity than that Matrix size SP1, no update Matrix size SP1, with update 200 400 600 800 1000 1200 Matrix size Paragon, no update 200 400 600 800 1000 1200 Matrix size Paragon, with update Figure 10. Speedup of the doubling-sequence tridiagonalization (routines DSBRDB and DSBRDT with over the SK-algorithm (LAPACK routine DSBTRD) with and without updating the matrix U (left and right pictures, resp. of the previous algorithms for banded matrices, which eliminated subdiagonals either one by one or all at once. The successive bandreduction (SBR) approach improves the scope for block operations. In particular, the update of the transformation matrix U can always be done with blocked Householder transformations. We also presented results that show that SBR approaches can provide better performance, by either using less memory to achieve almost the same speed, or by achieving higher performance. Our experience suggests that it is hard to provide a "rule of thumb" for selecting the parameters of an optimal bandreduction algorithm. While the flops count can be minimized by using the cost function (3), the actual performance of an implementation depends on the machine-dependent issues of floating-point versus memory access cost. In our experience, developing a more realistic performance model for advanced computer architectures is difficult (see, for example [Bischof and Lacroute 1990]), even for simpler problems. Another paper [Bischof et al. 1996] describes the implementation issues of a public-domain SBR toolbox enabling computational practioners to experiment with the SBR approach on problems of interest. --R Data Structures and Algorithms. Parallel tridiagonalization through two-step band reduction The WY representation for products of Householder matrices. An adaptive blocking strategy for matrix factorizations. The SBR toolbox - software for successive band reduction A framework for band reduction and tridiagonaliza- tion of symmetric matrices Tridiagonalization of a symmetric matrix on a square array of mesh-connected processors Block reduction of matrices to condensed forms for eigenvalue computations. Matrix Eigensystem Routines - EISPACK Guide Extension Matrix Computations (2nd Singular value decompositions with systolic arrays. Banded eigenvalue solvers on vector machines. A parallel algorithm for reducing symmetric banded matrices to tridiagonal form. Using level 3 BLAS in rotation based algorithms. A parallelizable eigensolver for real diagonalizable matrices with real eigenvalues. A new method for the tridiagonalization of the symmetric band matrix. On Jacobi rotation patterns. Bidiagonalization and symmetric tridiagonalization by systolic arrays. Journal of VLSI Signal Processing A storage-efficient WY representation for products of Householder transformations Tridiagonalization of a symmetric band matrix. Matrix Eigensystem Routines - EISPACK Guide (2nd ed --TR The WY representation for products of householder matrices Solution of large, dense symmetric generalized eigenvalue problems using secondary storage A storage-efficient WY representation for products of householder transformations An adaptive blocking strategy for matrix factorizations A parallel algorithm for reducing symmetric banded matrices to tridiagonal form Matrix computations (3rd ed.) Using Level 3 BLAS in Rotation-Based Algorithms Banded Eigenvalue Solvers on Vector Machines Band reduction algorithms revisited Algorithm 807 Data Structures and Algorithms Automatically Tuned Linear Algebra Software --CTR Daniel Kressner, Block algorithms for reordering standard and generalized Schur forms, ACM Transactions on Mathematical Software (TOMS), v.32 n.4, p.521-532, December 2006 Christian H. Bischof , Bruno Lang , Xiaobai Sun, Algorithm 807: The SBR Toolboxsoftware for successive band reduction, ACM Transactions on Mathematical Software (TOMS), v.26 n.4, p.602-616, Dec. 2000

blocked Householder transformations;symmetric matrices;tridiagonalization

365880

Exact Analysis of Exact Change.

We introduce the k-payment problem: given a total budget of N units, the problem is to represent this budget as a set of coins, so that any k exact payments of total value at most N can be made using k disjoint subsets of the coins. The goal is to minimize the number of coins for any given N and k, while allowing the actual payments to be made on-line, namely without the need to know all payment requests in advance. The problem is motivated by the electronic cash model, where each coin is a long bit sequence, and typical electronic wallets have only limited storage capacity. The k-payment problem has additional applications in other resource-sharing scenarios.Our results include a complete characterization of the k-payment problem as follows. First, we prove a necessary and sufficient condition for a given set of coins to solve the problem. Using this characterization, we prove that the number of coins in any solution to the k-payment problem is at least k HN/k, where Hn denotes the nth element in the harmonic series. This condition can also be used to efficiently determine k (the maximal number of exact payments) which a given set of coins allows in the worst case. Secondly, we give an algorithm which produces, for any N and k, a solution with a minimal number of coins. In the case that all denominations are available, the algorithm finds a coin allocation with at most (k+1)HN/(k+1) coins. (Both upper and lower bounds are the best possible.) Finally, we show how to generalize the algorithm to the case where some of the denominations are not available.

Introduction Consider the following everyday scenario. You want to withdraw N units of money from your bank. The teller asks you "how would you like to have it?" Let us assume that you need to have "exact change," i.e., given any payment request P - N , you should be able to choose a subset of your "coins" whose sum is precisely P . Let us further assume that you would like to withdraw your N units with the least possible number of coins. In this case, your answer depends on your estimate of how many payments you are going to make. In the worst case, you may be making N payments of 1 unit each, forcing you to take N coins of denomination 1. On the other extreme, you may need to make only a single payment P . In this case, even if you don't know P in advance, log coins are sometimes sufficient (as we explain later). In this article, we provide a complete analysis of the general question, which we call the k-payment problem: what is the smallest set of coins which enables one to satisfy any k exact payment requests of total value up to N . Motivation. In any payment system, be it physical or electronic, some transactions require payments of exact amounts. Forcing shops to provide change to a customer, if she does not possess the exact change, simply shifts the problem from the customers to the shops. The number of coins is particularly important in electronic cash (see, e.g., [2, 3]), because electronic coins are inherently long bit sequences and their handling is computationally intensive, while the typical "smart-card" used to store them has small memory space and computational power [7]. Another interesting application of the problem arises in the context of resource sharing. For concreteness, consider a communication link whose total bandwidth is N . When the link is shared by time-multiplexing, there is a fixed schedule which assigns the time-slots (say, cells in ATM lines) to the different connections. Typical schedules have small time slots, since assigning big slots to small requests entails under-utilization. It is important to note, however, that there is an inherent fixed overhead associated with each time slot (e.g., the header of an ATM cell). It would be therefore desirable to have a multiplexing schedule (with time slots of various sizes) which can accommodate any set of requests with the least number of slots. Similarly to the withdrawal scenario, an improvement upon the trivial N unit-slot solution can be achieved, if we know how many connections might be running in parallel. When we know a bound k on this number, and if we restrict ourselves to long-lived connections, the problem of designing a schedule naturally reduces to the k-payment problem. The k-payment problem: Definition. Formally, the problem is as follows. There are two parameters, the budget, denoted N , and the number of payments, denoted k. The problem is to find, for each i - 1, the number of i-coins (also called coins of denomination i), denoted c i , such that the following two requirements are satisfied. Budget compliance: k-partition: For any sequence of k payment requests, denoted N , there exists a way to exactly satisfy these payments using the coins. That is, there exist non-negative integers represents the number of i-coins used in the j-th payment), such that The problem can thus be broken into two parts as follows. The coin allocation problem is to to partition N into coins given N and k, i.e., determining the c i 's. The coin dispensing problem is how, given the c i 's, to actually make a payment. There are a few possible variants of the k-payment problem. First, in many systems, not all denominations are available (for example, we do not know of any system with 3-coins). Even in the electronic cash realm, denominations may be an expensive resource. (This is because each denomination requires a distinct pair of secret/public keys of the central authority; see, e.g., [1, 8].) Thus an interesting variant of the allocation problem is the restricted denominations version, where the set of possible solutions is restricted to only those in which c for a given allowed denomination set D. Also, one may consider the on-line coin dispensing problem, where the algorithm is required to dispense coins after each payment request, without knowledge of future requests, or the off-line version, where the value of all k payments is assumed to be known before the first coin is dispensed. Results. It turns out that the coin dispensing problem is easy: the greedy strategy works, even in the on-line setting. Most of our results concern the coin allocation problem, and we sometimes refer to this part of the problem as the k-payment problem. Our basic result is a simple necessary and sufficient condition for a sequence c 1 of coins to solve the allocation problem (Theorem 3.1). Using this characterization, we prove (in Theorem 3.6) a lower bound of kHN=k - k ln(N=k) on the number of coins in any solution for all N and k, where H n denotes the nth element of the harmonic series. The lower bound is the best possible in the sense that it is met with equality for infinitely many N 's and k's (Theorem 3.8). The characterization can be used to efficiently determine the maximal number k for which a given collection of coins solves the k payment problem (Corollary 3.9). Our next major result is an efficient algorithm which finds a solution for any N and k using the least possible number of coins. We first deal with the case where all denominations are allowed (Theorem 4.1). In this case the number of coins is never more than (Theorem 4.7). Similarly to our lower bound, the upper bound is the best possible in general (Theorem 4.8). Finally, using the same ideas in a slightly more refined way, we extend the algorithm to the general case of restricted denomination set (Theorem 5.1). Related work. To the best of our knowledge, the current work is the first to formulate the general k-payment problem, and hence the first to analyze it. One related classic combinatorial problem is k-partition, where the question is in how many ways can a natural number N be represented as a sum of k positive integers. This problem is less structured than the k-payment problem, and can be used to derive lower bounds (however, these bounds are suboptimal: see Section 2). The postage-stamp problem is also closely related: cast in our terms, the postage- stamp problem is to find a set of denominations which will allow to pay any request of value using at most h coins, so that N is maximized. The postage-stamp problem can be viewed as an inverse of the 1-payment problem: there is one payment to make, the number of coins is given, and the goal is to find a denomination set of a given size which will maximize the budget. We remark that the postage-stamp problem is considered a difficult problem even for a very small number of denominations. See, e.g., [10, 11, 4]. Another related question is change making [6], which is the problem of how to represent a given budget with the least number of coins from a given allowed denomination set. General change-making is (weakly) NP-hard. Kozen and Zaks [5], and Verma and Xu [14], study the question of which denominations sets allow one to use the greedy strategy for optimal change making. Organization. In Section 2 we introduce notation, give some preliminary observations and briefly discuss a few suboptimal results. In Section 3 we prove a characterization of the k- payment problem and a lower bound on the number of coins in any solution. In Section 4 we present and analyze an optimal algorithm for the unrestricted denomination case. In Section 5 we extend the algorithm for the case of restricted denominations. Finally, in Section 6 we give a short overview of the applications of our results for electronic cash. Notation and simple results In this section we develop some intuition for the k-payment problem by presenting a few simple upper and lower bounds on the number of coins required. The notation we shall use throughout this article is summarized in Figure 1. The solution S to which the c to should be clear from the context. For the remainder of this article, fix N and k to be arbitrary given positive integers. Note that we may assume without loss of generality that k - N , since payment requests of value 0 can be ignored. Let us now do some rough analysis of the k-payment problem. As already mentioned above, the case is trivial to solve: take c no better solution is possible since c 1 ! k would not satisfy k payments of value 1 each. The case of quite simple, at least when for an integer 1: we can solve it with coins of denominations Given any request P , we can satisfy it by using the coins which correspond to the ones in the binary representation of P . However, it is not immediately clear how can one generalize this solution to arbitrary N and k. Consider an arbitrary N : the usual technique of "rounding up" to the next power of 2 does Parameters of problem specification: the total budget. ffl k: the number of payments. ffl D: the set of allowed denominations. In the unrestricted case, Quantities related to solution specification S: the number of coins of denomination i (a.k.a. i-coins) in S. By convention, the budget allocated in S using coins of denomination i or less. Formally, ffl m: the largest denomination of a coin in S. Formally, always.) Quantities related to making a payment: refer to the respective quantities after a payment has been made. Standard quantities: . By convention, H Figure 1: Glossary of notation. not seem appropriate in the k-payment problem: can we ask the teller of the bank to round up the amount we withdraw just because it is more convenient for us? But let us ignore this point for the moment, and consider the problem of general k. If we were allowed to make the dubious assumption that we may enlarge N , then one simple solution would be to duplicate the solution for 1-payment k times, and let each payment use its dedicated set of coins. Specifically, this means that we allocate k 1-coins, k 2-coins, k 4-coins and so on, up to k 2 dlog 2 -coins. The result is approximately k log 2 N coins, and the guarantee we have based on this simplistic construction is that we can pay k payments, but for all we know each of these payments must be of value at most N . However, the total budget allocated in this solution is in fact kN , and thus it does not seem to solve the k-payment problem as stated, where the only limit on the value of payments is placed on their sum, rather than on individual values. As an aside, we remark that one corollary of our work (specifically, Theorem 3.1) is that if coin dispensing is done using the greedy strategy (see below), then the above "binary" construction for coin allocation indeed solves the general k-payment problem. More precisely, highest possible denomination use as many i-coins as possible 4 dispense j i-coins Figure 2: The greedy algorithm for coin dispensing. P is the amount to be paid. assume that a power of 2. Then the coin allocation algorithm allocates k coins of each denomination . Clearly, the number of coins is k log 2 and their sum is N . Coin dispensing is made greedily: at each point, the largest possible coin is used. description of the greedy dispensing algorithm is presented in Figure 2.) However, one should note that, perhaps surprisingly, our results also indicate that the binary algorithm is not the right generalization for 1: the best algorithm (described in Section yields a factor of about improvement, i.e., roughly 30% fewer coins. Let us now re-consider the question of a general N . Once we have an algorithm for infinitely many values of k and N , generalizing to arbitrary N and k is easy: find a solution for N 0 and dispense a payment of value . The remaining set of coins is a coin allocation of total budget N , and it can be used for k additional payments since the original set solved the 1)-payment problem. We close this section with a simple lower bound on the number of coins required in any solution to the k-payment problem. The bound is based on a counting argument, and we only sketch it here. For the number of distinct possible payment requests is allowed). Observe that the algorithm must dispense a different set of coins in response to each request. It follows that the number of coins in any solution must be at least log 2 (N 1). This argument can be extended to a general k, using the observation that the number of distinct responses of the algorithm (disregarding order), is at least p k (N ), the number of ways to represent N as a sum of k positive integers. Using standard bounds for partitions (see, e.g., [13]), and since the number of responses is exponential in the number of coins, one can conclude that the number of coins is \Omega\Gamma k log(N=k 2 )). 3 Problem characterization In this section we first prove a simple condition to be necessary and sufficient for a set of coins to correctly solve the k-payment problem. Using this result, we obtain a sharp lower bound on the number of coins in any solution to the k-payment problem. Finally, we outline an efficient algorithm which, given a set of coins S, determines the maximal k for which S is a solution of the k-payment problem. Please refer to Figure 1 for notation. 3.1 A Necessary and sufficient condition Theorem 3.1 S solves the k-payment problem if and only if T i - ki for all The theorem is proven by a series of lemmas below. The necessity proof is not hard: the intuition is that the "hardest" cases are when all payment requests are equal. The more interesting part is the sufficiency proof. We start by upper bounding m, the largest denomination in a solution. Lemma 3.2 If S solves the k-payment problem, then m - dN=ke : Proof: By contradiction. Suppose m ? dN=ke, and consider k payments of values bN=kc and dN=ke such that their total sum is N . Clearly, none of these payments can use m-coins, and since c m - 1 by definition, the total budget available for these payments is at most N \Gamma m, contradiction. The following lemma is slightly stronger than the condition in Theorem 3.1. We use this version in the proof of Theorem 3.6. Lemma 3.3 If S solves the k-payment problem, then T i - ki for all Proof: Let i - bN=kc. Then ki - N . Consider k payments of value i each such payment can be done only with coins of denomination at most i, hence T i - ki. We now turn to sufficiency. We start by showing that if the condition of Theorem 3.1 holds single payment of value up to N can be satisfied by S under the greedy algorithm. Lemma 3.4 Let P be a payment request, and suppose that in S we have that for some j, 1. 2. Then P can be satisfied by the greedy algorithm using only coins of denomination j or less. Proof: We prove, by induction on j, that the claim holds for j and any P . The base case is and hence there are at least P 1-coins, which can be used to pay any amount up to their total sum. For the inductive step, assume that the claim holds for j and all P , and consider j 1. Let a be the number of (j 1)-coins dispensed by the greedy algorithm, namely, Let R denote the remainder of the payment after the algorithm dispenses the (j 1). Note that (2) trivially holds after dispensing the (j + 1)-coins; we need to show that (1) holds as well. We consider two cases. If a = c j+1 , then using (1) we get and we are done for this case. If a ! c j+1 , then since the algorithm is greedy, it must be the case that R 1. On the other hand, by (2) we have that T j - j, and hence T j - R and we are done in this case too. The following lemma is the key invariant preserved by the greedy algorithm. It is interesting to note that while the algorithm proceeds from larger coins to smaller ones, the inductive proof goes in the opposite direction. Recall that "primed" quantities refer to the value after a payment is done. Lemma 3.5 If T i - ki holds for all after the greedy algorithm dispenses any amount up to Tm , T 0 holds for all Proof: By induction on i. For the claim is trivial. Assume that the claim holds for all is the amount dispensed using coins of denomination at most i). namely j is the smallest remaining denomination which is larger than i. Note that j is well defined since namely i is not the largest remaining coin. Next, note that since all the coins of denomination (whose sum is used by the algorithm, we have that Now, observe that since the algorithm is greedy, and since at least one j-coin was not used by the algorithm, it must be the case that the total amount dispensed using coins of denomination smaller than j is less than j, i.e., Using Eq. (1), we get that T i - Finally, using the assumption applied to We now complete the proof of the characterization. Proof (of Theorem 3.1): The necessity of the condition follows directly from Lemmas 3.2 and 3.3. For the sufficiency, assume that T i - ik for all m, and consider a sequence of up to k requests of total value at most N . After the l-th request is served by the greedy algorithm, we have, by inductive application of Lemma 3.5, that T 0 Moreover, by Lemma 3.4, any amount up to the total remainder can be paid from S, so long as which completes the proof. 3.2 A lower bound on the number of coins Using Theorem 3.1, we derive a lower bound on the number of coins in any solution to the k-payment problem. Theorem 3.6 The number of coins in any solution to the k-payment problem is at least We first prove a little lemma we use again in Theorem 4.7. Lemma 3.7 For any number j, Proof: Proof (of Theorem 3.6): By Lemma 3.7 and Lemma 3.3: bN=kc ki The lower bound of Theorem 3.6 is the best possible in general, as shown in the following theorem. Theorem 3.8 For any natural numbers are infinitely many N ? that there exists a solution for the k-payment problem with budget N with exactly kH bN=kc coins. Proof: Choose a natural number m such that m! ? km. In this case the solution with solves the k-payment problem by Theorem 3.1, and its total number of coins is precisely kH bN=kc . 3.3 Determining k for a given set Consider the "inverse problem:" we are given a set of coins, with c i coins of denomination i for each i, and the question is how many payments can we make using these coins, i.e., find k. Note that k is well defined: we say that k is 0 if there is a payment request which cannot be satisfied by the set, and k is never more than the total budget. Theorem 3.1 can be directly applied to answer such a question efficiently, as implied by the following simple corollary. Corollary 3.9 Let S be a given coin allocation. Then S solves the k-payment problem if and only if k - min fbT i =ic mg. 4 An optimal algorithm for unrestricted denominations We now present a coin allocation algorithm which finds a minimal solution to the k-payment for arbitrary N and k, assuming that all denominations are available. We first prove the optimality of the algorithm, and then give an upper bound on the number of coins it allocates. In Section 5 we generalize the algorithm to handle a restricted set of denominations. The algorithm. Given arbitrary integers N - k ? 0, the algorithm presented in Figure 3 finds an optimal coin allocation. Intuitively, the algorithm works by scanning all possible denominations for each i, the least number of i-coins which suffices to make T i - ki. The number of coins c i is thus an approximation of the i-th harmonic element multiplied by k. When the budget is exhausted, the remainder is added simply as a single addiotinal coin. 4.1 Optimality of Allocate Theorem 4.1 Let S denote the solution produced by Allocate. Then S solves the k-payment problem using the least possible number of coins. Allocate(N; 4 while denominations in order add i-coins but don't overflow 9 if N ? t add remainder Figure 3: Algorithm for optimal solution of the k-payment problem with unrestricted denominations Proof: Follows from Lemma 4.3 and Lemma 4.5 below. The important properties of the algorithm are stated in the following loop invariant. Lemma 4.2 Whenever Allocate executes line 4, the following assertions hold. (ii) if Proof: Line 4 can be reached after executing lines 1-3 or after executing the loop of lines 5-8. In the former case, we have and the lemma holds trivially. Suppose now that line 4 is reached after an iteration of the loop. We denote by t 0 the value of the variable t before the last execution of the loop. If lines 7 and 8 are not executed, then (i) holds trivially. If they are executed, then we have that and hence (i) holds after the loop is executed. Also note that t - l ki\Gammat 0 1), and therefore (iii) holds true after the execution of the loop. Finally, we prove that (ii) holds after executing the loop. If lines 7-8 are not executed, (ii) holds trivially. Suppose that lines 7-8 are executed, and that i t. In this case, by line which means that Therefore, by line 7, c l ki\Gammat , and hence l ki\Gammat 0 and we are done. Using Lemma 4.2 and Theorem 3.1, correctness is easily proven. We introduce some notation to facilitate separate handling of the remainder: i , for all i, is the value of the c i variable when line 9 is executed. . Lemma 4.3 S solves the k-payment problem. Proof: By (i) of Lemma 4.2, in conjunction with lines 9-10 of the code, we have that upon completion of the algorithm, By (ii) of Lemma 4.2, and since (by lines 4 and 10) the largest denomination is m - m , we have that T i - T denominations therefore, by Theorem 3.1, S solves the k-payment problem. Proving optimality takes more work. We first deal with the the coins allocated before line 9 is reached, and then show that even if the remainder was non-zero (and an additinal coin was allocated in line 10), the solution S produced by Allocate is still optimal. To this end, we fix an arbitrary solution T for the k-payment problem, and use the following additional notation: ffl d i is the number of i-coins in T . ffl U i is the budget allocated in T using coins of denomination i or less: U ffl n is the largest denomination of a coin in T : Formally, Note that by the definitions, the following holds for all i - 0: c The lemma shows that Allocate is optimal-ignoring the remainder. Lemma 4.4 c Proof: By contradiction. Suppose that l is the smallest such index, i.e., using Eq. (2), l. Hence, by Eq. (3), we have that d l ! c which implies, by integrality, that d l - c since T by (iii) of Lemma 4.2, we have ld l ic i.e., U l ! kl, contradiction to Theorem 3.1, since l ! n. We now prove that S is optimal even when considering the remainder. Lemma 4.5 d i . Proof: We consider two cases. If n ? m then using Lemma 4.4, and the fact that d n - 1 by definition of n, we get c and we are done for this case. So suppose n - m. Let We prove the lemma by showing that First, note that since by Lemma 4.4 we have using Eq. (3), we get On the other hand, since U Therefore, it follows from Eq. (2) that We now consider two subcases. If T reduces to nb and we are done for this subcase. Otherwise, and the proof of Lemma 4.5 is complete. 4.2 The number of coins allocated by Allocate Theorem 4.1 proved that the number of coins in the solution produced by Allocate is optimal. We now give a tight bound on that number in terms of N and k. There is a nice interpretation to the bound: it says that the worst penalty for having N and k which are not "nice" is equivalent to requiring an extra payment (i.e., solving the 1)-payment problem) for "nice" N an k. We remark that we do not know of any direct reduction which proves this result. First, we prove an upper bound on m, which is slightly sharper than the general bound of Lemma 3.2. Lemma 4.6 The largest denomination of a coin generated by Allocate satisfies m - l N Proof: By (iii) of Lemma 4.2 we have T 1)i, and in particular By (ii) of Lemma 4.2 we have that T have that T m. Using also Eq. (5), we get that 1, and by integrality, m - l N Theorem 4.7 The number of coins in the solution produced by Allocate is at most (k Proof: By (iii) of Lemma 4.2, and by integrality, T in conjunction with Lemmas 3.7 and 4.6, implies that c and therefore, The upper bound given by theorem 4.7 is the best possible in general, as proven in the following theorem. Theorem 4.8 For any natural numbers are infinitely many N ? that any solution for the k-payment problem for budget N requires at least (k coins. Proof: Choose a natural number m such that and divisible by 2; 3; these N and k, we have that the algorithm produces largest denomination and c It follows that the number of coins in this case is precisely 5 Generalization to a restricted set of denominations We now turn our attention to the restricted denomination case. In this setting, we are given a set D of natural numbers, and the requirement is that in any solution, c D. To get an optimal algorithm for an arbitrary set of denominations which allows for a solution, 1 we follow the idea of algorithm Allocate: ensure, with the least number of coins, that m. The treatment of the remainder is more complicated here, but has the same motivation: add the remainder with the least possible number of coins. We also have to make sure that the invariant maintained by the main loop is not broken, so we restrict the remainder allocation to use only denominations which were already considered. In fact, remainder allocation is precisely the change-making problem, solvable by dynamic program- ming. (For some allowed denomination sets [5, 14], the greedy strategy works too.) The main algorithm Allocate-Generalized is given in Figure 4. For completeness, we also include, in Figure 5, a description of a dynamic programming algorithm for optimal change-making. In Allocate-Generalized, and in the analysis, we use the following additional notation. ig, the largest allowed denomination smaller than i. ig, the smallest denomination larger than i. We now prove that Allocate-Generalized is correct and that the number of coins it allocates is optimal. We remark that the general arguments are similar to those for the unrestricted case, but are somewhat more refined here. Theorem 5.1 Let S denote the solution produced by Allocate-Generalized. Then S solves the k-payment problem with allowed denominations D using the least possible number of coins. Proof: Follows from Lemmas 5.3 and 5.8 below. We again use a loop invariant to capture the important properties of the algorithm. Lemma 5.2 Whenever Allocate-Generalized executes line 4, the following assertions hold. (ii) if Proof: Line 4 can be reached after executing lines 1-3 or after executing the loop of lines 5-10. In the former case and the lemma holds trivially. 1 Observe that the problem is solvable if and only if 1 2 D: if there are no 1-coins then certainly we cannot satisfy any payment request of value 1; and if 1 is allowed, then the trivial solution of N 1-coins works for all k. Allocate-Generalized(N; k; D) 4 while is the largest den. smaller than the next one in D l up to j 8 then if l kj \Gammat check if budget is exhausted 9 then c i / l kj \Gammat if not, add coins else goto 12 13 then D 0 / fd j d 2 D; d - ig add remainder without greater denominations Figure 4: Algorithm for optimal solution of the k-payment problem with allowed denominations D. In the latter case, let t 0 denote the value of the variable t before the last execution of the loop. If lines 9 and 10 are not executed, then (i) holds trivially. If they are executed, then holds after the loop is executed. Then, whether or not lines 9 and 10 are executed t - l kj \Gammat 0 thus (iii) is true after the execution of the loop. Finally, we show (ii) holds after executing the loop. If line 8 is not executed, or if line 11 is executed then (ii) holds trivially. Otherwise lines 9-10 are executed, and c l kj \Gammat , hence l kj \Gammat 0 The correctness of Allocate-Generalized is proven in the next lemma. Following the conventions of Section 4, we denote by S the allocation produced by Allocate-Generalized before line 12. We denote by c i the values in S corresponding to c i and T i in S, respectively. Lemma 5.3 S solves the k-payment problem with allowed denominations D. Proof: By (i) of Lemma 5.2, in conjunction with lines 12-14 of the code, we have that upon completion of the algorithm, be the highest denomination allocated by the algorithm. Then, by (ii) of Lemma 5.2, and the fact that Make-Change does not use coins with higher denominations (line 13), we have that for all 1 Therefore, by Theorem 3.1, S solves the k-payment problem. Make-Change(R;D) do if i 2 D 6 do for i / R downto 1 7 do if M 8 then for all 9 do if M [i \Gamma j] return M [R] Figure 5: Dynamic programming algorithm for finding least number of coins whose sum is R units using denomination set D. M is an array of multisets. We now turn to prove optimality of the algorithm. The analysis proceeds similarly to that of Section 4: We first show that the allocation of the coins up to the remainder (i.e., just before line 12 is executed), is optimal. We then prove that the handling of the remainder is optimal as well. The proof here is complicated by the fact that in order to ensure correctness, Allocate-Generalized allocates the remainder by using only denominations which were already considered in the main loop, and hence it is not immediately clear that the remainder is allocated optimally. For the remainder of this section, fix an arbitrary "competitor" solution T . We define for T notions analogous to those defined for the solution S produced by Allocate-Generalized. ffl d i is the number of i-coins in T . ffl U i is the budget allocated in T using coins of denomination i or less: U ffl n is the largest denomination of a coin in T : Formally, To facilitate treatment of the remainder, we fix an arbitrary subsulotion T of T which solves the k-payment problem for budget T . (This is possible since a solution for budget N is also a solution for any budget smaller than N .) We use the following additional definitions. jd . (Analogous to T i in S). is the largest denomination in T . is the largest denomination allocated up to line 12 by Allocate-Generalized. Hence, T Finally, we define As before, by the definitions, for all i - 0 we have d c d We start by proving that Allocate-Generalized produces an optimal solution-ignoring the remainder. Lemma 5.4 For all 1 d c Proof: Suppose that l is the smallest such index, i.e., using l. Hence, by Eq. (7), we have that d which implies, by integrality, that d l - c l by (iii) of Lemma 5.2. Therefore, since l U ld l id ic or U a contradiction to Theorem 3.1, since l Corollary 5.5 n - m . Proof: Suppose not. Then, by Lemma 5.4, U which is a contradiction to the fact that U Next, we prove that no solution can use a denomination larger than the largest one used by S. Lemma 5.6 n - m. Proof: We consider two cases. Suppose first that line 12 is reached after testing the condition line 4. Then m hence the remainder allocated by the dynamic algorithm is m. Note that n - max(n and therefore, n - max(m and we are done for this case. Suppose next that line 12 is reached from line 11. In this case we have and since T and hence or Now suppose for contradiction that n ? m. Then at least one coin of denomination allocated by T , hence N - we get from Eq. (8) that a contradiction to Theorem 3.1 since It follows that the remainder is allocated optimally (for any choice of a subsolution T ). Corollary 5.7 d c Proof: By Lemma 5.6, the amount of m must be allocated in T using only denominations used in S. The statement therefore follows from the optimality of dynamic programming used in Make-Change. We can now prove optimality of the full solution S. Lemma 5.8 d i . Proof: By Corollary 5.7, it is sufficient to prove that the number of coins in T is at least as much as in S . If n - m , then we are done by Lemma 5.4. It remains to consider the case of . We do it similarly to Lemma 4.5: define . With this notation, it is sufficient to show that By Lemma 5.4 we have using Eq. (7) we get U i(d Also, since U U c ic Hence, b. Thus, from Eq. (6) we get (d as required. 6 Exact payments in electronic cash In this section we discuss the issue of exact payments in the model of electronic cash (e-cash). Briefly, in this model we have a bank, a set of users and a set of shops. There are withdrawal and deposit protocols involving the bank and a user or shop, and a payment protocol, involving a shop and a user. It is required that the users are anonymous: given a coin, it should be impossible to infer which user withdrew it, even if the shops and bank collaborate. On the other hand, the identity of the user should be revealed if the user pays with the same coin more than once. The model has to be implemented with "electronic wallets" called smart cards which have a severely limited capacity for storage and computation [7]. There are a few ways to implement exact payments in e-cash. The simplest way is to use multiple coins. In this case, the results presented in this paper explain what coins should be withdrawn. Another approach is divisible coins [9, 8, 12]: arbitrary portions of the coin can be spent so long as their sum does not exceed the value of the coin. As expected, the size of a divisible coin and the computational resources (such as time and communication) required for its manipulation are greater than those of non-divisible coins. An intermediate approach is presented in [12]: a composite coin is divisible only to a set of prescribed set of subcoins. Subcoins can be used for payment and deposit, but cannot be divided further. Although the size of the composite coin grows linearly with the number of its constituent subcoins (for C subcoins, a typical size of the composite coin is 500 the size is still smaller than the size of arbitrarily divisible coins for a sufficiently small number of subcoins. The results of this paper are particularly useful for the composite coin if one can get a reasonable bound k on the number of exact payments, then composite coins are better than arbitrarily divisible coins in all respects for certain ranges of N and k. With current technology, the technique of [12] is better when k Acknowledgments We thank Mike Saks, Richard Stanley and Yishay Mansour for helpful discussions, Eric Bach for bringing the postage stamp problem to our attention, Rakesh Verma for providing us with a copy of [14], and Agnes Chan for valuable suggestions and comments. --R Untraceable off-line cash in wallets with observers Blind signatures for untraceable payments. Untraceable electronic cash. Construction of distributed loop networks. Optimal bounds for the change-making problem Knapsack Problems: Algorithms and Computer Implementations. Cryptographic smart cards. An efficient divisible electronic cash scheme. Universal electronic cash. On the postage stamp problem with 3 denominations. Associate bases in the postage stamp problem. Efficient Electronic Notions and Techniques. A Course in Combinatorics. On optimal greedy change making. --TR

coin allocation;change-making;exact change;electronic cash;k-payment problem

365883

Graph Searching and Interval Completion.

In the early studies on graph searching a graph was considered as a system of tunnels in which a fast and clever fugitive is hidden. The "classical" search problem is to find a search plan using the minimal number of searchers. In this paper, we consider a new criterion of optimization, namely, the search cost. First, we prove monotone properties of searching with the smallest cost. Then, making use of monotone properties, we prove that for any graph G the search cost of G is equal to the smallest number of edges of all interval supergraphs of G. Finally, we show how to compute the search cost of a cograph and the corresponding search strategy in linear time.

Introduction Search problems on graphs attract the attention of researchers from different fields of Mathematics and Computer Science for a variety of reasons. In the first place, this is the resemblance of graph searching to certain pebble games [18] that model sequential computation. The second motivation of the interest to the graph searching arises from the VLSI theory. Exploitation of game-theoretic approaches to some important parameters of graphs layouts such as the cutwidth [23], the topological bandwidth [22] and the vertex separation number [11] is very useful for the construction of efficient algorithms. Yet another reason is connections between graph searching, the pathwidth and the treewidth. These parameters play very important role in the theory of graph minors developed by Robertson and Seymour (see [1, 10, 29]). Also some search Department of Operations Research, Faculty of Mathematics and Mechanics, St.Petersburg State University, Bibliotechnaya sq.2, St.Petersburg, 198904, Russia, e-mail: fomin@gamma.math.spbu.ru The research of this author was partially supported by the RFBR grant N98-01-00934 y Department of Applied Mathematics, Faculty of Mathematics, Syktyvkar State Univer- sity, Oktyabrsky pr., 55, Syktyvkar, 167001, Russia, e-mail: golovach@ssu.edu.komi.ru. The research of this author was partially supported by the RFBR grant N96-00-00285. problems have applications in motion coordinations of multiple robots [30] and in problems of privacy in distributed environments with mobile eavesdroppers ('bugs') [13]. More information on graph searching and related problems one can find in surveys [1, 12, 25]. In the 'classical' node-search version of searching (see, e.g.[18]) at every move of searching a searcher is placed at a vertex or is removed from a vertex. Initially, all edges are contaminated (uncleared). A contaminated edge is cleared once both its endpoints are occupied by searchers. A clear edge e is recontaminated if there is a path without searchers leading from e to a contaminated edge. The 'classical' search problem is in finding the search program such that the maximal number of searchers used at any move is minimized. In this paper we are interested in another criterion of optimization. We are looking for node- search programs with the minimal sum (the sum is taken over all moves of the search program) of numbers of searchers. We call this criterion the search cost. Loosely speaking, the cost of a search program is the total number of 'man-steps' used in this program and the search cost is the cost of an optimal program. The reader is referred to section 2 for formal definitions of searching and its cost. One of the most important issues concerning searching is that of recontam- ination. In some search problems (see [2, 20]) the recontamination does not help to search a graph, i.e., if searchers can clear the graph then they can do it without recontamination of previously cleared edges. We establish the monotonicity of search programs of the smallest cost. To prove the monotonicity result we use special constructions named clews. Clews are closely related to crusades used by Bienstock and Seymour in [2] and the notion of clew's measure is related to the notion of linear width introduced by Thomas in [31]. This paper is organized as follows. In x 2 we give necessary definitions. In x 3 we introduce clews and prove the monotonicity of graph searching. In x 4 it is proved that for any graph G the search cost of G is equal to the smallest number of edges of an interval supergraph of G. In x 5 it is shown that the problem of computing the search cost is equivalent to the vertex separation sum problem and profile minimization problem. In x 6 we obtain some estimates of the search costs in terms of the vertex separation number and the sum bandwidth. In x 7 we show how to compute the search cost of graph's product. In x 8 we give a linear time algorithm determining the search cost of a cograph and the corresponding search program. 2 Statement of the problem We use the standard graph-theoretic terminology compatible with [6], to which we refer the reader for basic definitions. Unless otherwise specified, G is an undirected, simple (without loops and multiple edges) and finite graph with the vertex set V (G) and the edge set E(G); n denotes the order of G, i.e., n. The degree of a vertex v in G is denoted by deg(v) and the maximum degree of the vertices of a graph G by \Delta(G). A search program \Pi on a graph G is the sequence of pairs such that I. for II. for vertex incident with an edge in A j an edge in E(G) \Gamma A j i is in Z j III. for IV. (placing new searchers and clearing edges) for there is such that Z 1 is the set of all incident with v edges having one end in Z 2 V. (removing searchers and possible recontamination) for i is the set of all edges e 2 A 1 i such that every path containing e and an edge of E(G) \Gamma A 1 i , has an internal vertex in Z 2 We call this the search axioms. It is useful to treat Z 1 i as the set of vertices occupied by searchers right away after placing a new searcher at the ith step; i as the set of vertices occupied by searchers immediately before making the 1)th step; and A 1 i as the sets of cleared edges. The well known node search problem [18] is in finding \Pi with the smallest (this maximum can be treated as the maximum number of searchers used in one step). Let us suggest an alternative measure of search. We define the cost of \Pi to be j. One can interpret the cost of a search program as the total number of 'man-steps' used for the search or as the total sum that searchers earn for doing their job. The search cost of a graph G, denoted by fl(G), is the minimum cost of a search program where minimum is taken over all search programs on G. A search program if for each (recontamination does not occur when searchers are removed at the ith step). The monotone search cost of G, is the cost of the minimal (over all monotone search programs) search program on G. Notice that search programs can be defined not only for simple graphs but for graphs with loops and multiple edges as well. Adding loops and multiple edges does not change the search cost. 3 Monotone programs and clews Let G be a graph. For X ' E(G) we define V (X) to be the set of vertices which are endpoints of X and let We consider clews only in graphs of special structure. Let G 0 be obtained by adding a loop at each vertex of a graph G. A clew in G 0 is the sequence of subsets of E(G 0 ) such that 1. 2. for 3. for i the loop at v also belongs to X i . The measure of the clew is progressive Notice that if a clew (X Theorem 1 For any graph G and k - 0 the following assertions are equivalent: (ii) Let G 0 be obtained by adding a loop at each vertex of G. There is a clew in G 0 of measure - k. (iii) Let G 0 be obtained by adding a loop at each vertex of G. There is a progressive clew in G 0 of measure - k. Proof. (i) As mentioned above, be a search program on G 0 with the cost - k. We prove that A 2 is the clew of measure - k in G 0 . The third search axiom implies A 2 E(G). The second search axiom says that for every m is the clew if for every Suppose that for some inequality does not hold. Then there are vertices u; v such that Notice that loops e u at vertices u and v belongs to A 2 . From the fifth search axiom A 1 it follows that e . The latter contradicts the fourth axiom. Choose a clew (X such that and, subject to (1), First we prove that for i is the clew. Using (1), we get It is easy to check that jffij satisfies the submodular inequality Combining (3) and (4), we obtain the loop at v belongs to X Therefore, V (X we obtain that is the clew. Taking into account (5), (1) and (2), we get jX j. Thus we have X is the clew contradicting (2). Hence, (X progressive clew of measure - k in G 0 . We define the search program on G 0 setting Z 1 ng. Suppose that at the ith step searchers are placed at vertices of Z 1 i and all edges of X i are cleaned. Obviously no recontamination occurs by removing all searchers from vertices of Every edge of X is either the loop at v, or is incident with v and with a vertex of ffi (X . Then at the (i step the searcher placed at v cleans all edges of X Finally, 4 Interval graphs A graph G is an interval graph, if and only if one can associate with each vertex an open interval I on the real line, such that for all v; w 2 V (G), v 6= w: (v; w) 2 E(G), if and only if I v " I w 6= ;. The set of is called an (interval) representation for G. It is easy to check that every interval graph has an interval representation in which the left endpoints are distinct integers Such a representation is said to be canonical. A graph G is a supergraph of the graph G 0 if E(G). Let G be an interval graph and let I = fI v g v2V (G) be a canonical representation of G. The length of G with respect to I, denoted by l(G; I), is We define the length l(G) of an interval graph G as the minimum length over all canonical representations of G. For any graph G we define the interval length of G, denoted by il(G), as the smallest length over all interval supergraphs of G. We shall use the following property of canonical representation in the proof of Theorem 3. I be an interval graph of n vertices and I = fI , be its canonical representation such that For ng let P (i) be the set of intervals I v , containing i. Proof. (6) implies that every interval I contains br integers. Every ng belongs to jP (i)j intervals of I. Therefore, For every (the number of intervals I plus the number of intervals I From (8) it follows that which, combined with (7), proves Lemma 2. 2 Theorem 3 For any graph G and k ? 0 the following assertions are equivalent: (iii) there is an interval supergraph I of G such that jE(I)j - k. Proof. (i) then by Theorem 1 there is a monotone search program on G with cost - k. We choose " ! 1 and assign to each vertex v of G the interval (l "), where a searcher is placed on v at the l v th step and this searcher is removed from v at the r v th step, i.e., l and r g. After the nth step of the search program all edges of G are cleared, hence for every edge e of G there is a step such that both ends of e are occupied by searchers. Therefore, interval graph I with canonical representation is the supergraph of G. Since for sufficiently small " (in the left hand side equality each vertex v is counted br times), we see that immediately from Lemma 2. , be a canonical representation of the minimal length of an interval supergraph I of G. It is clear that r v ! n+1, Let us describe the following search program on G: ng we put Z 1 is the vertex assigned to the interval with the left endpoint Such actions of searchers do not imply recontamination because each path in I (and hence in G) from a vertex to a vertex u, l u a vertex of P (i 1). For each edge e of I (and hence for each edge of G) there is ng such that both ends of e belong to Z 1 . Then at the nth step all edges are cleared. The cost of the program advanced is 5 Linear layouts A linear layout of a graph G is a one-to-one mapping ng. There are various interesting parameters associated with linear layouts. For example, setting bw(G; E(G)g, one can define the bandwidth of G as is a linear layout of Gg: The bandwidth minimization problem for arises in different applications (see [8] for a survey). In some applications instead of taking the maximum difference it is more useful to find an 'average bandwidth'. The bandwidth sum of a graph G is is a linear layout of Gg: Define the profile [8] of a symmetric n \Theta n matrix as the minimum value of the sum n taken over all symmetric permutations of A, it being assumed that a ng. Profile may be redefined as a graph invariant p(G) by finding a linear layout f of G which minimizes the sum stands for 'is adjacent or equal to'. Notice, that bw(G; ug. Sum bandwidth and profile reductions are also relevant to the speedup of matrix computations; see [8] for further references. Also these problems arise in VLSI layouts. For a linear layout f of G we define and cw(G; i2f1;:::;ng If we 'draw' G with vertices in a straight line in the order given by f then (G; f) is the number of edges that cross the gap between i and i + 1. The cutwidth of G is is a linear layout of Gg: It is easy to check that for any layout f For U and there exists v 2 V (G) n U such that (u; v) 2 E(G)g: Ellis et al. [11] (see also [21]) studied the vertex separation of a graph. Let f be a linear layout of a graph G. Denote by S i (G; f ng, the set of vertices ig. The vertex separation with respect to layout f is vs(G; and the vertex separation of a graph G, we denote it by vs(G), is the minimum vertex separation over all linear layouts of G. Let us define an alternative 'average norm' of vertex separation. The vertex separation sum with respect to layout f is vs sum (G; and the vertex separation sum of a graph G [16] denoted by vs sum (G) is the minimum vertex separation sum over all linear layouts of G. Owing to Billionnet [3], for any graph G, the profile of G is equal to the smallest number of edges over all interval supergraphs of G. Theorem 4 For any graph G and k ? 0 the following assertions are equivalent: Proof. Because (i) , (iii) follows from [3] and Theorem 3, it remains to prove (i) , (ii). be a monotone search program on G with cost - k. Define a layout ng so that f(u) ! f(v) iff u accepts a searcher before v does. By the second search axiom for each i 2 ng jZ 2 and we conclude that vs sum (G; f) - k. ng be a linear layout such that vs sum (G; f) - k. We define the following subsets of V (G): ng we put Z 1 ng Z 2 For ng and i be the set of edges induced by [ i k . Note that for each edge (u; v) 2 E(G) there is 1g such that (v)). On this basis it is straightforward (as in Theorem 3) to prove that the sequence is the (monotone) search program of cost - k on G. 2 5.1 Complexity remark The problem of Interval Graph Completion: Instance: A graph E) and an integer k. Question: Is there an interval graph G is NP-complete even when G is stipulated to be an edge graph (see [14], Problem GT35). Interval Graph Completion arises in computational biology (see, e.g., [4]) and is known to be FPT [7, 17]. From Theorem 3 it follows immediately that the problem of Search Cost: deciding given a graph G and an integer k, whether fl(G) - k or not, is NP-complete even for edge graphs and that finding the search cost is FPT for a fixed k. An O(n 1:722 ) time algorithm was given in [19] for the profile problem for the case that G is a tree with n vertices. 6 Estimates of the search cost A split in a graph G is a partition of V of the vertex set of G such that jV i and the edges of G going from V 1 to V 2 induce a complete bipartite graph. Let V be a split of G and let be the vertices of the associated complete bipartite graph. The following proposition can be found in [25]. Proposition 5 Every monotone search program on G has a step at which all vertices from A 1 or all vertices from A 2 simultaneously carry a searcher. Corollary 6 Let us think that Proof. Suppose that the jth step is the first step of the monotone program \Pi at which all vertices of A 1 are occupied. Then at this step are placed in A 2 and n 1 searchers in A 1 . Therefore the cost of \Pi is at least2 (j As an illustration of the corollary we refer to a complete bipartite graph with bipartition (X; Y j. The monotone search program of cost 1(n is easily be constructed. First we place searchers on vertices of X and then on vertices of Y . Then by Corolary 6 Owing to Theorem 3 one can formulate the following proposition. Proposition 7 Let G be a graph on n vertices and m edges. Then and only if G is an interval graph, and and only if G is a complete graph. Further estimates of the search cost are obtained by making use of the vertex separation sum. Proposition 8 Let G be a graph on n vertices. Then2 Moreover, if G is connected then2 Proof. Let f be a linear layout of G such that vs sum It is readily shown that there are (G; for The proof of the second inequality is similar. Note that for any i 6= Makedon and Sudborough [23] obtain some bounds for the cutwidth in terms of the maximal degree and the search number. By virtue of (9) it is not surprising that there are strong connections between the bandwidth sum and the search cost. Proposition 9 For any linear layout f of G bw sum (G; f) - \Delta(G)vs sum (G; f). Therefore, bw sum (G) - \Delta(G)fl(G). Proof. The proof is apparent from (9), because for any i 2 ng cw i (G; f) - The line graph of a graph G is the graph with vertex set E(G), two vertices being adjacent iff they are adjacent (as edges) in G. be the line graph of G with Then Proof. Let f be an optimal (for the bandwidth sum) linear layout of G. Consider a linear layout g: E(G) of L(G) such that for any edges a = (u; v), a For ng we define Using we obtain vs sum (L(G); e Summing we get This implies that vs sum (L(G); g) - By (10) and Cauchy inequality, out Clearly, for every \Delta(G). If we combine this with (11) and (12), we get vs sum (L(G)) - \Delta(G)bw sum :7 Product of graphs The disjoint union of graphs G and H is the graph G - [H with the vertex set [V (H) and the edge set E(G) - (where - [ is the disjoint union on graphs and sets, respectively); We use G \Theta H to denote the following type of 'product' of G and H : G \Theta H is the graph with the vertex set V (G) - [V (G) and the edge set E(G) - The following theorem is similar to results on edge-search and node-search numbers [15, 25] and on the pathwidth and the treewidth [5] of a graph. Theorem 11 Let G Proof. (I) is trivial. (II). Let f be an optimal layout of G 1 \Theta G 2 , i.e. vs sum (G 1 \Theta G 2 k be the smallest number ensuring clarity's sake we suppose that be the 're- striction' of f to V (G 2 ), i.e. for any u; v Clearly, for any i 2 In addition, for any i 2 Consequently i=k\Gamman 1 Finally, For the other direction. Let f be an arbitrary layout of G 1 and let g be a layout of G 2 such that vs sum g)j. Define the layout h of by the rule ae It follows easily that j@S i and 1g. We conclude that Similarly, 8 Cographs Theorem 11 can be used to obtain linear time algorithms for the search cost and corresponding search strategy on cograph. Recall that a graph G is a cograph if and only if one of the following conditions is fulfilled: There are cographs G There are cographs G A similar algorithm for the treewidth and pathwidth of cographs was described in [5]. The main idea of the algorithm is in constructing a sequence of operations [' and `\Theta' producing the cograph G. With each cograph G one can associate a binary labeled tree, called the cotree TG . TG has the following properties: 1. Each internal vertex v of TG has a label label(v)2 f0; 1g. 2. There is a bijection - between the set of leaves of TG and V (G). 3. To each vertex v 2 (T G ) we assign the subgraph G v of G as follows: (a) If v is a leaf then G (b) If v is an internal vertex and label(v)= 0 then G [Gw , where u; w are the sons of v. (c) If v is an internal vertex and label(v)= 1 then G u; w are the sons of v. Notice that if r is the root of TG then G G. Corneil et al. [9] gave an determining whether a given graph G is a cograph and, if so, for constructing the corresponding cotree. Theorem 12 The search cost of a cograph given with a corresponding cotree can be computed in O(n) time. Proof. Let r be the root of TG . First we call COMPUTE-SIZE(r). This recursive procedure computes jV (G v )j for each procedure COMPUTE-SIZE(v: vertex); begin if v is a leaf of TG then else begin COMPUTE-SIZE(left son of v); COMPUTE-SIZE(right son of v); size(v):=size(left son of v)+size(right son of v); Then we call COMPUTE-GAMMA(r). procedure COMPUTE-GAMMA(v: vertex); begin if v is a leaf of TG then fl(v):=0 else begin COMPUTE-GAMMA(left son of v); COMPUTE-GAMMA(right son of v); if label(v)=0 then fl(v):=maxffl(left son of v), fl(right son of v)g else fl(v):= minfsize(left son of v) (size(left son of v)\Gamma1)/2 fl(right son of v), size(right son of v) (size(right son of v)\Gamma1)/2 son of v)g son of v)\Lambdasize(right son of v) By Theorem 11 are called once for each vertex of TG , so the time complexity of these procedures is O(n). 2 Theorem 13 Let G be a cograph of n vertices and e edges. The optimal search program on G can be computed in O(n Proof. Let TG be the cotree of G (this tree can be computed in O(n time [9]). Let r be the root of TG . First call COMPUTE-GAMMA(r) and COMPUTE-SIZE(r). Then call SEARCH(r). procedure SEARCH(v: vertex); begin if v is the root of TG then begin if v is a leaf of TG then begin else if label(v)=0 then begin first(left son of v):= first(v); last(left son of v):= first(v)+ size(left son of v)\Gamma1; first(right son of v):= first(v)+ size(left son of v); last(right son of v):= last(v); SEARCH(left son of v); SEARCH(right son of v) else ( label(v)=1) if size(left son of v) (size(left son of v)\Gamma1)/2 +fl(right son of v) ? size(right son of v) (size(right son of v)\Gamma1)/2 +fl(left son of v) then begin k:=first(v); for each vertex w 2 V (G) that is a leaf descendants of the right son of v begin place(w):=k; remove(w):=last(v); k:=k+1 end first(left son of v):= first(v)+ size(right son of v); last(left son of v):= last(v); SEARCH(left son of v) else begin k:=first(v); for each vertex w 2 V (G) that is a leaf descendants of the left son of v begin place(w):=k; remove(w):=last(v); k:=k+1 end first(right son of v):= first(v)+ size(left son of v); last(right son of v):= last(v); SEARCH(right son of v) For each leaf computes the numbers place(v) and remove(v). Clearly, the procedure is linear in size of the cotree. For the vertex of G at the place(v)th step a searcher is placed at - \Gamma1 (v) and at the remove(v)th step the searcher is removed from - \Gamma1 (v). Determining the sets and can be done with bucket sort in O(n) time. Define Z 1 computing the sets can be done in linear time. Finally, the edge sets A j i can be constructed in O(n+ e) time as follows. Put A 1 ;. For the vertex its neighbors and Using the characteristic vector of [ i i the addition of edges incident to v can be done in O(deg(v)) steps. We have proved that the sequence can be constructed in O(n+e) time. By Theorem 11 this sequence is the search program and the cost of this program is fl(G). 2 9 Concluding remarks In this paper, we introduced a game-theoretic approach to the problem of interval completion with the smallest number of edges. There are similar approaches to the pathwidth and treewidth parameters. The interesting problem is whether there is a graph-searching 'interpretation' of the fill-in problem. --R Graph searching Monotonicity in graph searching Basic graph theory: Paths and circuits A linear recognition algorithm for cographs The vertex separation and search number of a graph A graph-theoretic approach to privacy in distributed systems Computers and Intractability Tractability of parameterized completion problems on chordal and interval graphs: Minimum fill-in and physical mapping Searching and pebbling The profile minimization problem in trees Recontamination does not help to search a graph On minimizing width in linear layouts The complexity of searching a graph Some extremal search problems on graphs Graph minors - a survey Optimal algorithms for a pursuit-evasion problem in grids Tree decompositions of graphs. --TR --CTR Yung-Ling Lai , Gerard J. Chang, On the profile of the corona of two graphs, Information Processing Letters, v.89 n.6, p.287-292, 31 March 2004 Paolo Detti , Carlo Meloni, A linear algorithm for the Hamiltonian completion number of the line graph of a cactus, Discrete Applied Mathematics, v.136 n.2-3, p.197-215, 15 February 2004 Fedor V. Fomin , Dimitrios M. Thilikos, On the monotonicity of games generated by symmetric submodular functions, Discrete Applied Mathematics, v.131 n.2, p.323-335, 12 September Sheng-Lung Peng , Chi-Kang Chen, On the interval completion of chordal graphs, Discrete Applied Mathematics, v.154 n.6, p.1003-1010, 15 April 2006 Barrire , Paola Flocchini , Pierre Fraigniaud , Nicola Santoro, Capture of an intruder by mobile agents, Proceedings of the fourteenth annual ACM symposium on Parallel algorithms and architectures, August 10-13, 2002, Winnipeg, Manitoba, Canada Fedor V. Fomin , Dieter Kratsch , Haiko Mller, On the domination search number, Discrete Applied Mathematics, v.127 n.3, p.565-580, 01 May

interval graph completion;graph searching;linear layout;search cost;profile;cograph;vertex separation

365895

Probabilistic Models of Appearance for 3-D Object Recognition.

We describe how to model the appearance of a 3-D object using multiple views, learn such a model from training images, and use the model for object recognition. The model uses probability distributions to describe the range of possible variation in the object's appearance. These distributions are organized on two levels. Large variations are handled by partitioning training images into clusters corresponding to distinctly different views of the object. Within each cluster, smaller variations are represented by distributions characterizing uncertainty in the presence, position, and measurements of various discrete features of appearance. Many types of features are used, ranging in abstraction from edge segments to perceptual groupings and regions. A matching procedure uses the feature uncertainty information to guide the search for a match between model and image. Hypothesized feature pairings are used to estimate a viewpoint transformation taking account of feature uncertainty. These methods have been implemented in an object recognition system, OLIVER. Experiments show that OLIVER is capable of learning to recognize complex objects in cluttered images, while acquiring models that represent those objects using relatively few views.

Introduction Object recognition requires a model of appearance that can be matched to new images. In this paper, a new model representation will be described that can be derived automatically from a sample of images of the object. The representation models an object by a probability distribution that describes the range of possible variation in the object's appearance. Large and complex variations are handled by dividing the range of appearance into a conjunction of simpler probability distributions. This approach is general enough to model almost any range of appearance, whether arising from different views of a 3-D object or from different instances of a generic object class. The probability distributions of individual features can help guide the matching process that underlies recognition. Features whose presence is most strongly correlated with that of the object can be given priority during matching. Features with the best localization can contribute most to an estimate of the object's position, while features whose positions vary most can be sought over the largest image neighborhoods. We hypothesize initial pairings between model and image features, use them to estimate an aligning transformation, use the transformation to evaluate and choose additional pairings, and so on, pairing as many features as possible. The transformation estimate includes an estimate of its uncertainty derived from the uncertainties of the paired model and image features. Potential feature pairings are evaluated using the transformation, its uncertainty, and topological relations among features so that the least ambiguous pairings are adopted earliest, constraining later pairings. The method is called probabilistic alignment to emphasize its use of uncertainty information. Two processes are involved in learning a multiple-view model from training images (Fig. 1). First, the training images must be clustered into groups that correspond to distinct views of the object. Second, each group's members must be generalized to form a model view characterizing the most representative features of that group's images. Our method couples these two processes in such a way that clustering decisions consider how well the resulting groups can be generalized, and how well those generalizations describe the training images. The multiple-view model produced thus achieves a balance between the number of views it contains, and the descriptive accuracy of those views. Related research In recent years, there has been growing interest in modeling 3-D objects with information derived from a set of 2-D views (Breuel, 1992; Murase and Nayar, 1995). For an object that is even moderately complex, however, many qualitatively distinct views may be needed. Thus, a multiple-view representation may require considerably more space and complexity in matching than a 3-D one. Space requirements can be reduced somewhat by allowing views to share common structures (Burns and Riseman, 1992) and by merging similar views after discarding features too fine to be reliably discerned (Petitjean, Ponce, and Kriegman, 1992). In this paper, we develop a representation that combines nearby views over a wider range of appearance by representing the probability distribution of features over a range of Training Images Clusters Model Views Clustering Generalization Figure 1: Learning a multiple-view model from training images requires a clustering of the training images and a generalization of each cluster's contents. One method for improving the space/accuracy trade-off of a multiple-view representation is to interpolate among views. Ullman and Basri (1991) have shown that with three views of a rigid object whose contours are defined by surface tangent discontinuities, one can interpolate among the three views with a linear operation to produce other views under orthographic projection. If the object has smooth contours instead, six views allow for accurate interpolation. However, for non-rigid or generic models, a more direct form of sampling and linear interpolation can be more general while giving adequate accuracy, as described in this paper. There has also been recent development of methods using dense collections of local fea- tures, with rotational invariants computed at corner points (Schmid and Mohr, 1997). This approach has proved very successful with textured objects, but is less suited to geometrically defined shapes, particularly under differing illumination. The approach described in this paper can be extended to incorporate any type of local feature into the model represen- tation, so a future direction for improvement would be to add local image-based features. Initially, the system has been demonstrated with edge-based features that are less sensitive to illumination change. Other approaches to view-based recognition include color histograms (Swain and Bal- lard, 1991), eigenspace matching (Murase and Nayar, 1995), and receptive field histograms (Schiele and Crowley, 1996). These approaches have all been demonstrated successfully on isolated or pre-segmented images, but due to their more global features it has been difficult to extend them to cluttered and partially occluded images, particularly for objects lacking distinctive feature statistics. 2.1 Matching with uncertainty One general strategy for object recognition hypothesizes specific viewpoint transformations and tests each hypothesis by finding feature correspondences that are consistent with it. This strategy was used in the first object recognition system (Roberts, 1965), and it has been used in many other systems since then (Brooks, 1981; Bolles and Cain, 1982; Lowe, 1985; Grimson and Lozano-P-erez, 1987; Huttenlocher and Ullman, 1990; Nelson and Selinger, 1998). An example of this approach is the iterative matching in the SCERPO system (Lowe, 1987). A viewpoint transformation is first estimated from a small set of feature pairings. This transformation is used to predict the visibility and image location of each remaining model feature. For each of these projected model features, potential pairings with nearby image features are identified and evaluated according to their expected reliability. The best ranked pairings are adopted, all pairings are used to produce a refined estimate of the trans- formation, and the process is repeated until acceptable pairings have been found for as many of the model features as possible. This paper describes an enhanced version of iterative matching that incorporates feature uncertainty information. In a related approach, Wells (1997) has shown how transformation space search can be cast as an iterative estimation problem solved by the EM algorithm. Using Bayesian theory and a Gaussian error model, he defines the posterior probability of a particular set of pairings and a transformation, given some input image. In more recent work, Burl, Weber, and Perona (1998) provide a probabilistic model giving deformable geometry for local image patches. The current paper differs from these other approaches by deriving a clustered view-based representation that accounts for more general models of appearance, incorporating different individual estimates of feature uncertainty, and making use of a broader range of features and groupings. 2.2 Use of uncertainty information in matching Iterative alignment has been used with a Kalman filter to estimate transformations from feature pairings in both 2D-2D matching (Ayache and Faugeras, 1986) and 2D-3D matching (Hel-Or and Werman, 1995). Besides being efficient, this allows feature position uncertainty to determine transformation uncertainty, which in turn is useful in predicting feature positions in order to rate additional feature pairings (Hel-Or and Werman, 1995). However, this (partial) least-squares approach can only represent uncertainty in either image or model features, not both; total least squares can represent both, but may not be accurate in predicting feature positions from the estimated transformation (Van Huffel and Vandewalle, 1991). Most have chosen to represent image feature uncertainty; we have chosen to emphasize model feature uncertainty, which in our case carries the most useful information. 3 Model representation An object model is organized on two levels so that it can describe the object's range of appearance both fully and accurately. At one level, large variations in appearance are handled by subdividing the entire range of variation into discrete subranges corresponding to distinctly different views of the object; this is a multiple-view representation. At a second level, within each of the independent views, smaller variations are described by probability distributions that characterize the position, attributes, and probability of detection for individual features. The only form of appearance variation not represented by the model is that due to varying location, orientation, and scale of the object within the image plane. Two mechanisms accommodate this variation. One is the viewpoint transformation, which aligns a model view with an appropriate region of the image; we shall describe it in section 4. The other is the use of position-invariant representations for attributes, which allow feature attributes to be compared regardless of the feature positions. 3.1 Simplifying approximation of feature independence Our method lets each model view describe a range of possible appearances by having it define a joint probability distribution over image graphs. However, because the space of image graphs is enormous, it is not practical to represent or learn this distribution in its most general form. So instead, the joint distribution is approximated by treating its component features as though they were independent. This approximation allows the joint distribution to be decomposed into a product of marginal distributions, thereby greatly simplifying the representation, matching, and learning of models. One consequence of this simplification is that statistical dependence (association or co- among model features cannot be accurately represented within a single model view. Consider, for example, an object whose features are divided among two groups, only one of which appears in any instance. With its strongly covariant features, this object would be poorly represented by a single view. However, where one view cannot capture an important statistical dependence, multiple views can. In this example, two model views, each containing one of the two subsets of features, could represent perfectly the statistical dependence among them. By using a large enough set of views, we can model any object as accurately as we wish. For economy, however, we would prefer to use relatively few views and let each represent a moderate range of possible appearances. The model learning procedure described in section 6 gives a method for balancing the competing aims of accuracy and economy. 3.2 Model view representation A single model view is represented by a model graph. A model graph has nodes that represent features, and arcs that represent composition and abstraction relations among features. Each node records the information needed to estimate three probability distributions characterizing its feature: 1. The probability of observing this feature in an image depicting the modeled view of the object. This is estimated from a record of the number of times the model feature has been identified in training images by being matched to a similar image feature. 2. Given that this feature is observed, the probability of it having a particular position. This is characterized by a probability distribution over feature positions. We approximate this distribution as Gaussian to allow use of an efficient matching procedure based on least squares estimation. The parameters of the distribution are estimated from sample feature positions acquired from training images. 3. Given that this feature is observed, the probability of it having particular attribute values. This is characterized by a probability distribution over vectors of attribute values. Little can be assumed about the form of this distribution because it may depend on many factors: the type of feature, how its attributes are measured, possible deformations of the object, and various sources of measurement error. Thus we use a non-parametric density estimator that makes relatively few assumptions. To support this estimator, the model graph node records sample attribute vectors acquired from training images. 3.3 Model notation An object's appearance is modeled by a set of model graphs fG i g. A model graph G i is a tuple hF; R; mi, where F is a set of model features, R is a relation over elements of F , and m is the number of training images used to produce G i . A model feature j 2 F is represented by a tuple of the form ht j's type is represented by t j , whose value is one of a set of symbols denoting different types of features. The element m j specifies in how many of the m training images feature was found. The series A j contains the attribute vectors of those training image features that matched j. The dimension and interpretation of these vectors depend on j's type. The series contains the mean positions of the training image features that matched j. These positions, although drawn from separate training images, are expressed in a single, common coordinate system, which is described in the following section. From j's type t j , one can determine whether j is a feature that represents a grouping or abstraction of other features. If so then R will contain a single element, hk; l specifying j's parts as being l 1 through l n . The number of parts n may depend on j. Moreover, any l i may be the special symbol ?, which indicates that the part is not defined, and perhaps not represented in the model graph. 4 Coordinate systems and viewpoint transformations A feature's position is specified by a 2-D location, orientation, and scale. Image features are located in an image coordinate system of pixel rows and columns. Model features are located in a model coordinate system shared by all features within a model graph. Two different schemes are used to describe a feature's position in either coordinate system xy's The feature's location is specified by [x y], its orientation by ', and its scale by s. xyuv The feature's location is specified by [x y], and its orientation and scale are represented by the direction and length of the 2-D vector [u v]. We shall use the xy's scheme for measuring feature positions, and the xyuv scheme to provide a linear approach for aligning features in the course of matching a model with an image. They are related by Where it is not otherwise clear we shall indicate which scheme we are using with the superscripts xy's and xyuv . The task of matching a model with an image includes that of determining a viewpoint transformation that closely aligns image features with model features. The viewpoint trans- formation, T , is a mapping from 2-D image coordinates to 2-D model coordinates-it transforms the position of an image feature to that of a model feature. 4.1 Similarity transformations A 2-D similarity transformation can account for translation, rotation, and scaling of an ob- ject's projected image. It does not account for effects of rotation in depth, nor changes in perspective as an object moves towards or away from the camera. A 2-D similarity transformation decomposed into a rotation by ' t , a scaling by s t , and a translation by [x t y t ], in that order, can be expressed as a linear operation using the xyuv scheme, as Ayache and Faugeras (1986), among others, have done. The linear operation has two formulations in terms of matrices. We shall present both formulations here, and have occasion to use both in section 5. We shall develop the formulations by first considering the transformation of a point location from [x k y k ] to [x 0 k ]. We can write it as sin ' t cos ' t (1) Defining allows us to rewrite this as either (2) or Now consider a vector [u k v k ] whose direction represents an orientation and whose magnitude represents a length. When mapped by the same transformation, this vector must be rotated by ' t and scaled by s t to preserve its meaning. Continuing to use u , we can write the transformation of [u k v k ] as either or Equations 3 and 5 together give us one complete formulation of the transformation. We can write it with a matrix A k representing the position b being transformed, and a vector t representing the transformation: Equations 2 and 4 together give us another complete formulation. We can write it with a matrix A t representing the rotation and scaling components of the transformation, and a vector x t representing the translation components: Because it can be expressed as a linear operation, the viewpoint transformation can be estimated easily from a set of feature pairings. Given a model feature at b j and an image feature at b k , the transformation aligning the two features can be obtained as the solution to the system of linear equations b additional feature pairings, the problem of estimating the transformation becomes over-constrained; then the solution that is optimal in the least squares sense can be found by least squares estimation. We shall describe a solution method in section 5.3. 5 Matching and recognition methods Recognition requires finding a consistent set of pairings between some model features and some image features, plus a viewpoint transformation that brings the paired features into close correspondence. Identifying good matches requires searching among many possible combinations of pairings and transformations. Although the positions, attributes, and relations of features provide constraints for narrowing this search, a complete search is still impractical. Instead the goal is to order the search so that it is likely to find good matches sooner rather than later, stopping when an adequate match has been found or when many of the most likely candidates have been examined. Information about feature uncertainty can help by determining which model features to search for first, over what size image neighbourhoods to search for them, and how much to allow each to influence an estimate of the viewpoint transformation. 5.1 Match quality measure A match is a consistent set of pairings between some model and image features, plus a transformation closely aligning paired features. We seek a match that maximizes both the number of features paired and the similarity of paired features. Pairings are represented by image feature k, and e j =? if it matches nothing. H denotes the hypothesis that the modeled view of the object is present in the image. Match quality is associated with the probability of H given a set of pairings E and a viewpoint transformation T , which Bayes' theorem lets us write as There is no practical way to represent the high-dimensional, joint probability functions P(E j them by adopting simplifying assumptions of feature independence. The joint probabilities are decomposed into products of low-dimensional, marginal probability functions, one per feature: Y The measure is defined using log-probabilities to simplify calculations. Moreover, all positions of a modeled view within an image are assumed equally likely, so P(T With these simplifications the measure becomes log P(e log P(H), the prior probability that the object as modeled is present in the image, can be estimated from the proportion of training images used to construct the model. The remaining terms are described using the following notation for random events: ~ the event that model feature j matches image feature k; ~ e j =?, the event that it matches nothing; ~ the event that it matches a feature whose attributes are a; and ~ b, the event that it matches a feature whose position, in model coordinates, is b. There are two cases to consider in estimating the conditional probability, P(e for a model feature j. 1. When j is unmatched, this probability is estimated by considering how often j was found during training. We use a Bayesian estimator, a uniform prior, and the - m and recorded by the model: 2. When j is matched to image feature k, this probability is estimated by considering how often j matched an image feature during training, and how the attributes and position of k compare with those of previously matching features: P( ~ Viewpoint transformation Image coordinates Image feature position pdf Model coordinates Model feature position pdf Image feature position pdf Transformation space Figure 2: Comparison of image and model feature positions. An image feature's position is transformed from image coordinates (left) to model coordinates (right) according to an estimate of the viewpoint transformation. Uncertainty in the positions and the transformation are characterized by Gaussian distributions that are compared in the model coordinate space. P(~e j 6=?) is estimated as in (10). P(~ a estimated using the series of attribute vectors - recorded with model feature j, and a non-parametric density estimator described in (Pope, 1995). Estimation of P( ~ the probability that model feature j will match an image feature at position b k with transformation T , is described in Sect. 5.2. Estimates of the prior probabilities are based, in part, on measurements from a collection of images typical of those in which the object will be sought. From this collection we obtain prior probabilities of encountering various types of features with various attribute values. Prior distributions for feature positions assume a uniform distribution throughout a bounded region of model coordinate space. 5.2 Estimating feature match probability The probability that a model and image feature match depends, in part, on their positions and on the aligning transformation. This dependency is represented by the P( ~ term in (11). To estimate it, we transform the image feature's position into model coor- dinates, and then compare it with the model feature's position (Fig. 2). This comparison considers the uncertainties of the positions and transformation, which are characterized by Gaussian PDFs. Image feature k's position is reported by its feature detector as a Gaussian PDF in xy's image coordinates with mean b xy's k and covariance matrix C xy's k . To allow its transformation into model coordinates, this PDF is re-expressed in xyuv image coordinates using an approximation adequate for small ' and s variances. The approximating PDF has a mean, k , at the same position as b xy's k , and a covariance matrix C xyuv k that aligns the Gaussian envelope radially, away from the [u v] origin: l l and oe 2 l , oe 2 s and oe 2 ' are the variances in image feature position, scale and orientation estimates. T is characterized by a Gaussian PDF over [x t y t u t v t ] vectors, with mean t and covariance t estimated from feature pairings as described in Sect. 5.4. Using it to transform the image feature position from xyuv image to model coordinates again requires an approxima- tion. If we would disregard the uncertainty in T , we would obtain a Gaussian PDF in model coordinates with mean A k t and covariance A t C k A T . Alternatively, disregarding the uncertainty in k's position gives a Gaussian PDF in model coordinates with mean A k t and covariance A k C t A T . With Gaussian PDFs for both feature position and transformation, however, the transformed position's PDF is not of Gaussian form. At best we can approximate it as such, which we do with a mean and covariance given in xyuv coordinates by Model feature j's position is also described by a Gaussian PDF in xyuv model coordinates. Its mean b j and covariance C j are estimated from the series of position vectors - recorded by the model. The desired probability (that j matches k according to their positions and the transforma- tion) is estimated by integrating, over all xyuv model coordinate positions r, the probability that both the transformed image feature is at r and the model feature matches something at r: P( ~ Z r Here ~r j and ~r kt are random variables drawn from the Gaussian distributions N(b It would be costly to evaluate this integral by sampling it at various r, but fortunately the integral can be rewritten as a Gaussian since it is essentially one component in a convolution of two Gaussians: P( ~ where G(x; C) is a Gaussian with zero mean and covariance C. In this form, the desired probability is easily computed. 5.3 Matching procedure Recognition and learning require the ability to find at match between a model graph and an image graph that maximizes the match quality measure. It does not seem possible to find an optimal match through anything less than exhaustive search. Nevertheless, good matches can usually be found quickly by a procedure that combines qualities of both graph matching and iterative alignment. 5.3.1 Probabilistic alignment To choose the initial pairings, possible pairings of high level features are rated according to the contribution each would make to the match quality measure. The pairing hj; ki receives the rating log P(~e This rating favors pairings in which j has a high likelihood of matching, j and k have similar attribute values, and the transformation estimate obtained by aligning j and k has low variance. The maximum over T is easily computed because P(~e in T . Alignments are attempted from these initial pairings in order of decreasing rank. Each alignment begins by estimating a transformation from the initial pairing, and then proceeds by repeatedly identifying additional consistent pairings, adopting the best, and updating the transformation estimate with them until the match quality measure cannot be improved fur- ther. At this stage, pairings are selected according to how each might improve the match quality measure; thus hj; ki receives the rating This favors the same qualities as equation 12 while also favoring pairings that are aligned closely by the estimated transformation. In order for hj; ki to be adopted, it must rate at least as well as the alternative of leaving j unmatched, which receives the rating Significant computation is involved in rating and ranking the pairings needed to extend an alignment. Consequently, pairings are adopted in batches so that this computation need only be done infrequently. Moreover, in the course of an alignment, batch size is increased as the transformation estimate is further refined so that each batch can be made as large as possible. A schedule that seems to work well is to start an alignment with a small batch of pairings (we use five), and to double the batch size with each batch adopted. 5.4 Estimating the aligning transformation From a series of feature pairings, an aligning transformation is estimated by finding the least-squares solution to a system of linear equations. Each pairing hj; ki contributes to the system the equations A k is the matrix representation of image feature k's mean position, the transformation estimate, and b j is model feature j's mean position. U j is the upper triangular square root of j's position covariance (i.e., C weights both sides of the equation so that the residual error ~ e has unit variance. A recursive estimator solves the system, efficiently updating the transformation estimate as pairings are adopted. We use the square root information filter (SRIF) (Bierman, 1977) form of the Kalman filter for its numerical stability, and its efficiency with batched measure- ments. The SRIF works by updating the square root of the information matrix, which is the inverse of the estimate's covariance matrix. The initial square root, R 1 , and state vector, z 1 , are obtained from the first pairing hj; ki by With each subsequent pairing hj; ki, the estimate is updated by triangularizing a matrix composed of the previous estimate and data from the new pairing:6 4 R When needed, the transformation and its covariance are obtained from the triangular R i by back substitution: Verification Once a match has been found between a model graph and an image graph, it must be decided whether the match represents an actual instance of the modeled object in the image. A general approach to this problem would use decision theory to weigh prior expectations, evidence derived from the match, and the consequences of an incorrect decision. However, we will use a simpler approach that only considers the number and type of matching features and the accuracy with which they match. The match quality measure used to guide matching provides one indication of a match's significance. A simple way to accept or reject matches, then, might be to require that this measure exceeds some threshold. However, the measure is unsuitable for this use because its range differs widely among objects according to what high-level features they have. High-level features that represent groupings of low-level ones violate the feature independence assumption; consequently, the match quality measure is biased by an amount that depends on what high-level features are present in the model. Whereas this bias seems to have no adverse effect on the outcome of matching any one model graph, it makes it difficult to establish a single threshold for testing the match quality measure of any model graph. Thus the verification method we present here considers only the lowest-level features of the model graph-those that do not group any other model graph features. When counting paired model features, we weight each one according to its likelihood of being paired, thereby assigning greatest importance to the features that contribute most to the likelihood that the object is present. For model feature j, the likelihood of being paired, is estimated using statistics recorded for feature j. The count of each model feature is also weighted according to how well it is fit by image features. When j is a curve segment, this weighting component is based on the fraction of j matched by nearby image curve segments. The fraction is estimated using a simple approximation: The lengths of image curve segments matching j are totaled, the total length is transformed into model coordinates, and the transformed value is divided by the length of j. With s t denoting the scaling component of the viewpoint transformation T , and s j and s k denoting the lengths of j and k in model and image coordinates, respectively, the fraction of j covered by image curve segments is defined as curve segment. If we were to accept matches that paired a fixed number of model features regardless of model complexity, then with greater model complexity we would have an increased likelihood of accepting incorrect matches. For example, requiring that ten model features be paired may make sense for a model of twenty features, but for a model of a thousand fea- tures, any incorrect match is likely to contain at least that many "accidental" pairings. Thus we have chosen instead to require that some minimum fraction of the elements of C be paired. We define this fraction as A match hE; T i is accepted if Support(E; T ) achieves a certain threshold - . To validate this verification method and to determine a suitable value for - , we have measured the distribution of Support(E; T ) for correct and incorrect matches between various model graphs and their respective training image graphs. The distributions are well separated, with most correct matches achieving most incorrect matches achieving 6 Model learning procedure The learning procedure assembles one or more model graphs from a series of training images showing various views of an object. To do this, it clusters the training images into groups and constructs model graphs generalizing the contents of each group. We shall describe first the clustering procedure, and then the generalization procedure, which the clustering procedure invokes repeatedly. We use X to denote the series of training images for one object. During learning, the object's model M consists of a series of clusters X i ' X , each with an associated model graph - G i . Once learning is complete, only the model graphs must be retained to support recognition. 6.1 Clustering training images An incremental conceptual clustering algorithm is used to create clusters among the training images. Clustering is incremental in that, as each training image is acquired, it is assigned to an existing cluster or used to form a new one. Like other conceptual clustering algorithms, such as COBWEB (Fisher, 1987), the algorithm uses a global measure of over-all clustering quality to guide clustering decisions. This measure is chosen to promote and balance two somewhat-conflicting qualities. On one hand, it favors clusterings that result in simple, concise, and efficient models, while on the other hand, it favors clusterings whose resulting model graphs accurately characterize the training images. The minimum description length principle (Rissanen, 1983) is used to quantify and balance these two qualities. The principle suggests that the learning procedure choose a model that minimizes the number of symbols needed to encode first the model and then the training images. It favors simple models as they can be encoded concisely, and it favors accurate models as they allow the training images to be encoded concisely once the model has been provided. The clustering quality measure to be minimized is defined as L(M)+L(X j M), where L(M) is the number of bits needed to encode the model M, and L(X j M) is the number of bits needed to encode the training images X when M is known. To define L(M) we specify a coding scheme for models that concisely enumerates each of a model's graphs along with its nodes, arcs, attribute vectors and position vectors (see (Pope, 1995) for full details of the coding scheme). Then L(M) is simply the number of bits needed to encode M according to this scheme. To define L(X j M) we draw on the fact that given any probability distribution P(x), there exists a coding scheme, the most efficient possible, that achieves essentially P(x). Recall that the match quality measure is based on an estimate of the probability that a match represents a true occurrence of the modeled object in the image. We use this probability to estimate P(X j - the probability that the appearance represented by image may occur according to the appearance distribution represented by model graph - This probability can be computed for any given image graph X and model graph G i , using the matching procedure (Sect. 5.3) to maximize P(H used to estimate the length of an encoding of X given - E) The L u (X; E) term is the length of an encoding of unmatched features of X , which we define using a simple coding scheme comparable to that used for model graphs. Finally, we define L(X j M) by assuming that for any X 2 X i ' X , the best match between X and any - will be that between X and - (the model graph obtained by generalizing the group containing X). Then the length of the encoding of each X 2 X in terms of the set of model graphs M is the sum of the lengths of the encodings of each in terms of its respective model As each training image is acquired it is assigned to an existing cluster or used to form a new one. Choices among clustering alternatives are made to minimize the resulting L(M)+ evaluating an alternative, each cluster's subset of training images X i is first generalized to form a model graph - G i as described below. 6.2 Generalizing training images Within each cluster, training images are merged to form a single model graph that represents a generalization of those images. An initial model graph is formed from the first training image's graph. That model graph is then matched with each subsequent training image's graph and revised after each match according to the match result. A model feature j that matches an image feature k receives an additional attribute vector a k and position b k for its series - A j and - Unmatched image features are used to extend the model graph, while model features that remain largely unmatched are eventually pruned. After several training images have been processed in this way the model graph nears an equilibrium, containing the most consistent features with representative populations of sample attribute vectors and positions for each. 7 Experimental results In this section we describe several experiments involving a system implemented to test our recognition learning method. This system, called OLIVER, learns to recognize 3-D objects in 2-D intensity images. OLIVER has been implemented within the framework of Vista, a versatile and extensible software environment designed to support computer vision research (Pope and Lowe, 1994). Both OLIVER and Vista are written in C to run on UNIX workstations. The execution times reported here were measured on a Sun SPARCstation 10/51 processor. The focus of this research has been on the model learning and representation, which is independent of the particular features used for matching. To test the approach, we have chosen to use a basic repertoire of edge-based features. While some recent approaches to recognition have been based on image pixel intensities or image derivative magnitudes, the locations of intensity discontinuities may be more robust to illumination and imaging vari- ations. For example, the silhouette boundaries of an object on a cluttered background will have image derivatives of unknown sign and magnitude. The same is true for edges separating surfaces of different orientation under differing directions of illumination. Figure 3: Bunny training images. Images were acquired at 5 ffi intervals over camera elevations of 0 ffi to 25 ffi and azimuths of 0 ffi to 90 ffi . Shown here are three of the 112 images. In the following experiments, the lowest-level features are straight, circular and elliptical edge segments. An edge curve of any shape can be represented by approximating it with a series of primitive segments. Additional higher-level features represent groupings of these, such as junctions, groups of adjacent junctions, pairs of parallel segments, and convex regions. For full details on the derivation of these features, see (Pope, 1995). In the future, this set could be augmented with other features, such as those based on image derivatives, color, or texture, but the current features are suited for a wide range of objects. 7.1 Illustrative experiment The experiment described in this section demonstrates typical performance of the system in learning to recognize a complex object. The test object is a toy bunny shown in figure 3. Training images of the bunny were acquired at 5 ffi increments of camera elevation and azimuth over 25 ffi of elevation and 90 ffi of azimuth. Feature detection, including edge detection, curve segmentation, and grouping, required about seconds of CPU time per training image (we believe that much faster grouping processes are possible, but this was not the focus of the research). Figure 4 depicts some of the features found in one image, which include 4475 edgels, 81 straight lines and 67 circular arcs. During the first phase of clustering, the system divided the training images among 19 clusters. In the second phase, it reassigned two training images that remained the sole members of their clusters, leaving the 17 clusters shown in figure 5. Because this object's appearance varies smoothly with changes in viewpoint across much of the viewing range, it is not surprising that the clusters generally occupy contiguous regions of the viewsphere. When training images were presented to the system in other sequences, the system produced different clusterings than that shown in figure 5. However, although cluster boundaries varied, qualities such as the number, extent, and cohesiveness of the clusters remained largely unaffected. As this is a one-time batch operation, little effort was devoted to optimizing efficiency. Altogether, 19.4 hours of CPU time were required to cluster the training images and to in- (a) (b) (c) (d) Figure 4: Features of a bunny training image. Shown here are selected features found in the 0 training image (right image in figure 3). (a) Edgels. (b) Curve features. (c) L-junction features. (d) Parallel-curve features (depicted by parallel lines), Region features (depicted by rectangles), and Ellipse features (depicted by circles). duce a model graph generalization of each cluster. Figure 5 shows features of the model graph representing cluster C. Ellipses are drawn for certain features to show two standard deviations of location uncertainty. To reduce clutter and to give some indication of feature significance, they are drawn only for those features that were found in a majority of training images. Considerable variation in location uncertainty is evident. Some L-junction features have particularly large uncertainty and, consequently, they will be afforded little importance during matching. Figure 7 reports the results of matching the image graph for test image 1 with each of the bunny model graphs. For this test, match searches were allowed to examine all alignment hypotheses. Typically, there were 10-20 hypotheses examined for each pair of model and image graphs, and about five seconds of CPU time were needed to extend and evaluate each one. The matches reported here are those achieving the highest match quality measure. The model graph generalizing cluster D provides the best match with the image graph (as judged by each match's support measure). This is to be expected as the test image was acquired from a viewpoint surrounded by that cluster's training images. Moreover, other model graphs that match the image graph (although not as well) are all from neighbouring A A J J J A A J J P A A A B A J D D D D M K I O I Elevation Azimuth (a) (b) (c) (d) Figure 5: On the left are the training image clusters. Seventeen clusters, designated A through Q, were formed from the 112 training images. Contours delineate the approximate scope of the view classes associated with some of the clusters. On the right are selected features of the model graph obtained by generalizing the training images assigned to cluster C. Each feature is drawn at its mean location. (a) Curve features. (b) L-junction features. (c) Connected edge features. (d) Parallel curve, Region and Ellipse features. Figure test image 1. Left: Image. Right: Match of bunny model graph D with test image. MODEL MATCH WITH TEST IMAGE 1 MATCH WITH TEST IMAGE 2 GRAPH Correct Quality Pairings Support Correct Quality Pairings Support A 998 164 0.35 862 191 0.34 I 356 91 0.19 879 186 0.35 O 151 72 0.14 187 90 0.23 Figure 7: Each row documents the results of matching a model graph with the two image graphs, those describing bunny test images 1 and 2. Reported for each match are the fol- lowing. Correct: whether the match correctly identified most features of the object visible in the image, as judged by the experimenter. Quality: the match's match quality measure, Pairings: the number of image features paired. Support: the match's support mea- regions of the viewsphere. Image features included in the best match, that with model graph D, are shown in figure 6. For test image 2, shown in figure 8, additional clutter was present. Figure 7 reports the result of matching the image graph of test image 2 with each of the bunny model graphs. This time each match search was limited to 20 alignment hypotheses and about six seconds of CPU time were needed to extend and evaluate each hypothesis. Due to the additional clutter in the image, only model graph D correctly matched the image. This section has demonstrated the system's typical performance in learning models of complex objects, and in using those models to accomplish recognition. Figure 9 shows recognition of an even more complex object with significant occlusion. 7.2 Additional Experiments In this section, some additional experiments are briefly described in order to illustrate certain noteworthy aspects of the system's behaviour. Figure 8: Left: Bunny test image 2 with clutter and occlusion. Right: Match of bunny model graph D with test image 2. Figure 9: Example showing recognition of a complex object with substantial occlusion. Left: One of several training images. Right: Image curve features included in the match. 7.2.1 Effects of feature distribution When some regions of an object are much richer in stable features than others, those regions can dominate the matching processes that underlie learning and recognition. For example, most features of the shoe shown in figure 10 are concentrated near its centre. Moreover, as the shoe rotates about its vertical axis, features near the shoe's centre shift by small amounts while those near its heel and toe undergo much larger changes. Thus, when training images of the shoe are clustered during model learning, the many stable features near the shoe's centre are used to match training images over a large range of rotation, while the few variable features defining the heel and toe are dropped as being unreliable. The result is a model graph, like that shown in figure 11 (left), with relatively few features defining the shoe's extremities. If the dropped features are deemed important, we can encourage the system to retain them in the models it produces by setting a higher standard for acceptable matches. For example, requiring a higher Support measure ensures that matches will include more of Figure 10: Shoe training images. Images were acquired at 6 ffi intervals over camera elevations of 0 ffi to 12 ffi and azimuths of 0 ffi to 60 ffi . Shown here are three of the 33 images. Figure 11: Left: Curve features for a model that generalizes 14 training images (support threshold of 0.5). Right: Model that generalizes 7 training images (support threshold of 0.6). an object's features. Thus, fewer of those features will be judged unreliable and more will be retained by the model. Figure 11 (right) shows a model graph that was produced with a Support threshold of 0.6 rather than the usual value of 0.5; it provides somewhat more accurate representation of the shoe's heel and toe. Figure 12 shows this model graph being used for recognition. 7.2.2 Articulate objects Just as the system will use multiple views to model an object's appearance over a range of viewpoints, it will use additional views to model a flexible or articulate object's appearance over a range of configurations. In general, the number of views needed increases exponen- Figure 12: Shoe recognition example. Left: Test image. Right: Image curve features included in a match to shoe model. sail angleffi elevation, 0 sail angleffi elevation, 0 sail angle Figure 13: Boat training images. Images were acquired at elevations of 0 azimuths of 0 angles of 0 Shown here are three of the images. Figure 14: Boat model graphs. Shown here are curve features of model graphs that have been generalized from two clusters of boat training images. tially with the number of dimensions along which the object's appearance may vary. This could presumably be addressed by a part-based modeling and clustering approach that separated the independent model parts. The toy boat shown in figure 13 has a sail that rotates about the boat's vertical axis. Training images were acquired at camera elevations of 0 of angles of 0 . The system's learning procedure clustered these 120 images to produce 64 model views. Features of two of the model graphs are shown in figure 14. In comparison, only 13 views were needed to cover the same range of viewpoints when the sail angle was kept fixed at 0 ffi . 8 Conclusion We have presented a method of modeling the appearance of objects, of automatically acquiring such models from training images, and of using the models to accomplish recognition. This method can handle complex, real-world objects. In principle, it can be used to recognize any object by its appearance, provided it is given a sufficient range of training images, sufficient storage for model views, and an appropriate repertoire of feature types. The main features of the method are as follows: (a) Objects are modeled in terms of their appearance, rather than shape, to avoid any need to model the image formation process. This allows unusually complex objects to be modeled and recognized efficiently and reliably. (b) Appearance is described using discrete features of various types, ranging widely in scale, complexity, and specificity. This repertoire can be extended considerably, still within the framework of the approach, to accommodate a large variety of objects. (c) An object model represents a probability distribution over possible appearances of the object, assigning high probability to the object's most likely manifestations. Thus, learning an object model from training images amounts to estimating a distribution from a representative sampling of that distribution. (d) A match quality measure provides a principled means of evaluating a match between a model and an image. It combines probabilities that are estimated using distributions recorded by the model. The measure leads naturally to an efficient matching proce- dure, probabilistic alignment, used to accomplish both learning and recognition. (e) The model learning procedure has two components. One component identifies clusters of training images that ought to correspond to distinct model views. It does so by maximizing a measure that, by application of the minimum description length prin- ciple, combines the qualities of model simplicity and accuracy. The second component induces probabilistic generalizations of the images within each cluster. Working together, the two components construct a model by clustering training images, and, within each cluster, generalizing the images to form a model view. 8.1 Topics for further research Modeling a multifarious or highly flexible object with this approach may require an impractically large number of model views. For these objects, a more effective strategy may be first to recognize parts, and then to recognize the whole object as a configuration of those parts. The present method could perhaps be extended to employ this strategy by assigning parts the role of high level features. Speed in both learning and recognition tasks could be greatly improved by the addition of an indexing component, which would examine image features and suggest likely model views for the matching procedure to consider. Existing indexing methods (Beis and Lowe, 1999) could be used, with the attribute vectors of high-level features serving as index keys. Of course, more efficient methods for feature detection would also be important. Extending the feature repertoire would allow the method to work more effectively with a broader class of objects. It would be useful to have features representing additional groupings of intensity edges, such as symmetric arrangements and repeated patterns, and features representing local image regions with color or texture properties. Some challenging issues remain regarding how to organize a large collection of acquired models for greater efficiency. Savings in both storage and recognition time could be achieved by identifying parts or patterns common to several objects, factoring those parts out of their respective models, and recognizing the parts individually prior to recognizing their aggre- gates. Associating new feature types with some of the common parts and patterns would provide a means of automatically extending the feature repertoire and adapting it to the objects encountered during training. Furthermore, the same techniques of identifying and abstracting parts could be used to decompose flexible objects into simpler components, allowing those objects to be modeled with fewer views. Acknowledgments The authors would like to thank Jim Little, Bob Woodham, and Alan Mackworth for their ongoing comments on this research. 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object recognition;model indexing;appearance representation;visual learning;model-based vision;clustering

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On the Fourier Properties of Discontinuous Motion.

Retinal image motion and optical flow as its approximation are fundamental concepts in the field of vision, perceptual and computational. However, the computation of optical flow remains a challenging problem as image motion includes discontinuities and multiple values mostly due to scene geometry, surface translucency and various photometric effects such as reflectance. In this contribution, we analyze image motion in the frequency space with respect to motion discontinuities and translucence. We derive the frequency structure of motion discontinuities due to occlusion and we demonstrate its various geometrical properties. The aperture problem is investigated and we show that the information content of an occlusion almost always disambiguates the velocity of an occluding signal suffering from the aperture problem. In addition, the theoretical framework can describe the exact frequency structure of Non-Fourier motion and bridges the gap between Non-Fourier visual phenomena and their understanding in the frequency domain.

Introduction A fundamental problem in processing sequences of images is the computation of optical ow, an approximation to image motion dened as the projection of velocities of 3D surface points onto the imaging plane of a visual sensor. The importance of motion in visual processing cannot be under- stated: in particular, approximations to image motion may be used to estimate 3D scene properties and motion parameters from a moving visual sensor [21, 30, 31, 42, 51, 50, 1, 5, 38, 22, 54, 56, 34, 20, 16, 23], to perform motion segmentation [7, 40, 45, 36, 47, 14, 25, 8, 2, 46, 15], to compute the focus of expansion and time-to-collision [44, 41, 48, 24, 49, 9], to perform motion-compensated image encoding [10, 13, 35, 37, 39, 55], to compute stereo disparity [3, 12, 26, 28], to measure blood ow and heart-wall motion in medical imagery [43], and, recently, to measure minute amounts of growth in corn seedlings [6, 29]. 1.1. Organization of Paper This contribution addresses the problem of multiple image motions arising from occlusion and translucency phenomena. We present a theoretical framework for discontinuous optical ow in the Fourier domain. The concept of image velocity as a geometric function is described in Section 1. S. S. Beauchemin and J. L. Barron Section 2 is an analysis of occlusion in Fourier space with a constant model of velocity. Our approach focuses on the frequency structure of occluding surfaces and the theoretical results are constructed incrementally. For instance, a simple model of velocity is used to develop the structure of occlusion with sinusoidal signals which are then generalized to arbitrary signals. These theoretical results demonstrate that occlusion may be dierentiated from translucency and the motions associated with both the occluding and occluded surfaces can be discriminated. Section 3 is an investigation of the aperture problem and degenerate 1 signals, as they appear in the theoretical framework. For example, it is shown that the full velocity of a degenerate signal is almost always computable at the occlusion. Section 4 is a study of related issues such as translucency phenomena, Non-Fourier motion, generalized occlusion boundaries and phase shifts. Numerical experiments supporting the framework are presented. Results obtained with sets of sinusoidal signals created synthetically are compared with their corresponding theoretical predictions. Section 5 summarizes our results. 1.2. Contribution The motivation for the theoretical framework emanates from the observation that occlusion and translucency in the context of computing optical ow constitute di-cult challenges and threatens its precise computation. The theoretical results cast light on the exact structure of occlusion and translucency in the frequency domain. The results are essentially theoretical and stated in the form of Theorems and Corollaries. Relevant numerical experiments which support the theoretical results are presented. In addition, this contribution bridges what is seen as an important gap between Non-Fourier models of visual stimuli and optical ow methods in Computer Vision. In fact, Non-Fourier visual stimuli, to which belong translucency and occlusion eects, have been studied mainly with respect to the motion percept these stimuli elicit among human subjects [11, 52, 53]. However, more recently, it has been conjectured that a viable computational analysis of Non-Fourier motion could be carried out with Fourier analysis, since many Non-Fourier stimuli turn out to have simple frequency characterizations [19]. The results presented herein extend the concept of Non-Fourier stimuli such as occlusion and translucency from being not at all explained by its Fourier characteristics to the establishment of exact frequency models of visual stimuli exhibiting occlusions and translucencies. As a rst attempt to understand occlusion, the simplest set of controllable parameters were used, such as the structure of occlusion boundaries and the number of distinct frequencies for representing the occluding and occluded surfaces. A constant model of velocity was also used and no signal deformations (such as those created by perspective projection) were permitted. These preliminary results are extended to image signals composed of an arbitrary number of discrete frequencies. Dirichlet conditions are hypothesized for each signal thus allowing to expand them as complex exponential series. The potential use of the information-content of an occlusion boundary is outlined. Occluding boundaries contain a wealth of information that is not exploited by conventional optical ow frame- works, due to a theoretical void. It is shown that a degenerate occluding signal exhibiting a linear spectrum is supplemented by the linear orientation of its occluding boundary. These two spectra almost always yield the full velocity of an occluding signal suering from the aperture problem. The structure of occlusion when both signals are degenerate is also shown. It is demonstrated that this particular case collapses to a one-dimensional structure. The Corollaries show that additive translucency phenomena may be understood as a special case of the theoretical framework. In addition, the velocities associated with both the occluding and occluded signals may be identied as such, without the need of scenic information such as depth. 1.3. Image Motion Image motion is expressed in terms of the 3D motion parameters of the visual sensor and the 3D environmental points of the scene: let P (X; Y; Z) be an environmental point, x y z be the visu- On the Fourier Properties of Discontinuous Visual Motion 3 Y Z y x Image plane Environmental surface z y y x x Fig. 1. The geometry of the visual sensor. x y z are the instantaneous rotation and translation of the visual sensor. p(t) is the perspective projection of P(t) onto the imaging surface. al sensor's respective instantaneous rate of change in rotation and translation and P^ z the perspective projection of P onto the imaging surface (the focal length of the sensor is assumed to be 1), z is a normalized vector along the line-of- sight axis Z. The setup is shown in Figure 1. The instantaneous 3D velocity of P is given by P: (1) The relationship between the 3D motion parameters and 2D velocity that results from the projection of V onto the image plane can be obtained by temporally dierentiating p: _ Z Z _ Y Z Y _ Z Using x z Y; T y z X x Y y X) for substitution in (2), one obtains the image velocity equation [31]: xy y z x xy y x z Hence, image motion is a purely geometric quantity and, consequently, for optical ow to be exactly image motion, a number of conditions have to be satised. These are: a) uniform illumina- surface re ectance and c) pure translation parallel to the image plane. Realisti- cally, these conditions are never entirely satised in scenery. Instead, it is assumed that these conditions hold locally in the scene and therefore locally on the image plane. The degree to which these conditions are satised partly determines the accuracy with which optical ow approximates image motion. 1.4. Multiple Motions Given an arbitrary environment and a moving visual sensor, the motion eld generated onto the imaging plane by a 3D scene within the visual eld is represented as function (3) of the motion parameters of the visual sensor. Discontinuities in image motion are then introduced in (3) whenever the depth Z is other than single-valued and dierentiable 2 . The occurrence of occlusion causes the depth function to exhibit a discontinuity, whereas translucency leads to a multiple-valued depth function. 4 S. S. Beauchemin and J. L. Barron 1.5. Models of Optical Flow Generally, the optical ow function may be expressed as a polynomial in some local coordinate system of the image space of the visual sensor. It is assumed that the center of the neighborhood coincides with the origin of the local coordinate system. In this case, we may write the Taylor series expansion of a i th velocity about the origin as: r x;t=0 However, we simply adopt in what follows Fleet and Jepson's [18] constant model of optical ow denoted as: where a i is now the velocity vector. Hence, a 2D intensity prole I 0 translating with velocity yields the following spatiotemporal image intensity We use a negative translational rate in (5) without loss of generality and for mere mathematical convenience. 1.6. Signal Translation in the Frequency Domain Consider a signal I i (x) translating at a constant t). For this signal, the Fourier transform of the optical ow constraint equation is obtained with the dierentiation property as: F where i is the imaginary number, ^ I i (k) is the Fourier transform of I i (x) and -(k T a a Dirac delta function. Expression (7) yields a constraint on velocity. Simi- larly, the Fourier transform of a translating image signal I i (x; t) is obtained with the shift property as: Z Z I Z Z I e i!t dt which also yields the constraint k T a Hence, (7) and (8) demonstrate that the frequency analysis of image motion is in accordance with the motion constraint equation [18]. It is also observed that k T a in the frequency domain, an oriented plane passing through the origin, with normal vector a i descriptive of full velocity, onto which the Fourier spectrum of I i (x) lies. 1.7. Related Literature Traditionally, motion perception has been equated with orientation of power in the frequency domain. The many optical ow methods use what Chubb and Sperling term the Motion-From-Fourier- Components (MFFC) principle [11] in which the orientation of the plane or line through the origin of the frequency space that contains most of the spectral power gives the rate of image translation. The MFFC principle states that for a moving stimulus, its Fourier transform has substantial power over some regions of the frequency domain whose points spatiotemporally correspond to sinusoidal gratings with drift direction consonant with the perceived motion [11]. In addition, current models of human perception involve some frequency analysis of the imagery, such as band-pass l- tering and similar processes. However, some classes of moving stimuli which elicit a strong percept in subjects fail to show a coherent spatiotemporal frequency distribution of their power and cannot be understood in terms of the MFFC principle. Examples include drift-balanced visual stimuli [11], Fourier and Non-Fourier plaid superpositions [52], amplitude envelopes, sinusoidal beats and various multiplicative phenomena [19]. By drift-balanced it is meant that a visual stimulus with two (leftward and rightward, for example) or more dierent motions shows identical contents of Fourier power for each motion and therefore, according to the MFFC principle, should not elicit a coherent motion percept. However, some classes of drift-balanced stimuli dened by Chubb and Sperling do elicit strong coherent motion percept- On the Fourier Properties of Discontinuous Visual Motion 5 s, contrary to the predictions of the usual MFFC model. Sources of Non-Fourier motion also include the motion of texture boundaries and the motion of motion boundaries. For instance, transparency as considered by Fleet and Langley [19] is an example of Non-Fourier motion, as transparency causes the relative scattering of Fourier components away from the spectrum of the moving stimuli. In addition, occlusion, modeled as in (13), is another example of Non-Fourier motion which is closely related to the Theta motion stimuli of Zanker [53], where the occlusion window moves independently from both the foreground and the background, thus involving three independent velocities. It has been observed by Fleet and Langley that many Non-Fourier motion stimuli have simple characterizations in the frequency domain, namely power distributions located along lines or planes which do not contain the origin of the frequency space, as required by the MFFC idealization [19]. Occlusion and translucency being among those Non-Fourier visual stimuli, we develop their exact frequency representations, state their properties with respect to image motion (or optical ow), consider the aperture problem and include additive translucency phenomena within the theoretical framework. 1.8. Methodology To analyze the frequency structure of image signals while preserving representations that are as general as possible, an eort is made to only pose those hypotheses that would preserve the generality of the analysis to follow. We describe the assumptions and the proof techniques with which the theoretical results were obtained. Image Signals The geometry of visual scenes under perspective projection generally yields complex image signals. Conceptually, assumptions concerning scene structure should not be made, as they constrain the geometry of observable scenes. In addition, any measured physical signal, such as image intensi- ties, satises Dirichlet conditions. Such signals admit a nite number of nite discon- tinuities, are absolutely integrable and may be expanded into complex exponential series. Dirichlet conditions constitute the sum of assumptions made on image signals. Velocity On a local basis, constant models of signal translation may be adequate to describe velocity. However, linear models admit an increased number of deformations, such as signal dilation. Hence, the extent used for signal analysis may be larger with linear models. We considered a constant model of velocity, leaving deformations of higher order for further analysis. Occluding Boundaries Object frontiers and their projection onto the imaging plane are typically unconstrained in shape and are dicult to model on a large spatial scale. Simpler, local models appear to be more appropriate. The framework includes occlusion boundaries as locally straight edges, represented with step functions. This hypothesis only approximates reality and limits the analysis to local image regions. However, we outline in which way this hypothesis can be relaxed to include occlusion boundaries of any shape. Proof Techniques The Theorems and their Corollaries established in this analysis emanate from a general approach to modeling visual scenes exhibiting occlusion discontinuities or translucency. An equation which describes the spatio-temporal pattern of the superposition of a background and an occluding signal is established [17], in which a characteristic function describing the position of an occluding signal within the imaging space of the visual sensor is dened: within the occluding signal and two image signals I 1 (x) and I 2 (x), corresponding to the occluding and occluded signals respectively, are dened to form the over-all signal pattern: velocity. Note that the characteristic function describing the object has the same velocity as its corresponding 6 S. S. Beauchemin and J. L. Barron intensity pattern I 1 (x). In (10) are inserted the hypotheses made on its various components and the structure of occlusion in the frequency domain is developed. That is to say, signal structures are expanded into complex exponential series, such as: I n= ~c in e ix T Nk where I i (x) is the i th intensity pattern, c in are complex coe-cients, k i are fundamental are integers and I . Occlusion boundaries become locally straight edges, represented with step functions such as: where n 1 is a vector normal to the tangent of the occluding boundary. In addition, degenerate image signals under occlusion are inves- tigated, thus describing the aperture problem in the context of the framework. Whenever technically possible, the theoretical results were compared with numerical experiments using Fast Fourier Transforms operating on synthetically generated image sequences. Relevance of Fourier Analysis Many algorithms operating in the Fourier domain for which a claim of multiple motions capability is made have been developed [27]. However, this is performed without a complete knowledge of the frequency structure of occlusion phe- nomena. In addition, Non-Fourier spectra, including occlusion and translucency eects have been conjectured to have mathematically simple characterizations in Fourier space [19]. Consequently, the use of Fourier analysis as a local tool is justied as long as one realizes that it constitutes a global idealization of local phenomena. In that sense, Fourier analysis is used as a local tool whenever Gabor lters, wavelets or local Discrete Fourier Transforms are employed for signal analysis. Experimental Technique Given the theoretical nature of this contribution, the purpose of the numerical experiments is to verify the validity of the theoretical results. In order to accomplish this, the frequency content of the image signals used in the experiments must be entirely known to the experimenter, thus forbidding the use of natural image sequences. In addition, image signals with single frequency components are used in order to facilitate the interpretation of experiments involving 3D Fast Fourier transforms. The use of more complex signals impedes a careful examination of the numerical results and do not extend the understanding of the phenomena under study in any particular way. 2. Spectral Structure of Occlusion The analysis begins with the consideration of a simple case of occlusion consisting of two translating sinusoidal signals. These preliminary results are then generalized to arbitrary signals and the aperture problem is examined. 2.1. Sinusoidal Image Signals The case in which two sinusoidals play the role of the object and the background is rst considered. Let I i (x) be an image signal translating with velocity Its Fourier transform is Let I 1 (x) be occluding another image signal I 2 (x), with respective velocities v 1 (x; t) and v 2 (x; t). The resulting occlusion scene can then be expressed as: where U(x) is (12). The Fourier transform of (13) is: U(k) is the Fourier transform of a step function U(x) written as On the Fourier Properties of Discontinuous Visual Motion 7 THEOREM 1. Let I 1 (x) and I 2 (x) be cosine functions with respective angular frequencies k T t)). The frequency spectrum of the occlusion is: a) where a = a 1 a 2 and n 1 is a normal vector perpendicular to the occluding boundary. with n 1 as a vector normal to the occlusion boundary 1 as its negative reciprocal ( n y Theorem 1 is derived to examine occlusion with the simplest set of parameters, such as the form of occlusion boundaries, the number of distinct frequencies required to represent both the occluding and occluded image signals, and a constant model of velocity. Even with this constrained domain of derivation, a number of fundamental observations are made, such as: the occlusion in frequency space is formed of the Fourier transform of a step function convolved with every existing frequency of both the occluding and occluded sinusoidal signals and, the power content of the distortion term is entirely imaginary, forming lines of decreasing power which do not contain the origin, around the frequencies of both the occluding and occluded signals. Their orientation is parallel to the spectrum of the occluding signal, and the detection of their orientation allows to identify the occluding velocity, leaving the occluded velocity to be interpreted as such. We performed a series of experiments to graphically demonstrate the composition of a simple occlusion scene. To simplify the interpretation of the experiments, we used 1D sinusoidal signals composed of single frequencies. In addition, the signals are Gaussian-windowed in order to avoid the Gibbs phenomenon when computing their Fast Fourier Transforms (FFTs). Figure 2a), b) and c) show the components of a simple occlusion scene, pictured in 2d). Figure 2a) is the occluding signal with spatial frequency 2 and velocity 1:0, such that I 1 (x; and in 2b) is the occluded signal with spatial frequency8 and velocity 1:0, yielding I 2 (x; The occluding boundary in Figure 2c) is a 1D step function, written as and translates with a velocity identical to that of I 1 . The resulting occlusion scene in Figure 2d) is constructed with the following 1D version of (13): I where I 1 is (17), I 2 is (18) and is (19). Figures 2e) through h) show the amplitude spectra of gures 2a) through d) respectively, where it is easily observed the the spectrum of the step function (19) is convolved with each frequency of both sinusoidals. Further, Theorem 1 predicts Fourier spectra such as 2h) in their entirety as is demonstrated by the experiments in section 2.3. 2.2. Generalized Image Signals For this analysis to gain generality, we need to nd a suitable set of mathematical functions to represent physical quantities such as image signals that lend themselves to the analysis to follow and which do not impose unnecessary hypotheses on the structure of those signals. 8 S. S. Beauchemin and J. L. Barron d) Amplitude Spectrum of Sinusoid 1 Amplitude Spectrum of Sinusoid 2 Amplitude Spectrum of Step Function Amplitude Spectrum of Occlusion Fig. 2. (top): The composition of a simple 1D occlusion scene. a) The occluding sinusoidal signal with frequency 2and velocity 1:0. b) The occluded sinusoidal signal with frequency 2and velocity 1:0. c) The translating step function used to create the occlusion scene. d) The occlusion as a combination of a), b) and c). (center): Image plots of amplitude spectra and (bottom): amplitude spectra as 3D graphs. For this purpose we hypothesize that image signals satisfy Dirichlet conditions in the sense that for any interval x 1 x x 2 , the function f(x) representing the signal must be single-valued, have a nite number of maxima and minima and a - nite number of nite discontinuities. Finally, f(x) should be absolutely integrable in such a way that, within the interval, we obtain Z x2 In addition, any function representing a physical quantity satises Dirichlet conditions. Hence, those conditions can be assumed for visual signals without loss of generality and, in this context, the complex exponential series expansion, or Fourier series 1 converges uniformly to f(x). Theorem 2 generalizes Theorem 1 from sinusoidal to arbitrary signals. Theorems 1 and 2 introduce the approximation of occluding boundaries with step functions and, as surfaces of any shape may be imaged, the forms of their boundaries are typically unconstrained. On a local basis, however, as long as the spatial extent of analy- On the Fourier Properties of Discontinuous Visual Motion 9 THEOREM 2. Let I 1 (x) and I 2 (x) be 2D functions satisfying Dirichlet conditions such that they may be expressed as complex exponential series expansions: I 1 are integers, x are spatial coordinates, k are fundamental frequencies and c 1n and c 2n are complex coe-cients. Let I 1 (x; I and the occluding boundary be represented by: where n 1 is a vector normal to the occluding boundary. The frequency spectrum of the occlusion is: ~ where a = a 1 a 2 . sis remains su-ciently small, the approximation of occluding boundaries as straight-edged lines is su-cient and greatly simplies the derivation of the results. Also for simplicity, a constant model of velocity is adopted, which is thought of as a valid local approximation of reality [4, 32]. How- ever, the constraint on the shape of the occluding boundary may be removed while preserving the validity of most of the theoretical results, as we later demonstrate. As expected, the sum of properties identied in Theorem 1 hold for Theorem 2. For instance, it is found that the Fourier spectrum of the occluding boundary is convolved with every existing frequency of both the occluding and occluded signals in a manner consonant with its velocity. That is to say, its spectral orientation is descriptive of the motion of the occluding signal. Hence we state the following corollary: COROLLARY 1. Under an occlusion phe- nomenon, the velocities of the occluding and occluded signals can always be identied as such. Under occlusion, the spectral orientation of the occluding boundary is parallel to the plane descriptive of the occluding velocity and detecting the spectral orientation of the boundary amounts to identify the occluding velocity, leaving the occluded velocity to be considered as such. Figure 3 demonstrates the composition of a simple 2D occlusion scene and the Fourier spectra of its components. Figure 3a) is the occluding signal with spatial frequency ( 2 1:0; 1:0) such that I 1 (x; and Figure 3b) is the occluded signal with spatial frequency 8 ) and velocity (1:0; 1:0), yielding I 2 (x; S. S. Beauchemin and J. L. Barron y d) Fig. 3. (top): The composition of a simple 2D occlusion scene. a) The occluding sinusoidal signal with frequency ( 2; 2) and velocity ( 1:0; 1:0). b) The occluded sinusoidal signal with frequency ( 2; 2) and velocity (1:0; 1:0). c) The step function used to create the occlusion scene with normal vector ( d) The occlusion as a combination of a), b) and c). (bottom) e) through h): Image plots of corresponding amplitude spectra. The occluding boundary in Figure 3c) is a 2D step function identical to (12) and translates with a velocity which equals that of I 1 , the occluding signal. The resulting occlusion scene in Figure 3d) is constructed with (13). Figures 3e) through show the 3D amplitude spectra of Figures 3a) through d), respectively. In the experiments with 2D signals depicted in Figure 4, the spatial frequencies of the occluding and occluded signals are k T Only the velocities and the orientation of the occlusion boundary vary. The velocities of the occluding and occluded signals and the occlusion boundary normal vectors, from left to right in Figure 3, are a) a T and n T p2 ); c) a T and n T As per Theorem 1, the spectral extrema located at the spatiotemporal frequencies of both signals and t the constraint planes k T a The oblique spectra intersecting the peaks are the convolutions of the spectrum of the step function with the frequencies of both signals and t lines described by the intersection of planes These spectra are parallel to the constraint plane of the occluding signal and are consonant with its velocity. Theorem 2 is the generalization of Theorem 1 from sinusoidal to arbitrary signals and its geometric interpretation is similar. For instance, frequencies t the constraint planes of the occluding and occluded signals, dened as k T a In the distortion term, the Dirac - function with arguments (k 1 and k T a 1 +! a T Nk 2 represent a set of lines parallel to the constraint plane of the occluding signal and, for every discrete frequency exhibited by both signals, there is a frequency spectrum tting the lines given by the intersection of planes k T a On the Fourier Properties of Discontinuous Visual Motion 11 y d) @ @ @ a 1 a 2 @ @ @ a 2 a 1 a 1 a 2 @ @ @ a 1 a 2 Fig. 4. Four cases of predicted and computed Fourier spectra of occlusion scenes. In all cases the frequencies of the occluding and the occluded signals are ( 2; 2) and ( 2; 2). (top): a) Occluding and occluded velocities a and a Occluding and occluded velocities a Occluding and occluded velocities boundary normal (1:0; 0). d) Occluding and occluded velocities Computed FFTs of corresponding occlusion scenes. (bottom) i) through l): Fourier spectra predicted by theoretical results. and The magnitudes of these spectra are determined by their corresponding scaling functions c 1n [(k Theorem 2 reveals useful constraint planes, as the power spectra of both signals peak within planes k T a and the constraint planes arising from the distortion are parallel to the spectrum of the occluding signal I 1 (x; t). 3. The Aperture Problem: Degenerate Cases In the Fourier domain, the power spectrum of a degenerate signal is concentrated along a linear rather than a planar structure. To see this, consider a 1D signal moving with a constant model of velocity in a 2D space, in the direction of the gradient normal n i and with speed s S. S. Beauchemin and J. L. Barron The Fourier transform of this signal is given by i is the negative reciprocal of n i . The their intersection forms a linear constraint onto which the spectrum of the degenerate signal re- sides. Therefore, the planar orientation describing full velocity is undetermined. However, the presence of an occlusion boundary disambiguates the measurement of a degenerate occluding signal in most cases as a straight-edged occlusion boundary provides one constraint on normal velocity and so does its corresponding degenerate occluding sig- nal. Since these structures have an identical full velocity, these constraints should be consistent with it, allowing to form a system of equations to obtain full velocity. For instance, consider the Fourier transform of a translating occluding degenerate signal expressed as its complex exponential series expansion: Z I 1 c in -(k nk 1 is the normal of the signal, s 1 is its speed is the fundamental frequency. Ad- ditionally, consider the Fourier transform of the occluding boundary with normal vector n 2 and speed The convolution of (4) and (29) yields the following spectrum: c 1n -(k nk 1 c 1n Expression (30) allows to derive two directional vectors, tting the spectra of the degenerate occluding signal and boundary respectively, which are d T cross product yields a vector a 1 normal to the planar structure containing both spectra, which is the full velocity of the degenerate occluding sig- nal. The constraints on normal velocities form the following system of equations a T a T and its solution, obtained by dividing d 1 d 2 with its third component, is which is full velocity when a constant model is used. This system has a unique solution if and only if n 1 s 2 and (31) has no unique solution. Thus, we state the following Theorem: THEOREM 3. The full velocity of a degenerate occluding signal is obtainable from the structure of the Fourier spectrum if and only if its normal is dierent from the normal of the occlusion boundary. We performed experiments with degenerate signals as shown in Figure 5. An occluding degenerate sinusoidal pattern with spatial orientation translating with normal velocity depicted in Figure 5a). The pattern was generated according to I 1 (x; As can be seen from its Fourier transform 5e), the frequency content is composed of two - functions from which only a normal velocity estimate can be obtained by computing the orientation of the line passing through the spectral peaks and the origin of the frequency space. Figure 5b) shows the occluding signal and the occlusion edge combination. The normal vector to the edge is 1:0). The Fourier transform is shown in 5f), where the spectrum of the edge is convolved with the peaks of the signal. In this case, the full velocity of the degenerate signal is obtained by computing the normal vector to the plane containing the entire spectrum On the Fourier Properties of Discontinuous Visual Motion 13 y d) Fig. 5. Cases of degenerate occluding signals. (top): a) Occluding signal with normal Occluding signal and boundary with normal Occluded signal with normal d) Complete occlusion scene. (bottom) e) through h): Corresponding frequency spectra. and the origin of the frequency space. Figure 5c) shows the occluded signal with spatial orientation translating with normal velocity This pattern was generated according to I 2 (x; and its frequency content appears in 5g). The complete occlusion scene is shown in 5d) and the corresponding frequency content is depicted in 5h). To disambiguate the normal velocity of the occluding signal, it is rst necessary to identify the occluding velocity. This is accomplished by nding a line that is parallel to the spectral orientation of the Fourier transform of the occluding edge and that also contains the frequency content of one signal. In this case, this signal is said to be occluding, and the normal to the plane containing its frequency spectrum, including the spectrum of the occluding edge convolved with its discrete fre- quencies, yields a full velocity measurement. 4. Related Considerations In this section we consider the relationship between additive translucency and the theoretical framework, the eects of occluding edges away from the origin of the spatiotemporal domain, occluding boundaries of various shapes and the relevancy of the theoretical model with respect to Non-Fourier motions such as Zanker's Theta motions [53]. 4.1. Translucency Transmission of light through translucent material may cause multiple motions to arise within an image region. Generally, this eect is depicted on the image plane as function of the density of the translucent material [17]. Under the local assumption of spatially constant f( 1 ) with translucency factor ', (35) is reformulated as a weighted super- 14 S. S. Beauchemin and J. L. Barron x a) b) c) d) Fig. 6. The composition of an additive transparency scene. a): First sinusoidal signal with frequency k sinusoidal signal with frequency k Transparency created with the superposition of rst and second sinusoidal signal. d): Frequency spectrum of transparency. position of intensity proles, written as where I 1 (v 1 (x; t)) is the intensity prole of the translucent material and I 2 (v 2 (x; t)) is the intensity prole of the background. With I 1 (v 1 (x; t)) and I 2 (v 2 (x; t)) satisfying Dirichlet conditions, the frequency spectrum of (36) is written as: Hence, with respect to its frequency structure, translucency may be reduced to a special case of occlusion for which the distortion terms vanish. Figure 6 shows the Fourier transform of an additive translucency composed of two sinusoidals. 4.2. Phase Shifts For reasons of simplicity and clarity, in each Theorem and numerical result, the occluding boundary contained the origin of the coordinate system. We generalize this by describing the occlusion boundary as where y 0 is the y-axis intercept. The Fourier spectrum of such a boundary includes a phase shift and is written as: Equation (39) can be further simplied as: The Fourier spectrum of the boundary is to be convolved with the complex exponential series expansions of the occluding and occluded signals and subsequently with the Fourier transform of the Gaussian window. In the case of the occluding signal, the convolution with the the shifted occlusion boundary can be written as: and, similarly for the occluded signal: These convolutions are combined together as before to obtain the Fourier spectrum of occlusion On the Fourier Properties of Discontinuous Visual Motion 15 d) Fig. 7. Phase shifts from occluding edge. (top): a) y through h): Corresponding frequency spectra. The relative magnitude between the occluding and occluded signals depend on their respective visible areas under the Gaussian envelope. For instance, The frequencies of the occluding signal dominate over those of the occluded signal in e), and vice versa in h). with an occluding boundary not containing the origin of the space. We conducted experiments with 1D image signals and shifted the occlusion point with dieren- t values of y 0 in (38). As observed in Figure 7, these phase shifts do not alter the structure of occlusion in frequency space. The variations in the amplitude spectra are due to the Gaussian windowing of the occlusion scene. For instance, the frequency peaks of the occluding signal in Figure 7e) show more power than those of the occluded signal, owing to the fact that the signal is dominant within the Gaussian window. The contrary is observed when the occluded signal occupies most of the window, as shown in Figure 7h). 4.3. Generalized Occluding Boundaries Typically, occlusion boundaries are unconstrained in shape, yielding a variety of occluding situations. Under the hypothesis that the motion of the occluding boundary is rigid on the image plane, we can derive the frequency structure of such occlusion events. For instance, consider a generalized occlusion boundary represented by the characteristic function (x) in the coordinates of the image plane and the Fourier transforms of the complex exponential series expansions of both the occluding and occluded signals I 1 and I 2 . Substituting these terms into (13) yields the following Fourier spectrum from which it is easily observed that the spectrum of the occluding boundary is repeated at every non-zero frequency of both signals. The spectrum S. S. Beauchemin and J. L. Barron y d) Fig. 8. Generalized occluding boundaries. a), b) and c): Images from a sequence in which the occluding pattern moves with velocity a T= ( 1:0; 1:0). Spatial frequency of the sinusoidal texture within the circular boundary is k T= ( 2; 2). d): The frequency spectrum of the sequence, where the plane contains the spectrum of the boundary convolved with the frequency of the texture. occupies a plane descriptive of full velocity and can be used to perform such measurements. Figure 8 shows an experiment where the occluding signal is within a circular occlusion boundary. The signal and boundary are moving at a constant velocity a T and the occluded signal is a background of constant intensity. Figures 8a) through c) show the motion of the occluding region while Figure 8d) is the frequency spectrum of the sequence, from which we observe the spectrum of the circular boundary and the peaks representing the frequencies of the occluding sinusoidal texture are conned to a planar region fully descriptive of the image motion. 4.4. Non-Fourier Motion Non-Fourier motion is characterized by its inability to be explained by the MFFC principle. In other terms, such motions generate power distributions that are inconsistent with translational mo- tion. Sources of Non-Fourier motion include such phenomena as translucency and occlusion and, in particular, Zanker's Theta motion stimuli involving occlusion [53]. This category of motion is described by an occlusion window that translates with a velocity that is uncorrelated with the velocities of the occluding and occluded signals. For 1D image signals, such an occlusion scene can be expressed as: As Zanker and Fleet [53, 19], we model the occlusion window with a rectangle function in the spatial coordinate as x x0 x x0 x x0 Such a function has a non-zero value in the interval 2 ] and zero otherwise. We then write the Fourier transform of the occlusion scene as: KX KX c 2n -(k nk and the phase shift from x 0 in (45) . The spectra -(kv 3 and are consonant with the motion of the occluding window and represent a On the Fourier Properties of Discontinuous Visual Motion 17 x x a) b) c) d) Fig. 9. Examples of Theta Motion. a): Velocities of occlusion window, occluding and occluded signals are v Frequency spectrum of a). c): Velocities of occlusion window, occluding and occluded signals are Frequency spectrum of c). case of Non-Fourier motion, as they do not contain the origin. We performed two experiments with Theta motions as pictured in Figure 9. It is easily observed that the spectrum of the sinc function is convolved with each frequency of both signals and that its orientation is descriptive of the velocity of the win- dow. As expected, the visible peaks represent the motions of both signals in the MFFC sense. 5. Conclusion Retinal image motion and optical ow as its approximation are fundamental concepts in the eld of vision. The computation of optical ow is a challenging problem as image motion includes discontinuities and multiple values mostly due to scene geometry, surface translucency and various photometric eects such as surface re ectance. In this contribution, we analyzed image motion in frequency space with respect to motion discontinuities and surface translucence. The motivation for such a study emanated from the observation that the frequency structure of occlusion, translucency and Non-Fourier motion in frequency space was not known. The results cast light on the exact structure of occlusion, translucency, Theta mo- tion, the aperture problem and signal degeneracy for a constant model of image motion in the frequency domain with related geometrical properties Appendix Proof Method of Theorem 2 The Fourier transform of the complex exponential series expansion of a 2D signal is: I Z ~X n= ~c in e ix T Nk i e ik T x dx n= ~c in -(k Nk i ) and the Fourier transform of 2D step function under constant velocity is: Z where n i is a vector normal to the occlusion boundary. Introducing (A1) and (A2) into the Fourier transform of (13) under constant velocity and solving the convolutions leads to Theorem 2. S. S. Beauchemin and J. L. Barron Notes 1. Signals that are termed as degenerate have a spatially constant intensity gradient or, in other words, a unique texture orientation. This phenomenon is generally referred to as the aperture problem which arises when the Fourier spectrum of I i (x) is concentrated on a line rather than on a plane [18, 33]. Spatiotemporally, this depicts the situation in which I i (x; t) exhibits a single orientation. In this case, one only obtains the speed and direction of motion normal to the orientation, noted as v?i (x; t). If many normal velocities are found in a single neighborhood, their respective spectra t the plane 0 from which full velocity may be obtained. 2. This assertion assumes dierentiable sensor motion. --R Determining three-dimensional motion and structure from optical ow generated by several moving objects A fast obstacle detection method based on optical ow. Disparity analysis of images. Performance of optical ow techniques. The feasibility of motion and structure from noisy time-varying image velocity information Optic ow to measure minute increments in plant growth. A model for the detection of motion over time. Motion segmentation and qualitative dynamic scene analysis from an image sequence. A split-merge parallel block-matching algorithm for video displacement estimation Stereo correspondence from optical ow. The sampling and reconstruction of time-varying imagery with application in video systems On the detection of motion and the computation of optical ow. Obstacle detection by evaluation of optical ow Measurement of Image Velocity. Computation of component image velocity from local phase information. Computational analysis of non-fourier motion The use of optical ow for autonomous navigation. Optical motions and space perception: An extension of gibson's analysis. Subspace methods for recovering rigid motion 2: Algorithm and implemen- tation Recovery of ego-motion using image stabilization Direct computation of the focus of expan- sion Segmentation of frame sequences obtained by a moving observer. Mixture models for optica ow computation. Vertical and horizontal disparities from phase. A computer algorithm for reconstructing a scene from two projections. The interpretation of a moving retinal image. An iterative image- registration technique with an application to stereo vision Directional selectivity and its use in early visual processing. The accuracy of the computation of optical ow and of the recovery of motion parameters. A video encoding system using conditional picture-element replenishment Scene segmentation from visual motion using global optimization. Advances in picture coding. Motion recovery from image sequences using only Motion compensated television coding: Part 1. Motion and structure from optical ow. Motion estimation from tagged mr image sequences. How do we avoid confounding the direction we are looking and the direction we are moving. Movement detectors of the correlation type provide sucient information for local computation of 2d velocity multiple motions from optical ow. Edge detection and motion detection. Bounds on time-to-collision and rotational component from rst-order derivatives of image ow A fast method to estimate sensor translation. Uniqueness and estimation of three-dimensional motion parameters of rigid objects with curved surfaces Estimating three-dimensional motion parameters of a rigid planar patch 2: Singular value decomposition Coherence and transparency of moving plaids composed of fourier and non-fourier gratings Theta motion: A paradoxical stimulus to explore higher-order motion extraction An error-weighted regularization algorithm for image motion- eld estima- tion Automatic feature point extraction and tracking in image sequences for unknown image motion. --TR Edge detection and motion detection Scene segmentation from visual motion using global optimization Bounds on time-to-collision and rotational component from first-order derivatives of image flow Obstacle detection by evaluation of optical flow fields Stereo correspondence from optic flow Computation of component image velocity from local phase information The feasibility of motion and structure from noisy time-varying image velocity information Techniques for disparity measurement On the Detection of Motion and the Computation of Optical Flow Three-dimensional motion computation and object segmentation in a long sequence of stereo frames Subspace methods for recovering rigid motion I Motion recovery from image sequences using only first order optical flow information Motion segmentation and qualitative dynamic scene analysis from an image sequence Performance of optical flow techniques The use of optical flow for the autonomous navigation Measurement of Image Velocity The Accuracy of the Computation of Optical Flow and of the Recovery of Motion Parameters A Fast Method to Estimate Sensor Translation multiple motions from optical flow A Fast Obstacle Detection Method based on Optical Flow --CTR Weichuan Yu , Gerald Sommer , Steven Beauchemin , Kostas Daniilidis, Oriented Structure of the Occlusion Distortion: Is It Reliable?, IEEE Transactions on Pattern Analysis and Machine Intelligence, v.24 n.9, p.1286-1290, September 2002 Abhijit S. Ogale , Yiannis Aloimonos, A Roadmap to the Integration of Early Visual Modules, International Journal of Computer Vision, v.72 n.1, p.9-25, April 2007 Weichuan Yu , Gerald Sommer , Kostas Daniilidis, Multiple motion analysis: in spatial or in spectral domain?, Computer Vision and Image Understanding, v.90 n.2, p.129-152, May

image motion;aperture problem;occlusion;optical flow;non-Fourier motion

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Logic Based Abstractions of Real-Time Systems.

When verifying concurrent systems described by transition systems, state explosion is one of the most serious problems. If quantitative temporal information (expressed by clock ticks) is considered, state explosion is even more serious. We present a notion of abstraction of transition systems, where the abstraction is driven by the formulae of a quantitative temporal logic, called qu-mu-calculus, defined in the paper. The abstraction is based on a notion of bisimulation equivalence, called , n-equivalence, where is a set of actions and n is a natural number. It is proved that two transition systems are , n-equivalent iff they give the same truth value to all qu-mu-calculus formulae such that the actions occurring in the modal operators are contained in , and with time constraints whose values are less than or equal to n. We present a non-standard (abstract) semantics for a timed process algebra able to produce reduced transition systems for checking formulae. The abstract semantics, parametric with respect to a set of actions and a natural number n, produces a reduced transition system , n-equivalent to the standard one. A transformational method is also defined, by means of which it is possible to syntactically transform a program into a smaller one, still preserving , n-equivalence.

Introduction In this paper we address the problem of verifying systems in which time plays a fundamental role for a correct behaviour. We refer to the Algebra of Timed Processes (ATP) [22] as a formalism able both to model time dependent systems and to prove their properties. ATP is an extension of traditional process algebras which can capture discrete quantitative timing aspects with respect to a global clock. We express the semantics of this language in terms of labeled transition systems where some transitions are labeled by the special action , called time action. Such an action represents the progress of time and can be viewed as a clock tick. One widely used method for verication of properties is model checking [8, 7, Model checking is a technique that proves the correctness of a system specication with respect to a desired behavior by checking whether a structure, representing the specication, satises a temporal logic formula describing the expected behavior. Most existing verication techniques, and in particular those dened for concurrent calculi, like CCS [21], are based on a representation of the system by means of a labeled transition system. In this case, model checking consists in checking whether a labeled transition system is a model for a formula. When representing systems specications by transition systems, state explosion is one of the most serious problems: often we have to deal with transition systems with an extremely large number of states, thus making model checking inapplicable. Moreover, when in system specications quantitative temporal information (expressed by clock ticks) is considered, state explosion is even more serious, the reason for this being that a new state is generated for every clock tick. Fortunately, in several cases, to check the validity of a property, it is not necessary to consider the whole transition system, but only an abstraction of it that maintains the information which \in uences" the property. This consideration has been used in the denition of abstraction criteria for reducing transition systems in order to prove properties e-ciently. Abstraction criteria of this kind are often based on equivalence relations dened on transition systems: minimizations with respect to dierent notions of equivalence are in fact used in many existing verication environments (see, for instance, [10, 13, 16]). In this paper we present a notion of abstraction of transition systems, where the abstraction is driven by the formulae of a quantitative temporal logic. This logic, which we call qu-mu-calculus, is similar to the mu-calculus [19], in particular to a variant of it [4], in which the modal operators are redened to include the denition of time constraints. Many logics have been dened to deal with time aspects, see, for example [1{3, 14, 15, 20]. A fundamental feature of qu-mu- calculus is that its formulae can be used to drive the abstraction: in particular, given the actions and the time constraints occurring in the modal operators of a formula of the qu-mu-calculus, we use them in dening an abstract (reduced) transition system on which the truth value of is equivalent to its value on the standard one. The abstraction is based on a notion of bisimulation equivalence between transition systems, called h; ni-equivalence, where is a set of actions (dierent from the time action ) and n is a natural number: informally, two transition systems are h; ni-equivalent i, by observing only the actions in and the paths composed of time actions shorter than or equal to n, they exhibit the same behaviour. Some interesting properties of such an equivalence are presented. We prove that two transition systems are h; ni-equivalent if and only if they give the same truth value to all formulae such that the actions occurring in the modal operators are contained in , and with time constraints whose values are less than or equal to n. Thus, given a formula , with actions in and maximum time constraint n, we can abstract the transition system to a smaller one (possibly the minimum) h; ni-equivalent to it, on which can be checked. In the paper we present a non-standard (abstract) semantics for the ASTP [22] language, dening abstract transition systems. ASTP is the sequential subset of ATP; actually, this is not a limitation: our abstract semantics is easily applicable to the concurrent operators and its ability in reducing the transition system can be suitably investigated also on the concurrent part. The abstract semantics can be usefully exploited as a guide in implementing an algorithm to build the reduced system. We also present a set of syntactic rewriting rules which can transform a process into a smaller one, while preserving h; ni-equivalence. This syntactic reduction can be used as a rst step of the reduction process, before applying the abstract semantics. After the preliminaries of Section 2, we introduce our logic in Section 3 and the abstract semantics in Section 4. Section 5 describe the syntactic transformations and Section 6 concludes the paper. Preliminaries 2.1 The Algebra of Timed Processes Let us now quickly recall the main concepts about the Algebra of Timed Processes [22], which is used in the specication of real-time concurrent and distributed systems. For simplicity, we consider here only the subset of ATP, called ASTP (Algebra of Sequential Timed Processes), not containing parallel operators. The syntax of sequential process terms (processes or terms for short) is the following where ranges over a nite set of asynchronous actions A :::g. We denote by A the set A [ fg, ranged over by :. The action (time action) is not user-denable and represents the progress of time. x ranges over a set of constant names: each constant x is dened by a constant denition x def We denote the set of process terms by P . The standard operational semantics [22] is given by a relation ! P AP , where P is the set of all processes: ! is the least relation dened by the rules in Table 1. Rule Act manages the prexing operator: p evolves to p by a transition labeled by . The operator behaves as a standard nondeterministic choice for processes with asynchronous initial actions (rule Sum 1 and the symmetric one which is not shown). Moreover, if p and q can perform a action reaching respectively can perform a action, reaching p 0 q 0 (rule Sum 2 ). The process bpc(q) can perform the same asynchronous initial actions as p (rule Delay 1 ). Moreover bpc(q) can perform a action, reaching the process q (rule Delay 2 ). Finally, rule Con says that a constant x behaves as p if x denition. Note that there is no rule for the process 0, which thus cannot perform any move. In the following we use :p to denote the term b0c(p); this process can perform only the action and then becomes the process p. Moreover we dene n :p (n > 1) as: Act Delay 1 Con Table 1. Standard operational semantics of ASTP A labeled transition system (or transition system for short) is a quadruple is a set of states, A is a set of transition labels (actions), is the initial state, and ! T S A S is the transition relation. If Given a process p, we write p exist such that is the empty sequence. Given p 2 S, we denote the set of the states reachable from p by ! T with R !T A g. Given a process p and a set of constant denitions, the standard transition system for p is dened as p). Note that, with abuse of nota- tion, we use ! for denoting both the operational semantics and the transition relation among the states of the transition system. On ASTP processes equivalence relations can be dened [22], based on the notion of bisimulation between states of the related transition systems. Example 1. Let us consider a vending machine with a time-dependent behavior. The machine allows a user to obtain dierent services: a soft drink immediately after the request; a coee after a delay of a time unit; a cappuccino after a delay of two time units; a cappuccino with chocolate after a delay of three time units. Moreover, it is possible to recollect the inserted coin, only if requested within one time unit. The ASTP specication of the machine is: V=coin brecollect money V c( coee :(collect coee V ) choc cappuccino 3 :(collect choc cappuccino V ) soft drink collect soft drink V ) The standard transition system for the vending machine has 14 states and transitions. 3 Quantitative temporal logic and abstractions In order to perform quantitative temporal reasoning, we dene a logic, that we call qu-mu-calculus, which is an extension of the mu-calculus [19] and in particular of the selective mu-calculus [4]. The syntax is the following, where Z ranges over a set of variables: hi R;n j Z: j Z: where { R A ; { n 2 N , where N is the set of natural numbers; n is called time value. In hi R;<n and [] R;<n it must be n > 0. The satisfaction of a formula by a state p of a transition system, written p is dened as follows: any state satises tt and no state satises ff; a state sat- are the quantitative modal oper- ators. The informal meaning of the operators is the following: hi R;<n is satised by a state which can evolve to a state satisfying by executing , not preceded by actions in R [ fg, within n time units. [] R;<n is satised by a state which, for any execution of occurring within units and not preceded by actions in R [ fg, evolves to a state satisfying . hi R;n is satised by a state which can evolve to a state satisfying by executing , not preceded by actions in R[ fg, after at least n time units. [] R;n is satised by a state which, for any execution of occurring after at least n time units and not preceded by actions in R [ fg, evolves to a state satisfying . As in standard mu-calculus, a xed point formula has the form Z: (Z:) where Z (Z) binds free occurrences of Z in . An occurrence of Z is free if it is not within the scope of a binder Z (Z). A formula is closed if it contains no variables. Z: is the least x point of the recursive equation Z: is the greatest one. We consider only closed formulae. The precise denition of the satisfaction of a closed formula by a state p of a transition system T is given in Table 2. It uses the relation =) ;n Denition 1 (=) ;n relation). Given a transition system a set of actions A , and n 2 N , we dene the relation =) ;n such that, for each 2 is the number of actions occurring in -. By T q we express the fact that it is possible to pass from p to q by executing a (possibly empty) sequence of actions not belonging to and containing exactly k , followed by the action in . A transition system T satises a formula i its initial state satises . An ASTP process p satises a formula i S(p) satises . Example 2. Examples of properties concerning the vending machine described in the previous section are the following: \it alway holds that, after a coin has been inserted, a soft drink may be collected within two time units". \ it is not possible to recollect the inserted coin after more than one time unit". 3.1 Formula driven equivalence A formula of the qu-mu-calculus can be used to dene a bisimulation equivalence between transition systems. The bisimulation is dened by considering only the asynchronous actions occurring in the quantitative operators belonging where, for each m, Z are dened as: where the notation [ =Z] indicates the substitution of for every free occurrence of the variable Z in . Table 2. Satisfaction of a formula by a state to the formula, and the maximum time value of the quantitative operators occurring in the formula. Thus all formulae with the same set of occurring actions and the same maximum time value dene the same bisimulation. Given a set A of actions and a time value n, a h; ni-bisimulation relates states p and q if: i) for each path starting from p, containing k < n time actions and no action in and ending with 2 , there is a path starting from q, containing exactly k time actions and no action in and ending with 2 , such that the reached states are bisimilar, and ii) for each path starting from p, containing k n time actions and no action in and ending with 2 , there is a path starting from q, containing m n (possibly m 6= actions and no action in and ending with 2 , such that the reached states are bisimilar. Denition 2 (h; ni-bisimulation, h; ni-equivalence). A and n 2 N . B, is a binary relation on S T S such that rBq implies: - T and are h; ni-equivalent ( T ;n ) i there exists a h; ni-bisimulation containing the pair (p; p 0 ). Fig. 1. Examples of h; ni-equivalence Example 3. Consider the transition systems illustrated in Figure 1. T1 is hfag; ni- equivalent to T 2, with while T1 is not hfag; ni-equivalent to T 2, with n 3. Moreover T1 is not hfa; bg; ni-equivalent to T 2, for every n 2 N . The following proposition holds, relating equivalences with dierent and n. Proposition 1. For each Proof. See Appendix A. In order to relate h; ni-equivalence with quantitative temporal properties, we introduce the following denition, concerning equivalences based on sets of formulae Denition 3 (logic-based equivalence). Let T and be two transition sys- tems, and a set of closed formulae. The logic-based equivalence is dened by: Given a formula of the qu-mu-calculus, we dene the set of occurring actions in and the maximum time value of . Denition 4. (O(), max()). Given a formula of the qu-mu-calculus, the set O() of the actions occurring in is inductively dened as follows: The maximum time value of the modal operators occurring in (max()) is inductively dened as follows: The following theorem states that h; ni-equivalent transition systems satisfy the same set of formulae with occurring actions in and maximum time value less than or equal to n. Theorem 1. Let systems and let A and n 2 N . where closed formula of the qu-mu-calculus such that O() and max() ng: Proof. See Appendix A. 4 Abstract transition systems and abstract semantics In this section, in order to reduce the number of states of a transition system for model checking, we dene an abstraction of the transition system on which a formula can be equivalently checked. First we dene the notion of time path. time path is an acyclic path composed only of actions and such that each state (but the rst one) has only one input transition and each state (but the last one) has only one output transition. Denition 5. (time path) Let transition system and path each path p 1 that { holds that p i { 8i, 1 i < n, 6 { 8i, 1 < i n, 6 9q 6= p i such that q Given an ASTP process p and a pair h; ni, we dene an abstract transition system for p by means of a non-standard semantics which consists of a set of inference rules that skip actions not in and produce time paths not longer than n. The abstract transition system is h; ni-equivalent to the standard transition system of p. The non standard rules are shown in Table 3 (the symmetric rules of Sum 1 and Sum 2 are not shown). They use a transition relation ! m ;n parameterized by an integer m n. The ideas on which the semantics is based are the following: { the actions in are always performed (rules Act 1 , Delay 4 and Sum 1 ) { the actions not in are skipped: when an action not in is encountered, a \look-ahead" is performed in order to reach either an action in or a time action (rules Act 2 , Delay 3 and Sum 2 ); { when a time action is encountered, it is skipped only if the process we reach by this action can perform a sequence of n time units. In order to count the time units we use the superscript of ! m ;n q occurs when an action belonging to can be executed after m time actions starting from p. In fact, in order to generate the transition p ;n q , we rst prove that q ;n q 0 for some q 0 (rules Delay 1 and Delay 2 , Sum 3 and Sum 4 ). Successive applications of Delay 2 and Sum 4 allow us to skip all time actions in a sequence but the last n ones. Note that in the premises of rules Delay 3 , Delay 4 , Sum 1 , Sum 2 Sum 3 and Sum 4 the standard operational relation ! is used, in order to know the rst action of the process and consequently to respect the standard behavior of the operators, which is dierent depending on whether the rst action is a time action or not. The following proposition characterizes the transitions of the non-standard semantics Proposition 2. Let A and n 2 N . For each ASTP process p, 1. p ;n q implies 2 and 2. p Proof. By induction on depth of inference. The proposition states that there are two kinds of transitions: the rst one represents the execution of action 2 and is characterized by the superscript 0; the second one represents the execution of a action, and is characterized by a The following result holds, relating the paths composed of time actions of the standard transition system with those of the non-standard one: Proposition 3. Let A and n 2 N . For each ASTP process p, 1. j n and p 2. j n and p ;n q. Proof. See Appendix A. The proposition states that, whenever there is a path in the standard semantics composed of less than or equal to n time actions, followed by an action in , a path with the same number of time actions occurs in the abstract system, while every path with more than n time actions in the standard system corresponds to a path with exactly n time actions in the abstract system. Now we formally dene the notion of abstract transition system. Denition 6 (abstract transition system). For each ASTP process p, given A and n 2 N the abstract transition system for p is dened as where q only if 9j:q The following theorem holds, stating that the transition system dened by the non-standard semantics is a suitable abstraction of the standard one. Theorem 2. Let A and n 2 N . For each ASTP process p, 1. the transitions of N ;n (p) are labeled only either by actions in or by ; 2. the length of each time path in N ;n (p) is less than or equal to n; 3. S(p) ;n N ;n (p). Proof. See Appendix A. Note that, if the abstract transition system N ;0 (p) for a process p does not contain transitions labeled by time actions and expresses only the precedence properties between the asynchronous actions in . The following proposition relates h; ni-equivalences with dierent and n. It says that h; ni-equivalence is preserved by keeping a larger and a greater n. Proposition 4. Let ; 0 A . For each ASTP process p, Proof. By Proposition 1 and by Theorem 2, point 3. Delay 1 Delay 3 ;n r ;n r ;n r Con Table 3. Non-standard operational semantics for ASTP Example 4. Recall the vending machine of Example 1. Let us suppose that we have to verify the following two formulae, expressed in Example 2. The formula 1 can be checked on the abstract transition system N 1 ;n1 (V ), with collect soft drinkg; and 2. states and 14 transitions. On the other hand 2 can be checked on N 2 ;n2 (V ), with states and 13 transitions. 5 Syntactic reduction In this section we investigate a syntactic approach to the reduction of transition systems, still based on the formula to be checked. Given a process p and a property , it is possible to perform syntactic transformations which reduce the size of p (in terms of number of operators), based on the actions and the time values occurring in . The transformations are h; ni-equivalence preserving, that is can be equivalently checked on the transformed process. The syntactic reduction can be used independently from the semantic abstraction dened in the previous section. Table 4. Transformation rules The h; ni-equivalence preserving transformations are shown in Table 4 in the form of rewriting rules: p 7! q means \rewrite p as q". Rule R 1 allows deleting an asynchronous action not in , while rules R 2 and R 3 cancel time actions from sequences of time actions. R 2 deletes m n time actions from a sequence of m ones (if m > n), it can only be applied if the sequence is not the operand of a summation, and this is ensured by imposing that the sequence is prexed by an asynchronous action. When handling summations, R 3 is ap- plied, which deletes n time actions from both operands. Note that, in order to preserve h; ni-equivalence, in all cases the transformed term must be guarded by an asynchronous action. The following theorem states the correctness of the transformations. Theorem 3. Let A ; n 2 N and q be an ASTP process. If q i Proof. See Appendix A. Other rules could be dened, performing further reductions. However, every syntactic method, being static, cannot perform all possible simplications, since it cannot \know" the behavior of the process at \run time". A semantic approach, like that described in the preceding section, based on an abstract semantics, can be in general more precise. On the other hand, compared with the semantic approach, the syntactic one has the advantage of being less complex in time, since it only analyzes the source code, without executing the program. Though the semantic and syntactic reductions are independent, they can be protably combined. Given a process p, rst it can be syntactically transformed into a process q, and then an abstract transition system can be built for q using the abstract semantics. Example 5. Recall the vending machine in Example 1. Let us suppose that we have to verify the property 2 of Example 2. If we apply the transformation rules to the vending machine, with fcoin; moneyg and n we obtain the following reduced process: cappuccino :(collect cappuccino V 2) choc cappuccino :(collect choc cappuccinoV 2) The formula 2 can be checked on the standard transition for V 2, which has states and 14 transitions. Moreover, 2 can be checked on the abstract transition system N 2 ;n2 (V 2), obtained applying to V 2 the abstract semantics. states and 9 transitions. Note that applying rst the syntactic reduction and after the abstract semantics produces a transition system smaller than the one obtained with the abstract semantics applied to the initial process 6 Conclusions In this paper we have presented an approach to the problem of the reduction of the number of states of a transition system. Many abstraction criteria for system specications not including time constraints have been dened, see for example [4, 6, 9, 11, 12]. For real-time systems the work [17] denes abstractions for transition systems with quantitative labels, but there, the abstraction is not driven by the property to be proved. We have introduced an abstract semantics for ASTP processes in order to formally dene the abstract transition system. Our abstract semantics is easily applicable to the concurrent operator: Appendix B shows the extension of the semantics to cope with this operator. The abstract semantics can be used to design a tool for automatically building an abstract transition system. In the implementation, some care must be taken to manage innite loops which can occur in the look-ahead process. The syntactic reductions are easily implementable. The degree of reduction performed by the abstract semantics depends on the size of the set of actions and on the bound n. In particular, the reduction can be signicant either when the set is a small subset of A or when the bound n is small with respect to the length of the time paths in the standard transition system. Obviously, no reduction is performed if and n is greater than the longest time path in the standard transition system. --R Model Checking via Reachability Testing for Timed Automata. Logics and Models of Real Time: A Survey. A Really Temporal Logic. Selective mu-calculus: New Modal Operators for Proving Properties on Reduced Transition Systems Selective mu-calculus and Formula-Based Equivalence of Transition Systems Property Preserving Simu- lations Automatic Veri Model Checking and Abstraction. The NCSU Concurrency Workbench. Generation of Reduced Models for Checking Fragments of CTL. Abstract Interpretation of Reactive Systems. Aboard AUTO. CADP A Protocol Validation and Veri Concept of Quanti Symbolic Model Checking for real-time Systems Results on the propositional mu-calculus From Timed Automata to Logic - and Back The Algebra of Timed Processes Local Model Checking for Real-Time Systems --TR Automatic verification of finite-state concurrent systems using temporal logic specifications Communication and concurrency Verifying temporal properties of processes A really temporal logic Symbolic model checking for real-time systems The algebra of timed processes, ATP Abstract interpretation of reactive systems Selective mu-calculus and formula-based equivalence of transition systems From Timed Automata to Logic - and Back Concept of Quantified Abstract Quotient Automaton and its Advantage Model Checking via Reachability Testing for Timed Automata Generalized Quantitative Temporal Reasoning Property Preserving Simulations Generation of Reduced Models for Checking Fragments of CTL Local Model Checking for Real-Time Systems (Extended Abstract) Validation and Verification Toolbox The NCSU Concurrency Workbench Logics and Models of Real Time Real-Time and the Mu-Calculus (Preliminary Report) Selective MYAMPERSANDmicro;-calculus

temporal logic;ATP;state explosion

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Empirical Studies of a Prediction Model for Regression Test Selection.

AbstractRegression testing is an important activity that can account for a large proportion of the cost of software maintenance. One approach to reducing the cost of regression testing is to employ a selective regression testing technique that 1) chooses a subset of a test suite that was used to test the software before the modifications, then 2) uses this subset to test the modified software. Selective regression testing techniques reduce the cost of regression testing if the cost of selecting the subset from the test suite together with the cost of running the selected subset of test cases is less than the cost of rerunning the entire test suite. Rosenblum and Weyuker recently proposed coverage-based predictors for use in predicting the effectiveness of regression test selection strategies. Using the regression testing cost model of Leung and White, Rosenblum and Weyuker demonstrated the applicability of these predictors by performing a case study involving 31 versions of the KornShell. To further investigate the applicability of the Rosenblum-Weyuker (RW) predictor, additional empirical studies have been performed. The RW predictor was applied to a number of subjects, using two different selective regression testing tools, DejaVu and TestTube. These studies support two conclusions. First, they show that there is some variability in the success with which the predictors work and second, they suggest that these results can be improved by incorporating information about the distribution of modifications. It is shown how the RW prediction model can be improved to provide such an accounting.

Introduction Regression testing is an important activity that can account for a large proportion of the cost of software maintenance [5, 17]. Regression testing is performed on modified software to provide confidence that the software behaves correctly and that modifications have not adversely impacted the software's quality. One approach to reducing the cost of regression testing is to employ a selective regression testing technique. A selective regression testing technique chooses a subset of a test suite that was used to test the software before modifications were made, and then uses this subset to test the modified software. 5 Selective regression testing techniques reduce the cost of regression testing if the cost of selecting the subset from the test suite together with the cost of running the selected subset of test cases is less than the cost of rerunning the entire test suite. Empirical results obtained by Rothermel and Harrold on the effectiveness of their selective regression testing algorithms, implemented as a tool called DejaVu, suggest that test selection can sometimes be effective in reducing the cost of regression testing by reducing the number of test cases that need to be rerun [24, 26]. However, these studies also show that there are situations in which their algorithm is not cost-effective. Furthermore, other studies performed independently by Rosenblum and Weyuker with a different selective 1 Department of Computer and Information Science, Ohio State University. 2 Department of Information and Computer Science, University of California, Irvine. 3 Department of Computer Science, Oregon State University. 5 A variety of selective regression testing techniques have been proposed (e.g., [1, 3, 4, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 21, 26, 27, 28, 29, 30]). For an overview and analytical comparison of these techniques, see [25]. regression testing algorithm, implemented as a tool called TestTube [8], also show that such methods are not always cost-effective [23]. When selective regression testing is not cost-effective, the resources spent performing the test case selection are wasted. Thus, Rosenblum and Weyuker argue in [23] that it would be desirable to have a predictor that is inexpensive to apply but could indicate whether or not using a selective regression testing method is likely to be worthwhile. With this motivation, Rosenblum and Weyuker [23] propose coverage-based predictors for use in predicting the cost-effectiveness of selective regression testing strategies. Their predictors use the average percentage of test cases that execute covered entities-such as statements, branches, or functions-to predict the number of test cases that will be selected when a change is made to those entities. One of these predictors is used to predict whether a safe selective regression testing strategy (one that selects all test cases that cover affected will be cost-effective. Using the regression testing cost model of Leung and White [19], Rosenblum and Weyuker demonstrate the usefulness of this predictor by describing the results of a case study they performed involving 31 versions of the KornShell [23]. In that study, the predictor reported that, on average, it was expected that 87.3% of the test cases would be selected. Using the TestTube approach, 88.1% were actually selected on average over the 31 versions. Since the difference between these values is very small, the predictor was clearly extremely accurate in this case. The authors explain, however, that because of the way their selective regression testing model employs averages, the accuracy of their predictor might vary significantly in practice from version to version. This is particularly an issue if there is a wide variation in the distribution of changes among entities [23]. However, because their predictor is intended to be used for predicting the long-term behavior of a method over multiple versions, they argue that the use of averages is acceptable. To further investigate the applicability of the Rosenblum-Weyuker (RW) predictor for safe selective regression testing strategies, we present in this paper the results of additional studies. We applied the RW predictor to subjects developed by researchers at Siemens Corporate Research for use in studies to compare the effectiveness of certain software testing strategies [14]. For the current paper we used both DejaVu and TestTube to perform selective regression testing. In the following sections we discuss the results of our studies. Background: The Rosenblum-Weyuker Predictor Rosenblum and Weyuker presented a formal model of regression testing to support the definition and computation of predictors of cost-effectiveness [23]. Their model builds on work by Leung and White on modeling the cost of employing a selective regression testing method [19]. In both models, the total cost of regression testing incorporates two factors: the cost of executing test cases, and the cost of performing analyses to support test selection. A number of simplifying assumptions are made in the representation of the cost in these models: 1. The costs are constant on a per-test-case basis. 2. The costs represent a composite of the various costs that are actually incurred; for example, the cost associated with an individual test case is a composite that includes the costs of executing the test case, storing execution data, and validating the results. 3. The cost of the analyses needed to select test cases from the test suite has a completely negative impact on cost-effectiveness, in the sense that analysis activities drain resources that could otherwise be used to support the execution of additional test cases. 4. Cost-effectiveness is an inherent attribute of test selection over the complete maintenance life-cycle, rather than an attribute of individual versions. As in Rosenblum and Weyuker's model [23], we let P denote the system under test and let T denote the regression test suite for P , with jT j denoting the number of individual test cases in T . Let M be the selective regression testing method used to choose a subset of T for testing a modified version of P , and let E be the set of entities of the system under test that are considered by M . It is assumed that T and E are non-empty and that every syntactic element of P belongs to at least one entity in E. The Rosenblum-Weyuker (RW) model defined covers M (t; e) as the coverage relation induced by method for P and defined over T \Theta E, with covers M (t; e) true if and only if the execution of P on test case t causes entity e to be exercised at least once. Rosenblum and Weyuker specify meanings for "exercised" for several kinds of entities of P . For example, if e is a function or module of P , e is exercised whenever it is invoked, and if e is a simple statement, statement condition, definition-use association or other kind of execution subpath of P , e is exercised whenever it is executed. Letting E C denote the set of covered entities, the RW model defined E C as follows: with jE C j denoting the number of covered entities. Furthermore, covers M (t; e) can be represented by a 0-1 matrix C, whose rows represent elements of T and whose columns represent elements of E. Then, element C i;j of C is defined to be: ae 1 if covers M (i; Finally, CC was the cumulative coverage achieved by T (i.e., the total number of ones in the 0-1 matrix): As a first step in computing a predictor for safe strategies when a single entity had been changed, Rosenblum and Weyuker considered the expected number of test cases that would have to be rerun. Calling this average NM they defined: Rosenblum and Weyuker emphasized that this predictor was only intended to be used when the selective regression testing strategy's goal was to rerun all affected test cases. A slightly refined variant of NM was defined using E C rather than E as the universe of entities. Then the fraction of the test suite that must be rerun was denoted -M , the predictor for jT M j=jT j: Rosenblum and Weyuker discussed results of a case study in which test selection and prediction results were compared for 31 versions of the KornShell using the TestTube selective regression testing method. As mentioned above, in this study, the test selection technique chose an average of 88.1% of the test cases in the test suite over the 31 versions, while the predicted value was 87.3%. They concluded that, because the difference between these values was very small, their results indicated the usefulness of their predictor as a way of predicting cost-effectiveness. 3 Two New Empirical Studies of the Rosenblum-Weyuker Predic- tor The results of the Rosenblum-Weyuker case study were encouraging for two reasons: 1. The difference between the predicted and actual values was insignificant. 2. Because a large proportion of the test set would have to be rerun for regression testing, and it could be quite expensive to perform the analysis necessary to determine which test cases did not need to be rerun, it would often be cost-effective to use the predictor to discover this and then simply rerun the test suite rather than selecting a subset of the test suite. Nevertheless, this study involved a single selective regression testing method applied to a single subject pro- gram, albeit a large and widely-used one for which there were a substantial number of actual production versions. In order to obtain a broader picture of the usefulness of the RW predictor, we conducted additional studies with other subject software and other selective regression testing methods. In particular, we performed two new studies with two methods, DejaVu and TestTube, applied to a suite of subject programs that have been used in other studies in the testing literature. 3.1 Statement of Hypothesis The hypothesis we tested in our new studies is the hypothesis of the Rosenblum-Weyuker study: Hypothesis: Given a system under test P , a regression test suite T for P , and a selective regression testing method M , it is possible to use information about the coverage relation covers M induced by M over T and the entities of P to predict whether or not M will be cost-effective for regression testing future versions of P . The previous and current studies test this hypothesis under the following assumptions: 1. The prediction is based on a cost metric that is appropriate for P and T . Certain simplifying assumptions are made about costs, as described in Section 2. 2. The prediction is performed using data from a single version of P to predict cost-effectiveness for all future versions of P . 3. "Cost-effective" means that the cumulative cost over all future versions of P of applying M and executing the test cases in T selected by M is less than the cumulative cost over all future versions of P of running all test cases in T (the so-called retest-all method). Lines of Number of Number of Test Pool Average Test Program Code Functions Modified Versions Size Suite Size Description separation schedule2 printtokens2 printtokens1 replace 516 21 replacement Table 1: Summary of subject programs. 3.2 Subject Programs For our new studies, we used seven C programs as subjects that had been previously used in a study by researchers at Siemens Corporate Research [14]. Because the researchers at Siemens sought to study the fault-detecting effectiveness of different coverage criteria, they created faulty modified versions of the seven base programs by manually seeding the programs with faults, usually by modifying a single line of code in the base version, and never modifying more than five lines of code. Their goal was to introduce faults that were as realistic as possible, based on their experience. Ten people performed the fault seeding, working "mostly without knowledge of each other's work'' [14, p. 196]. For each base program, Hutchins et al. created a large test pool containing possible test cases for the program. To populate these test pools, they first created an initial set of black-box test cases "according to good testing practices, based on the tester's understanding of the program's functionality and knowledge of special values and boundary points that are easily observable in the code" [14, p. 194], using the category partition method and the Siemens Test Specification Language tool [2, 20]. They then augmented this set with manually-created white-box test cases to ensure that each executable statement, edge, and definition-use pair in the base program or its control flow graph was exercised by at least test cases. To obtain meaningful results with the seeded versions of the programs, the researchers retained only faults that were "neither too easy nor too hard to detect" [14, p. 196], which they defined as being detectable by at least three and at most 350 test cases in the test pool associated with each program. Table presents information about these subjects. For each program, the table lists its name, the number of lines of code in the program, the number of functions in the program, the number of modified (i.e., fault seeded) versions of the program, the size of the test pool, the average number of test cases in each of the 1000 coverage-based test suites we generated for our studies, and a brief description of the program's function. We describe the generation of the 1000 coverage-based test suites in greater detail below. 3.3 Design of the New Studies In both studies, our analysis was based on measurements of the following variables: Independent Variable: For each subject program P , test suite T for P and selective regression testing method M , the independent variable is the relation covers M (t; e) defined in Section 2. Dependent Variables: For each subject program P , test suite T for P and selective regression testing method M , there are two dependent variables: (1) the cost of applying M to P and T , and (2) the cost of executing P on the test cases selected by M from T . We then used the model described in Section 2 plus our measurements of the dependent variables to perform analyses of cost-effectiveness. Each study involved different kinds of analysis, as described below. For both studies, we used the Siemens test pools from which we selected smaller test suites. In particular, we randomly generated 1000 branch-coverage-based test suites for each base program from its associated test pool. 6 To create each test suite T applied the following algorithm: 1. Initialize T i to OE. 2. While uncovered coverable branches remain, (a) Randomly select an element t from T using a uniform distribution, (b) Execute P with t, recording coverage, (c) Add t to T i if it covered branches in P not previously covered by the other test cases in T i , 3. If T i differs from all other previously generated test suites, keep T i , and increment a new T i . Step 3 of the procedure ensures that there are no duplicate test suites for a program, although two different test suites may have some test cases in common. For both of our studies, we computed the cost measures (dependent variables) for both DejaVu and TestTube, and we compared this information to test selection predictions computed using the RW predictor. To gather this information, we considered each base program P with each modified version P i and each test suite T j . For each P and each T j , we computed the following: , the percentage of test cases of T j that the RW predictor predicts will be selected by DejaVu when an arbitrary change is made to P ; , the percentage of test cases of T j that the RW predictor predicts will be selected by TestTube when an arbitrary change is made to P ; , the percentage of test cases of T j actually selected by DejaVu for the changes made to create P i from P ; and , the percentage of test cases of T j actually selected by TestTube for the changes made to create P i from P . Finally, we used these values to evaluate the accuracy of the RW predictor, as described in detail in the following sections. The goal of our first study was to determine the accuracy, on average, of the RW predictor for the subject programs, modified versions, and test suites for each of the selective regression testing approaches we con- sidered. We therefore used the regression test selection information described above to compute the average 6 Because our studies focused on the cost-effectiveness of selective regression testing methods rather than the fault-detecting effectiveness of coverage criteria, the realism of the modifications made to the Siemens programs is not a significant issue. What is more important is that they were made independently by people not involved in our study, thereby reducing the potential for bias in our results. percentages of test cases selected by DejaVu and TestTube over all versions P i of P . For each P and each we computed the following: P jversions of P j jversions of P j (1) P jversions of P j jversions of P j (2) The first step in our investigation was to see how much the percentage of test cases actually selected by each of the methods differed from the predicted percentage of test cases. For this analysis, we needed the following two additional pieces of data, which we computed for each P and each and D TestTube j represent the deviations of the percentages of the test cases predicted by the RW predictor for T j from the average of the actual percentages of test cases selected by the respective method for all versions P i of P . Because it is possible for D DejaVu j and D TestTube j to lie anywhere in the range [-100, 100], we wanted to determine the ranges into which the values for D DejaVu j and D TestTube j actually fell. Thus, we rounded the values of D DejaVu j and D TestTube j to the nearest integer I , and computed, for each I such that \Gamma100 - I - 100, the percentage of the rounded D values with value I . For each P , using each of its values, the result was a set H r is the range value; \Gamma100 - r - 100; prd is the percentage of rounded D DejaVu j values at rg (5) Similarly, for each P , using each of its D TestTube j values, the result was a set H TestTube : r is the range value; \Gamma100 - r - 100; prd is the percentage of rounded D TestTube j values at rg (6) These sets essentially form a histogram of the deviation values. In the ideal case of perfect prediction, all deviations would be zero, and therefore each graph would consist of the single point (0, 100%). 3.3.2 Study 2 In Study 1, we treated the RW predictor as a general predictor in an attempt to determine how accurate it is for predicting test selection percentages for all future versions of a program. In the earlier KornShell study [23], it was determined that the relation covers M (t; e) changes very little during maintenance. In particular, Rosenblum and Weyuker found that the coverage relation was extraordinarily stable over the 31 versions of KornShell that they included in their study, with an average of only one-third of one percent of the elements in the relation changing from version to version, and only two versions for which the amount of change exceeded one percent. For this reason, Rosenblum and Weyuker argued that coverage information from a single version might be usable to guide test selection over several subsequent new versions, thereby saving the cost of redoing the coverage analysis on each new version. However, in circumstances where the coverage relation is not stable, it may be desirable to make predictions about whether or not test selection is likely to be cost-effective for a particular version, using version-specific information. The goal of our second study was therefore to examine the accuracy of the RW predictor as a version-specific predictor for our subject programs, modified versions, and test suites. The intuition is that in some cases it might be important to utilize information that is known about the specific changes made to produce a particular version. We considered each base program P , with each modified version P i and test suite T j , as we had done in Study 1, except that we did not compute averages over the percentages of test cases selected over all versions of a program. Instead, the data sets for this study contain one deviation for each test suite and each version of a program. As in Study 1, the first step in our investigation was to see how much the percentage of test cases actually selected by each of the methods differed from the predicted percentage of test cases. For this analysis, we needed the following additional pieces of data, which we computed for each P , each P i , and each and D TestTube i;j represent the deviations of the percentages of the test cases predicted by the RW predictor for T j from the actual percentages of test cases selected by the respective method for the versions P i of P . As in Study 1, to determine the ranges into which the values for D DejaVu i;j and D TestTube i;j actually fell, we rounded the values of D DejaVu i;j and D TestTube i;j to the nearest integer I and computed, for each I such that \Gamma100 - I - 100, the percentage of the rounded D values with value I . For each P , using each of its D DejaVu i;j values, the result was a set H r is the range value; \Gamma100 - r - 100; prd is the percentage of rounded D DejaVu i;j values at rg (9) Similarly, for each P , using each of its D TestTube i;j values, the result was a set H TestTube : r is the range value; \Gamma100 - r - 100; prd is the percentage of rounded D TestTube i;j values at rg (10) 3.4 Threats to Validity There are three types of potential threats to the validity of our studies: (1) threats to construct validity, which concern our measurements of the constructs of interest (i.e., the phenomena underlying the independent and dependent variables); (2) threats to internal validity, which concern our supposition of a causal relation between the phenomena underlying the independent and dependent variables; and (3) threats to external validity, which concern our ability to generalize our results. 3.4.1 Construct Validity Construct validity deals directly with the issue of whether or not we are measuring what we purport to be measuring. The RW predictor relies directly on coverage information. It is true that our measurements of the coverage relation are highly accurate, but the coverage relation is certainly not the only possible phenomenon that affects the cost-effectiveness of selective regression testing. Therefore, because this measure only partially captures that potential, we need to find other phenomena that we can measure for purposes of prediction. Furthermore, we have relied exclusively on the number of test cases selected as the measure of cost reduction. Care must be taken in the counting of test cases deemed to be "selected", since there are other reasons a test case may not be selected for execution (such as the testing personnel simply lacking the time to run the test case). In addition, whereas this particular measure of cost reduction has been appropriate for the subjects we have studied, there may be other testing situations for which the expense of a test lab and testing personnel might be significant cost factors. In particular, the possibility of using spare cycles might affect the decision of whether or not it is worthwhile to use a selective regression testing method at all in order to eliminate test cases, and therefore whether or not a predictor is meaningful. 3.4.2 Internal Validity The basic premises underlying Rosenblum and Weyuker's original predictor were that (1) the cost-effectiveness of a selective regression testing method, and hence our ability to predict cost-effectiveness, are directly dependent on the percentage of the test suite that the selective regression testing method chooses to run, and that (2) this percentage in turn is directly dependent on the coverage relation. In this experiment we take the first premise as an assumption, and we investigate whether the relation between percentage of tests selected and coverage exists and is appropriate as a basis for prediction. The new data presented in this paper reveal that coverage explains only part of the cost-effectiveness of a method and the behavior of the RW predictor. Future studies should therefore attempt to identify the other factors that affect test selection and cost effectiveness. 3.4.3 External Validity The threats to external validity of our studies are centered around the issue of how representative the subjects of our studies are. All of the subject programs in our new studies are small, and the sizes of the selected test suites are small. This means that even a selected test suite whose size differs from the average or the predicted value by one or two elements would produce a relatively large percentage difference. (The results of Study 1 are therefore particularly interesting because they show small average deviations for most of the subject programs.) For the studies involving the Siemens programs, the test suites were chosen from the test pools using branch coverage, which is much finer granularity than TestTube uses, suggesting that there is a potential "mismatch" in granularity that may somehow skew the results. More generally, it is reasonable to ask whether our results are dependent upon the method by which the test pools and test suites were generated, and the way in which the programs and modifications were designed. We view the branch coverage suites as being reasonable test suites that could be generated in practice, if coverage-based testing of the programs were being performed. Of course there are many other ways that testers could and do select test cases, but because the test suites we have studied are a type of suite that could be found in practice, results about predictive power with respect to such test suites are valuable. The fact that the faults were synthetic (in the sense that they were seeded into the Siemens programs) may also affect our ability to investigate the extent to which change information can help us predict future changes. In a later section we will introduce a new predictor that we call the weighted predictor. This predictor depends on version-specific change information. Because it seemed likely that conclusions drawn using synthetic changes would not necessarily hold for naturally occurring faults, we did not attempt to use the Siemens programs and their faulty versions to empirically investigate the use of the weighted predictor. Nevertheless, the predictor itself is not dependent on whether the changes are seeded or naturally occurring, and thus our results provide useful data points. 4 Data and Results 4.1 Study 1 Figure presents data for D DejaVu j (Equation (3)) and D TestTube j (Equation (4)), with one graph for each subject program P ; the Appendix gives details of the computation of this data using one of the subject programs, printtokens2, as an example. Each graph contains a solid curve and a dashed curve. The solid curve consists of the connected set of points H DejaVu j (Equation (5)), whereas the dashed curve consists of the connected set of points H TestTube j (Equation (6)). Points to the left of the "0" deviation label on the horizontal axes represent cases in which the percentage of test cases predicted was less than the percentage of test cases selected by the tool, whereas points to the right of the "0" represent cases in which the percentage of test cases predicted was greater than the percentage of test cases selected by the tool. To facilitate display of the values, a logarithmic transformation has been applied to the y axes. No smoothing algorithms were applied to the curves. For all P and T j , the D DejaVu j were in the range [-20,33] and the D TestTube j were in the range [-24,28]. However, as we shall see in Figure 2, these ranges are a bit misleading because there are rarely any significant number of values outside the range (-10,0] or [0,10), particularly for TestTube. The graphs show that, for our subjects, the RW predictor was quite successful for both the DejaVu and TestTube selection methods. The predictor was least successful for the printtokens2 program for which it predicted an average of 23% more test cases than DejaVu actually selected. This was the only deviation that exceeded 10% using the DejaVu approach. For schedule1, the prediction was roughly 9% high, on average, compared to the DejaVu-selected test suite. DejaVu selected an average of roughly 10% more test cases than predicted for schedule2, 7% more for totinfo, 7% more for tcas, 3% more for printtokens1, and 4% fewer for replace than the RW predictor predicted. For TestTube, the predictor also almost always predicted within 10% of the actual average number of test cases that were actually selected. The only exception was for the totinfo program, for which the average deviation was under 12%. For the other programs, the average deviations were 5% for the printtokens1 program, 5% for the printtokens2 program, 4% for the replace program, 7% for the tcas program, 10% for schedule1 and 1% for schedule2. We consider these results encouraging, although not as successful as the results described by Rosenblum and Weyuker for the KornShell case study. Recall that in that study there were a total of 31 versions of KornShell, a large program with a very large user base, with all changes made to fix real faults or modify functionality. None of the changes were made for the purpose of the study. Another way to view the data for this study is to consider deviations of the predicted percentage from the totinfo schedule2 schedule1 printtokens2printtokens1 -60legend actual test selection which a level of deviation occurred vertical axes - percentage of test suites for solid curve - DejaVu results dashed curve - TestTube results horizontal axes - deviation in predicted versus Figure 1: Deviation between predicted and actual test selection percentages for application of DejaVu and TestTube to the subject programs. The figure contains one graph for each subject program. In each graph, the solid curve represents deviations for DejaVu, and the dashed curve represents deviations for TestTube. tcas schedule1 schedule2 totinto printtokens2 printtokens1 replace [10%, 20%) [20%, 30%) Predicted and Actual Test Selection Percentages Absolute Value Deviation Between Figure 2: Absolute value deviation between predicted and actual test selection percentages for application of DejaVu and TestTube to the subject programs. The figure contains two bars for each subject program: the left bar of each pair represents the absolute value deviation of DejaVu results from the RW predictor, and the right bar represents the absolute value deviation of TestTube results from the RW predictor. For each program P , these numbers are averages over all versions P i of P . Each bar represents 100% of the test suites T j , with shading used to indicate the percentage of test suites whose deviations fell within the corresponding range. actual percentage without considering whether the predicted percentage was greater or less than the actual percentage selected. These deviations constitute the absolute value deviation. To compute the absolute value deviation, we performed some additional computations: For each P and each T j , we first computed AbsD DejaVu j j. We then tabulated the percentage of the AbsD DejaVu j and the AbsD TestTube j that fell in each of the ranges [0%,10%), [10%,20%), :::, [90%,100%]. Figure depicts these results as segmented bar graphs. The figure contains two bars for each subject program: the left bar of each pair represents the absolute value deviation of DejaVu results from the RW predictor, and the right bar represents the absolute value deviation of TestTube results from the RW predic- tor. For each program P , these numbers are averages over all versions P i of P . Each bar represents 100% of the test suites T j , with shading used to indicate the percentage of test suites whose deviations fell within the corresponding range. For instance, in the case of printtokens2, 100% of the test suites showed less than 10% deviation for TestTube, whereas for DejaVu, 14% of the test suites showed deviations between 10% and 20%, 82% showed deviations between 20% and 30%, and 4% showed deviations between 30% and 40%. The results of this study show that for many of the subject programs, modified versions, and test suites, the absolute value deviation for both DejaVu and TestTube was less than 10%. In these cases, the RW model explains a significant portion of the data. However, in a few cases, the absolute value deviation was significant. For example, as mentioned above, for printtokens2, the absolute value deviation from the predictor for DejaVu was between 20% and 30% for more than 80% of the versions. One additional feature of the data displayed in Figure 1 bears discussion. For all programs other than printtokens2, the curves that represent deviations for DejaVu and TestTube are (relatively) close to one another. For printtokens2, in contrast, the two curves are disjoint and (relatively) widely separated. Examination of the code coverage data and locations of modifications for the programs reveals reasons for this difference. Sixteen of the nineteen printtokens2 functions are executed by a large percentage (on average over 95%) of the test cases in the program's test pool; the remaining three functions are executed by much lower percentages (between 20% and 50%) of the test cases in that test pool. All modifications of printtokens2 occur in the sixteen functions that are executed by nearly all test cases. Thus, the actual test selections by TestTube, on average, include most test cases. The presence of the latter three functions, and the small number of test cases that reach them, however, causes a reduction in the average number of test cases per function, and causes the function-level predictor to under-predict by between 0% and 10% the number of test cases selected by TestTube. Even though nearly all test cases enter nearly all functions in printtokens2, several of these functions contain branches that significantly partition the paths taken by test cases that enter the functions. Thus, many of the statements in printtokens2 are actually executed by fewer than 50% of the test cases that enter their enclosing functions. When modifications occur in these less-frequently executed statements, DejaVu selects much smaller test suites than TestTube. (For further empirical comparison of TestTube and DejaVu, see [22].) This is the case for approximately half of the modified versions of printtokens2 utilized in this study. However, the presence of a large number of statements that are executed by a larger proportion of the test cases causes the average number of test cases per statement to exceed the number of test cases through modified statements. The end result is that the statement-level predictor over-predicts the number of test cases selected by DejaVu by between 5% and 27%. Of course, the precise locations of modifications in the subjects directly affect the results. Therefore, the fact that all changes were synthetic is of concern when trying to generalize these results. 4.2 Study 2 Like Figure 1, Figure 3 contains one graph for each subject program. The graphs also use the same notation as was used in Figure 1, using a solid curve to represent the percentage of occurrences of D DejaVu i;j over deviations for all test suites T j and using a dashed curve to represent the percentage of occurrences of over deviations for all test suites T j . Again, a logarithmic transformation has been applied to the y axes. Figure 4 depicts these results as a segmented bar graph, in the manner of Figure 2. The results of this study show that, for the subject programs, modified versions, and test cases, the deviations and absolute value deviations for individual versions for both DejaVu and TestTube are much greater than in Study 1. This is not surprising because in this study the results are not averaged over all versions as they were in Study 1. For example, consider tcas, printtokens1, and replace. In Study 1, the average absolute value deviation from the predicted percentage for each of these programs is less than 10% using either DejaVu or TestTube. However, when individual versions are considered, the percentage of test schedule1 tcas schedule2 printtokens2 replace legend horizontal axes - deviation in predicted versus actual test selection a level of deviation occurred vertical axes - percentage of test suites for which solid curve - DejaVu results dashed curve - TestTube results Figure 3: Version-specific absolute value deviation between predicted and actual test selection percentages for application of DejaVu and TestTube to the subject programs. The figure contains one graph for each subject program. In each graph, the solid curve represents deviations for DejaVu, and the dashed curve represents deviations for TestTube. tcas schedule1 schedule2 totinto printtokens2 printtokens1 replace [10%, 20%) [20%, 30%) Version-Specific Absolute Value Deviation Between Predicted and Actual Test Selection Percentages Figure 4: Version-specific absolute value deviation between predicted and actual test selection percentages for application of DejaVu and TestTube to the subject programs. The figure contains two bars for each subject program: the left bar of each pair represents the absolute value deviation of DejaVu results from the RW predictor, and the right bar represents the absolute value deviation of TestTube results from the RW predictor. For each program P , these numbers are averages over all versions P i of P . Each bar represents 100% of the test suites T j , with shading used to indicate the percentage of test suites whose deviations fell within the corresponding range. cases selected by DejaVu for these programs varies significantly, up to 64%, from the percentages predicted. Deviations and absolute value deviations for the other subjects show similar differences. In Figure 3, the range of deviations can be seen. In most cases there are at least a few versions that have a small number of instances for which the deviations are significant. The bar graphs in Figure 4 show more clearly how frequently these large absolute value deviations occur. In Figure 3, the data for printtokens2 is again particularly interesting. In this case, the curve for TestTube is peaked and relatively narrow, whereas the curve for DejaVu is nearly flat and relatively wide. As discussed in the preceding section, for both techniques, these differences reflect differences in the degree of variance in the coverage relations at the statement and function level, as well as differences in the location of modifications. In this case, however, considering prediction on a version-specific basis causes the deviation in prediction at the statement level, where the variance in coverage is large, to be flat. Lack of variance in coverage at the function level prevents the TestTube curve from being flat. 5 Improved Predictors As discussed in Section 3.4, the assumptions underlying our measurement of costs may pose threats to the validity of our conclusions about predictions of cost-effectiveness using the RW predictor. Furthermore, in some of the subject programs of our studies, there was significant absolute deviation of the results of the selective regression testing tools (DejaVu and TestTube) with respect to test selection values predicted by the RW predictor. Therefore, we believe that there may be factors affecting cost-effectiveness that are not being captured by the RW predictor. These factors, if added to the model, could improve the accuracy of both general and version-specific predictors. The RW predictor accounts for test coverage but does not account for the locations of modifications. Therefore, one obvious refinement would be to incorporate information about modifications into the predictor. We saw in Study 2 that the specific changes made to create a particular version may have significant effects on the accuracy of prediction in practice. Thus, we believe that an extended weighted predictor might be more accurate for both general and version-specific prediction. Such a predictor would incorporate information about the locations of the changes and weight the predictor accordingly. To this end, in this section we extend the RW predictor by adding weights that represent the relative frequency of changes to the covered entities. For each element e is the relative frequency with which e j is modified, and it is defined such that 1. The original, unweighted RW model, discussed in Section 2, computes the expected number of test cases that would be rerun if a single change is made to a program. To do this, the model uses the average number of test cases that cover each covered entity This average is referred to as N C M . The weighted analogue of N C M is a weighted average, WN C which we define as follows: WN C where C i;j is defined as before: ae 1 if covers M (i; Note that the inner sum represents the total number of test cases covered by e multiplying that sum by weighted contribution to the total number of test cases selected overall. For this weighted average, the fraction of the test suite T that must be rerun, denoted by \Pi M , is given as follows: Note that the original, unweighted RW predictor, -M , is a version of this weighted predictor in which w j is 1 (which represents an assumption that each entity is equally likely to be changed): Figure 5: Coverage Pattern A. Figure Coverage Pattern B. To see the impact of the difference between -M and \Pi M , consider coverage patterns A and B, shown respectively in Figures 5 and 6, where each dot represents an entity and each closed curve a test case. Assume that the patterns are generalized over a large number of entities, n. As discussed by Rosenblum and Weyuker [23], the value of -M predicted for each pattern is 2=n. In Pattern A, the test cases are distributed evenly over the entities, and thus, \Pi M and -M are the same, and yield the exact number of test cases that would be selected by either DejaVu or TestTube, regardless of the relative frequency of changes (and hence, regardless of the values assigned to the w j ). In Pattern B, the test cases are not distributed evenly over the entities, and in contrast with Pattern A, the RW predictor never predicts the exact fraction selected for any changed entity, and it is significantly inaccurate for a change to the "core" element of that pattern. Suppose, however, that instead of assuming that the frequency of change is equal for all entities, we had information about the relative frequency of modifications to individual entities. In this case, using the weighted predictor, we could compute a more accurate estimate of the fraction of the test suite that would be selected. For example, if we knew that changes are always made to two of the non-core entities (with one changed exactly as often as the other) and that no other entities are ever changed, then the weights would be 1=2 for the two changed entities and 0 for all other entities. And thus, we would predict that (for the case of a single entity change) 1=n of the test suite would be selected, rather than 2=n as predicted by the unweighted predictor. 5.1 Improved General Prediction Provided we can obtain values for weights that accurately model the distribution of future modifications to a program, we can use the weighted predictor, \Pi M , to improve general prediction. One approach is to utilize change history information about the program, often available from configuration management systems. Assuming that the change histories do accurately model the pattern of future modifications (a result suggested by the work of Harrison and Cook [11]), we can use this information to compute weights for \Pi M . If the change histories are recorded at the module level, \Pi M can be used as well to predict the percentage of test cases selected on average by a tool, such as TestTube, that considers module-level changes to the system. If the change histories are recorded at the statement level, \Pi M can be used to predict the percentage of test cases selected on average by a tool, such as DejaVu, that considers statement-level changes to the system. In either case, the weighted predictor can be used to incorporate data that may account for change-location information, without performing full change analysis. Thus, it can be used to assess whether it will be worthwhile to perform all of the analysis needed by a selective regression testing tool. In practice, weights may be collected and assumed to be fixed over a number of subsequent versions of a program, or they may be adjusted as change history information becomes available. In this context, an important consideration involves the extent to which weights collected at a particular time in the history of a program can continue to predict values for future versions of that program, and the extent to which the accuracy of predictions based on those weights may decrease over time. Future empirical study of this issue is necessary. 5.2 Improved Version-Specific Prediction We can also use the weighted predictor, \Pi M , as a version-specific predictor. For this version-specific predictor, one approach computes the w i using the configuration management system. We assign a weight of 1=k to each entity that has been changed (where k is the total number of entities changed), and we assign a weight of 0 to all other entities in the system. Using these weights, \Pi M computes the exact percentage of test cases that will be selected by a test selection tool that selects at the granularity of the entities. For example, if the entities are modules, then \Pi M will predict the exact percentage of test cases that will be selected by a test selection tool, such as TestTube, that considers changes at the module level. If the entities are statements, then \Pi M will predict the exact percentage of test cases that will be selected by a test selection tool, such as DejaVu, that considers changes at the statement level. If the cost of determining the number of test cases that will be selected is cheaper than the cost of actually selecting the test cases, this approach can be cost-effective. It is worth noting that Rosenblum and Weyuker found, in their experiments with KornShell, that it was typically not necessary to recompute the coverage relation frequently, because it remained very stable over the 31 versions they studied. If this is typical of the system under test, then this should make version-specific predictors extremely efficient to use and therefore provide valuable information about whether or not the use of a selective regression testing strategy is likely to be cost-effective for the current version of the system under test. An alternative approach assumes that method M can be supplemented with an additional change analysis capability that is more efficient but less precise than M 's change analysis. This supplementary change analysis is used during the critical phase of regression testing - after all modifications have been made to create P 0 , the new version of P . 7 The results of the supplementary change analysis can be used to assign weights to the entities in the system, which are then used for prediction as described above. Using the weighted predictor, \Pi M , as a version-specific predictor will be especially appropriate for test suites whose test cases are not evenly distributed across the entities, such as the case illustrated by Pattern B, where test selection results for specific versions may differ widely from average test selection results over a sequence of versions. 6 Conclusions In this paper we presented results from new empirical studies that were designed to evaluate the effectiveness and accuracy of the Rosenblum-Weyuker (RW) model for predicting cost-effectiveness of a selective regression testing method. The RW model was originally framed solely in terms of code coverage information, and evaluated empirically using the TestTube method and a sequence of 31 versions of KornShell. In the new studies, two selective regression testing methods were used (TestTube and DejaVu), and seven different programs were used as subjects. For the experimental subjects we used in the new studies, the original RW model frequently predicted the average, overall effectiveness of the two test selection techniques with an accuracy that we believe is acceptable given that the cost assumptions underlying the RW model are quite realistic for our subjects. However, the predictions of the model occasionally deviated significantly from observed test selection results. Moreover, when this model was applied to the problem of predicting test selection results for particular modified versions of the subject programs, its predictive power decreased substantially, particularly for DejaVu. These results suggest that the distribution of modifications made to a program can play a significant role in determining the accuracy of a predictive model of test selection. We therefore conclude that to achieve improved accuracy both in general, and when applied in a version-specific manner, prediction models must account for both code coverage and modification distribution. In response to this result, we showed how to extend the Rosenblum-Weyuker predictor to incorporate information on the distribution of modifications. However, to judge the efficacy of this extended predictive model in practice, we require additional experimentation. For this purpose, the subjects used in the studies reported in this paper will not suffice. Rather, we require versions of a program that form a succession of changes over their base versions, as the versions of KornShell did. We are currently building a repository of such programs and versions that, when complete, will provide subjects suitable for further empirical investigation of predictive models for regression testing in general, and of our weighted predictor in particular. Future studies must be directed not only toward further validation of the RW predictor and the improved predictors described in this paper, but toward the development of a more realistic cost model for regression 7 Rothermel and Harrold divide regression testing into two phases for the purpose of cost analysis. During the preliminary phase, changes are made to the software, and the new version of the software is built. During the critical phase, the new version of the software is tested prior to its release to customers [25]. Test Version 4 41.7 18.2 25.0 30.0 38.5 37.5 36.4 ::: 35.7 Version 5 8.3 9.1 12.5 10.0 7.7 12.5 9.1 ::: 7.1 Version 6 16.7 45.5 25.0 20.0 30.8 25.0 18.2 ::: 14.3 Version 7 41.7 18.2 25.0 30.0 38.5 37.5 36.4 ::: 35.7 Version 8 66.7 81.8 62.5 70.0 76.9 75.0 81.8 ::: 71.4 Version 9 41.7 18.2 25.0 30.0 38.5 37.5 36.4 ::: 35.7 33.3 Table 2: Partial data used in the computation of graphs in Figure 1. testing. This will require extensive field studies of existing large systems in order to create a better picture of the different factors driving cost-effectiveness, such as test suite size, test case execution times, testing personnel costs, and the availability of spare machine cycles for regression testing. Acknowledgments This work was supported in part by a grant from Microsoft, Inc., by NSF under NYI award CCR-9696157 to Ohio State University, CAREER award CCR-9703108 to Oregon State University, and Experimental Software Systems award CCR-9707792 to Ohio State University and Oregon State University, and by an Ohio State University Research Foundation Seed Grant. Thanks to Huiyu Wang who performed some of the experiments, to Panickos Palletas and Qiang Wang who advised us on the statistical analysis, to Jim Jones who performed some of the statistical analysis, and to Monica Hutchins and Tom Ostrand of Siemens Corporate Research for supplying the subject programs and other data necessary for the experimentation. Appendix Details of the Computation of Data for Figures 1 and 3 To compute the data used for the graphs in Figure 1, we used a procedure described in Section 3.3.1. As further explanation, we give details of the computation of that data for one subject program, printtokens2. For our experiments, we used 1000 coverage-based test suites, T 1 ; :::T 1000 . Table 2 shows data for a subset of these test suites: . For each test suite T j , we used the RW predictor to predict the number of test cases that would be selected by DejaVu when an arbitrary change is made to printtokens2. We then used this number to determine - DejaVu j , the percentage of test cases that the RW predictor predicts will be selected by DejaVu when an arbitrary change is made to printtokens2. The first row of Table 2 gives these percentages for We had ten versions of printtokens2 (see Table 1). We next ran DejaVu on these ten versions, with each of the 1000 test suites, and, for each version i and test suite j, recorded the number of test cases selected. We then used this number to compute, for each i and j, S DejaVu i;j , the percentage of test cases selected. The ten rows for Versions these percentages. For example, from the table, we can see that, for ranges from 8.3% to 66.7%. Using Equation (1), we then computed the S DejaVu j for each test suite T j . Table 2 gives these percentages for each T j . We then used Equation (3) to compute, for each T j , the difference between the percentage predicted by the RW predictor and the average percentage selected by Dejavu (i.e., D DejaVu j Table 2 shows that, for the D DejaVu j range from 18.9% to 26.1%. Finally, we created the set, H DejaVu (Equation (5)). The ordered pairs in this set are obtained by first rounding the percentages of the D DejaVu j , then determining the number of those rounded percentages that have range value \Gamma100 - r - 100, and then determining the percentage of those percentages that occur for each value of r. Thus, for printtokens2, D DejaVu 6 rounds to 19, D DejaVu 3 , and D DejaVu 7 round to rounds to 22, D DejaVu 4 rounds to 23, D DejaVu 1000 rounds to 24, and D DejaVu 2 rounds to 26. Thus, there will be ordered pairs in H DejaVu with first coordinates 19, 21, 22, 21, 24, and 26, and the number of rounded percentages for T 1 :::T 1000 are used to compute the percentage of times (among the 1000 test suites) each percentage occurs, which is then used in the computation of the second coordinates of the ordered pairs. We used these ordered pairs to plot the solid curve for printtokens2 in Figure 1. We used a similar approach to obtain the data for Figure 3 except that we did not compute the averages of the deviations. To compute the data used for the graphs in Figure 3, we used a procedure described in Section 3.3.2. As further explanation, we give details of the computation of that data for one subject program, printtokens2. Table 3 shows data for a subset of the 1000 coverage-based test suites: . For each test suite T i , we used the RW predictor to predict the number of test cases that would be selected by DejaVu when an arbitrary changes is made to printtokens2. We then used this number to determine - DejaVu j , the percentage of test cases that the RW predictor predicts will be selected by DejaVu when an arbitrary change is made to printtokens2. The first row of Table 3 gives these percentages for Next, we ran DejaVu on the ten versions of printtokens2, and, for each version, recorded the number of test cases selected. We then used this number to compute S DejaVu i;j 1:::10, the percentage of test cases selected. The ten rows for Versions in Table 3 give these percentages. We then used Equation (7) to compute, for each version i and each T j , D DejaVu i;j , the difference between the percentage predicted by the RW predictor and the percentage selected by Dejavu. Table 3 shows that, for percentages range from -26.6% to 52.5%. Finally, we created the set, H DejaVu (Equation (9)). These ordered pairs are obtained by first rounding the percentages of the D DejaVu i;j , determining the number of those rounded percentages that have range value then determining the percentage of those percentages that occur for each value of r. For example, for printtokens2, D DejaVu 1;2 , D DejaVu 8;2 , and D DejaVu 8;6 round to \Gamma20 and D DejaVu 4;5 , D DejaVu 7;5 , D DejaVu 4;7 , D DejaVu 7;7 , D DejaVu 9;7 , and D DejaVu 10;7 round to 20. Thus, H DejaVu contains ordered pairs with \Gamma20 and 20 as the first coordinates, and the number of rounded percentages for T 1 :::T 1000 are used to compute the percentage of times (among the 1000 * 10 test-suite/version pairs) each percentage occurs, and used in the computation of the second coordinates of the ordered pairs. Test Version 4 41.7 18.2 25.0 30.0 38.5 37.5 36.4 ::: 35.7 Version 5 8.3 9.1 12.5 10.0 7.7 12.5 9.1 ::: 7.1 Version 6 16.7 45.5 25.0 20.0 30.8 25.0 18.2 ::: 14.3 Version 7 41.7 18.2 25.0 30.0 38.5 37.5 36.4 ::: 35.7 Version 8 66.7 81.8 62.5 70.0 76.9 75.0 81.8 ::: 71.4 Version 9 41.7 18.2 25.0 30.0 38.5 37.5 36.4 ::: 35.7 Version 3 38.1 43.4 28.0 44.0 34.9 42.6 37.8 ::: 33.4 Version 4 13.1 43.4 28.0 24.0 19.5 17.6 19.6 ::: 19.1 Version 6 38.1 16.1 28.0 34.0 27.2 30.1 37.8 ::: 40.5 Version 7 13.1 43.4 28.0 24.0 19.5 17.6 19.6 ::: 19.1 Version 8 \Gamma11:9 \Gamma20:2 \Gamma9:5 \Gamma16:0 \Gamma18:9 \Gamma19:9 \Gamma25:8 ::: \Gamma16:6 Version 9 13.1 43.4 28.0 24.0 19.5 17.6 19.6 ::: 19.1 Table 3: Partial data used in the computation of graphs in Figure 3. We used this procedure to obtain the data for the rest of the graphs in Figure 3 for DejaVu, and used a similar procedure to obtain the data for the graphs for TestTube. --R Incremental regression testing. Automatic generation of test scripts from formal test specifi- cations On the limit of control flow analysis for regression test selection. Incremental program testing using program dependence graphs. Software Testing Techniques. Semantics guided regression test cost reduction. A system for selective regression testing. A methodology for retesting modified software. An approach to regression testing using slicing. Insights on improving the maintenance process through software measure- ment An incremental approach to unit testing during maintenance. Techniques for selective revalidation. Experiments on the effectiveness of dataflow- and controlflow-based test adequacy criteria of program modifications and its applications in software main- tenance A methodology for test selection. Insights into regression testing. A study of integration testing and software regression at the integration level. A cost model to compare regression test strategies. The category-partition method for specifying and generating functional tests Using dataflow analysis for regression testing. An empirical comparison of regression test selection techniques. Using coverage information to predict the cost-effectiveness of regression testing strategies Empirical studies of a safe regression test selection technique. Analyzing regression test selection techniques. Logical modification oriented software testing. An approach to software fault localization and revalidation based on incremental data flow analysis. A regression test selection tool based on textual differencing. A method for revalidating modified programs in the maintenance phase. --TR --CTR David Notkin, Longitudinal program analysis, ACM SIGSOFT Software Engineering Notes, v.28 n.1, p.1-1, January Alessandro Orso , Nanjuan Shi , Mary Jean Harrold, Scaling regression testing to large software systems, ACM SIGSOFT Software Engineering Notes, v.29 n.6, November 2004 Amitabh Srivastava , Jay Thiagarajan, Effectively prioritizing tests in development environment, ACM SIGSOFT Software Engineering Notes, v.27 n.4, July 2002 John Bible , Gregg Rothermel , David S. Rosenblum, A comparative study of coarse- and fine-grained safe regression test-selection techniques, ACM Transactions on Software Engineering and Methodology (TOSEM), v.10 n.2, p.149-183, April 2001 Hyunsook Do , Gregg Rothermel, An empirical study of regression testing techniques incorporating context and lifetime factors and improved cost-benefit models, Proceedings of the 14th ACM SIGSOFT international symposium on Foundations of software engineering, November 05-11, 2006, Portland, Oregon, USA Mary Jean Harrold, Testing: a roadmap, Proceedings of the Conference on The Future of Software Engineering, p.61-72, June 04-11, 2000, Limerick, Ireland

regression test selection;software maintenance;selective retest;regression testing

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A flexible message passing mechanism for objective VHDL.

When defining an object-oriented extension to VHDL, the necessary message passing is one of the most complex issues and has a large impact on the whole language. This paper identifies the requirements for message passing suited to model hardware and classifies different approaches. To allow abstract communication and reuse of protocols on system level, a new, flexible message passing mechanism proposed for Objective VHDL will be introduced.

Introduction A (hardware) system can be described in the object-oriented fashion as a set of interacting or communicating (concurrent) objects. In consequence to encapsulation the objects tend to be relatively autonomous and only loosely coupled with their environment [13]. Further, an object should contain most of the elements it needs to perform its functionality. This property provides high potential to reuse objects in other environments than the original one. On the other hand the specialization and structural decomposition 1. This work has been funded as part of the OMI-ESPRIT Project REQUEST under contract 20616 of objects requires communication among the objects. The communication enables objects to use services of other ob- jects, to inform other objects about something, or in concurrent domain to synchronize with each other. Such a communication mechanism is called message passing in the object-oriented domain. The basic idea is that objects are able to send and receive messages to provide or get some information. Of course, it is desirable that a message passing mechanism preserves as much as possible of the loose coupling of the object with its environment in order to obtain universally reusable objects. Generally, for the communication of concurrent be- haviours/processes two mechanisms can be used [6]. The first is communication using shared memory and the second the message passing via channels or communication pathways. For the communication of concurrent objects the latter choice seems to be more appropriate because passing messages is one of the basic concepts of the object-oriented paradigm. Figure 1 shows an abstract picture of message passing. Object X in the role of a client needs a service of object Y in the role of a server. To invoke the corresponding method of Y, X sends a message via a communication pathway to Y. The message exchange is controlled by a protocol. After Y has received the message, the correct method has to be invoked. This functionality is provided by a dispatcher. As indicated by the figure we will consider the dispatching as part of the message passing mechanism. After execution of the method return values-if any-are replied. Figure 1: Message passing send message to invoke method return values protocol communication pathways Object Y attributes methods Object X attributes methodsCopyright 1998 EDAA. Published in the Proceedings of Date' 98, February 23-25, 1997 in Paris, France. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works, must be obtained from EDAA. In the sequential domain often the terms message passing and method call are used as synonyms. But in concurrent domain both terms have different semantics. While with the method call the invoked method needs the computational thread of the caller to execute the method, invocation of a method via message passing does not need the computational thread of the caller because the target object can use its own. 3 Aspects of message passing In this chapter several aspects of message passing will be considered separately. This consideration shows on the one hand the design space for message passing and on the other hand the large impact to the whole language. 3.1 Consistency to language Message passing is an integral part of an object-oriented language. The relationship between message passing and the other parts of the language is bilateral. While it is indispensable that message passing is consistent to the other object-oriented concepts, these concepts can be used for the specification of message passing. If for example an language embodies different encapsulation concepts (classes, objects) message passing has to take this in account-at least its implementation. A class concept based on abstract data types needs another realization of the message passing compared to a class concepts which represents (structural) hardware-components. The message passing mechanism proposed for Objective VHDL in this paper will give an example how the object-oriented concepts like classification, inheritance, and polymorphism can be used to implement it. Another desired feature of message passing is the con- sitency with the techniques of object-oriented modelling. This means consistency in terms of refinement and extensi- bility. If a model is extended by components which need a more refined or different way for communication, the message passing mechanism must be adaptable and by object-oriented means. Further, consistency means that the abstraction level of message passing fits to the abstraction levels the whole language is intended for. The programming interface should be abstract and easy to use. For example, the appropriate encapsulation of the transmission of messages by send/re- ceive methods. Abstraction and encapsulation, however, shouldn't be considered in isolation because they have large impact on other aspects of message passing (e.g. flexibility, simula- tion/synthesis). 3.2 Communication pathways Passing a message from one object to another and performing the communication protocol requires a communication pathway which interconnects the objects. If communication is restricted to 1:1 relation 2 the target object can be identified directly by the communication path- way. But the abstraction and definition of communication pathways differs significantly in literature. In case a method call has the semantics of a procedure/ function call, the procedure/function call mechanism and the name of the target object/method can abstractly be considered as the communication pathway. If the communicating objects represent hardware com- ponents, in VHDL terms-entities, another representation of communication pathways is necessary. In [14] some kind of identifier for an entity object, called a handle which can be exchanged among entities, is proposed. If an object has the handle of another object, the handle can be used to address the other object to pass a message. So the handle can be considered as the communication pathway. This solution is abstract because this communication pathways have no direct physical representation and flexible because it allows to establish communication pathways during runt- ime. Generally, it allows to send messages to objects which are dynamically generated during runtime. The dynamic generation of objects, however, might be a powerful feature for system design, but it is really far away from hardware. Another solution proposed in [12][9][11] is to use the VHDL mechanism to exchange data between components, i.e. to interconnect components by signals. From the send- er's point of view the target object can be addressed by the port which connects both objects. Of course, this approach isn't as flexible as the handle solution to address compo- nent-objects, but it is very close to hardware. Although signals can be used for communication, there is still a gap between the abstraction provided by VHDL signals and communication in object-oriented sense. 3.3 Protocol In software, sending a message to an object has the semantics of a procedure or function call 3 . Results can be given back by assignments [12]. In hardware message passing among concurrent components/objects needs specialized protocols. Of course, the necessity of protocols results in a much tighter relation between the objects than it is desired by object-oriented paradigm (cf. Chapter 2). Sending a message needs the knowledge and the ability to perform the target object's communication protocol. 2. A 1:1 relation not necessarily means a point to point communication because an object can consist of concurrent processes. 3. Without consideration of distributed programs. Figure 2: Encapsulation of an object by protocol From a communication point of view the protocol can be seen as the encapsulation of an object (Figure 2). In the following subchapters several aspects of a protocol for message passing will be illuminated 3.3.1 Abstraction Since the object-oriented paradigm addresses modelling on higher abstraction levels, the details of the communication protocol should be encapsulated and the specialization of the protocol has to correspond with the ab- straction. The protocol should be applied through an abstract interface. 3.3.2 Flexibility A universal message passing mechanism for hardware design needs the possibility to integrate different protocols for message exchange. An abstract model of an MMU on system level may need another protocol than a simple register on RT level. Further, in perspective of a top down design methodology, it should be possible to refine a protocol towards more detail according to the description level. The same need occurs if co-simulation of abstract models together with already synthesized models is desired. The flexibility, however, to choose or refine a new protocol is at the expense of encapsulation of the protocol. Even if the interface to use a protocol can be encapsulated it is not possible to hide the protocol by the language completely 3.3.3 Synchronization In concurrent object-oriented domain objects need to synchronize with each other to describe their behaviour dependent on the state of other objects. We would like to differ three synchronization modes: . synchronous . asynchronous . data-driven With synchronous communication the sender object needs to wait until a receiver object is ready to receive a message. In most synchronous communication mechanisms [7] the sender object/process is blocked from the moment of sending a message request until the service which is intended to be invoked by the message is finished and the results are given back. With asynchronous communication the sender does not wait for the readiness of the server object to receive a message. In order to avoid the loss of messages, this mode requires queuing of messages within the communication pathway or the receiver. An additional advantage of such queues is the potential flexibility to dequeue messages in another order than FIFO. On the other hand the message queues can have large impact on simulation and synthesis aspects. Generally, asynchronous communication is non-blocking [3]. So the sender object can continue its computations directly after sending. The return of results requires to send explicitly a message from the server to the client. The data-driven synchronization [5] allows a sender object to run its computations until the results of a previously sent message request are needed. In this case the sender has to wait until these results will be provided by the server object. In literature the initial sending of a message request is described to be asynchronously [3]. However, a synchronous (but non-blocking!) sending of a message request is also conceivable. In summary, the data-driven synchronization allows more flexibility than standard synchronous/asynchronous communication. 3.3.4 Concurrency To be consistent with (Objective) VHDL, a message passing mechanism must preserve and support the concurrency provided by (Objective) VHDL. The relation between communication and concurrency is ambivalent. On the one hand concurrent objects/process- es are the reason for the necessity of message passing, on the other hand the message passing mechanism can restrict the concurrency. If an object contains only one process (dispatcher process) which receives requests and dispatches them, all requests will be sequentialized. For concurrent object-oriented languages it is expected that the objects can have own activity and can run in paral- lel. But besides the concurrency of parallel running objects there can be concurrency inside the objects if they contain concurrent processes. This intra-object concurrency may allow an object to execute requests for services in parallel. For example, a dual-ported RAM allows concurrent read and write operations. This can be modelled with concurrent dispatching processes. However, allowing parallel method execution raises the potential problem of nondeterministic behaviour of the object, due to concurrent access to the same attributes. But VHDL already provides mechanisms to solve concurrent access to signals and variables. dispatching dispatching dispatching Methods protocol protocol protocol protocol object communication pathways Concurrent (write) access to signals can be handled by resolution functions. With variables the proposed protected types can be used [15]. So at least atomic access to variables can be ensured. But the problem with nondeterminism is still unsolved because the value of a shared variable depends on the activation order of the accessing processes. A possibility to avoid nondeterminsim is to ensure that concurrent methods have only access to exclusive at- tributes. This can be implemented by grouping all methods which have access to the same attributes. Each group gets an own dispatching process and maybe a queue. So only methods without access conflicts can be executed in parallel 3.4 Simulation/Synthesis efficiency Simulation efficiency and synthesizability are general aspects for the quality of object-oriented extensions of a HDL. These are being influenced by implementation decisions of message passing/protocol and communication frequency between objects. A complex protocol enriched with a lot of detailed timing informations and unlimited message buffers to avoid the loss of messages may decrease the simulation speed significantly. Moreover, without limitation of maximal buffer size the protocol is not translatable into hardware. 4 Classification scheme In the above chapters some of the aspects a message passing mechanism for an object-oriented HDL has to deal with have been proposed and discussed. These can be used to define a classification scheme for different message passing mechanisms. The first criterion for classification is the flexibility of the message passing mechanism. The following cases will be distinguished: . flexible: different protocols are possible and can be refined . fixed: the protocol is not modifiable. . semi-flexible: one protocol, with potential for refinement or different not-refinable protocols. The second criterion is the ability of an object to accept and perform several requests concurrently. It should be only distinguished between: . yes: it is possible. . no: it is not possible. The third criterion is the synchronization of the message passing. It can be: . blocking: the sender is always blocked until the return values are received. . non-blocking: the sender continues execution after sending a request. Maybe at a certain point of execution he has to wait for the results. . both: depending on the kind of message or on the kind of method invocation (method call vs. message pass- ing, cf. Introduction) blocking or non blocking communication is possible. The last criterion is whether there are queues to buffer messages if the receiver is busy. It will be distinguished between . no: no queues are provided. . one: one queue per object. . many: more than one queues per object. 5 Other OO-VHDL approaches During the last years several proposals for an object-oriented extension to VHDL have been made [1][2][4][12][14]. All the proposals need to define message passing to be object-oriented. Because not all of the proposals can be considered here, two typical but completely different proposals are selected to describe their message passing mechanisms. Because this cannot be done isolated, it is necessary to introduce the core concepts of the proposals up to a certain degree. But it is outside the scope of the paper to draw a complete picture of the special proposals. The Vista approach [14] introduces a new design unit called Entity Object (EO) which is based on the VHDL entity and it's architecture. In addition to the entity the EO may contain method specifications called operation speci- fications. Operations are similar to procedures, but they are visible outside the EO. In contrast to procedures operations can have a priority and a specified minimal execution time. For invocation of operations the EOs are not interconnected with explicit communication pathways (signals). To address an EO, each EO has an accompanying handle which is a new predefined type that can be stored in signals or variables and can be part of composite types. Handles can be exchanged to make the corresponding EO addressable for other objects. A special handle to address an object itself (self) is predefined. The parent class can be addressed by the new keyword 'super'. A message send request is performed by a send statement which includes the handle of the target EO, the name of the operation, and the parameter values. Each EO has one queue to buffer incoming messages. The messages in the queue are dequeued by their priority. Messages with the same priority are dequeued by FIFO. One queue per EO means that concurrent requests are se- flexibility parallel methods (per object) synchronization queues (per object) flexible fixed semi-flexible yes no blocking non-blocking both no one many Table 1: Classification scheme quentialized. If an EO needs to invoke an own operation (send self), this request will not be queued. It will be treated like a procedure call and immediately executed. This mechanism avoids deadlocks with recursive method calls. Messages can be of blocking (default) or non-blocking mode (immediate). But immediate messages are restricted to have in-parameters only. The blocking mode cannot be changed during inheritance. For EO synchronization a rendezvous concept is pro- vided. Accept and select statements similar to Ada are used to establish a rendezvous. Finally, it should be remarked that the proposed message passing mechanism is neither sythesizable nor it is intended to be. The abstract concepts of dynamic communication pathways represented by the handle concept and the unlimited message buffers have no counterpart in hardware. Another approach was developed at Oldenburg University [12]. It is based on the VHDL type concept. Records are used to represent the objects. To allow a record to be expandable by inheritance, it is marked as a tagged record. The corresponding methods, which are simple procedures, must be defined in the same design unit (package). Because the methods cannot be formally encapsulated by the object it belongs to, the parameter list of each method contains a parameter of the object's type which assigns the method to the object. For each tagged record a corresponding class-wide type exists, which is the union of all types derived from the tagged record. With the attribute 'CLASS the class-wide type can be referenced. Polymorphism is based on the class-wide types. If a method is called with an actual of class-wide type (mode in, inout), the actual type is determined during runtime and the correct method will be invoked Inter-process communication can be modelled consistently with VHDL by signals. Abstraction and expandability are supported by use of polymorphic signals (class-wide type). Sending a message to another object has the semantics of a procedure/function call or in our terminology a method call. The requested method is executed by the sender which is blocked until the end of the request. Several methods of an object can be performed concurrently if they are requested by different processes. But in case of concurrent assignment to a signal (instantiation of a tagged record), resolution functions are necessary. Even if in [12] a special protocol mechanism is proposed other protocols can be integrated because the protocol is not built into the language. The classification here refers to the proposed master-slave protocol. 6 Objective VHDL Objective VHDL is the object-oriented extension to VHDL developed in the EC-Project REQUEST 4 [8][10]. Objective VHDL combines the structural object approach [14] with the type object approach [12]. Both language extensions have shown their suitability for hardware design. The structural objects are usual VHDL entities. Attributes and methods of an entity class are declared within the declarative part of the entity. They correspond to VHDL object declarations (signals, shared-variables and constants) and procedure or function declarations respec- tively. The implementation of the methods follows in the corresponding architecture. Single inheritance for entities and architectures is provided but no polymorphism on entity objects. A type class consists of a declaration and a definition, likewise. Class types are declared like usual types but the declaration of attributes and methods are assembled between the new 'is class' and `end class' constructs. The implementation of the declared methods or private methods, which are not declared in the interface, follows in the corresponding class body. As well as for the entity classes, single inheritance is provided for the type classes. Each class type has an associated class-wide type, marked by a new attribute 'CLASS. The class-wide type is the union of the type itself and all derived subclasses. Similar to the [12] approach class-wide types are used to realize polymorphism on type classes. A variable or signal of class-wide type T'CLASS can hold any instantiation of a class which is derived from T or T itself. Calling a method of a directly visible instantiation of a class type has the semantics of a simple procedure/function call (blocking). Calling a method of an entity object or a type object instantiated in another entity is more difficult. A method of an entity cannot be called directly because the entity encapsulates the procedures/methods completely. The only interface to the outer world are the ports and generics of the entity. Breaking this encapsulation would 4. Objective VHDL is defined in [9]. The Objective VHDL Language Reference Manual is intended as an extension to the VHDL LRM. Although the current status of Objective VHDL within the REQUEST project is stable minor changes in the language are possible in future. flexibility parallel methods (per object) synchronization queues (per object) fixed no both one Table 2: Classification of [14] flexibility parallel methods (per object) synchronization queues (per object) flexible (yes) blocking no Table 3: Classification of [12] change VHDL. A solution to that problem, which was chosen in [14], was to introduce a new design unit-the Entity Object where the methods (operations) of an EO are visible outside the EO. Due to the additional implementation costs for a new design unit, this solution was discarded for Objective VHDL. 6.1 Message passing mechanism To provide the flexibility to use an appropriate communication for message exchange, the message passing is not fixed in the language definition. Nevertheless, a way to implement a flexible message passing will be shown by usage of the other object-oriented features. The main ideas will be now described in more detail. Basically, the message passing mechanism consists of three parts: . the communication structure which defines the connection of objects with communication pathways (Fig- ure 3), . protocol and messages for exchange (Figure 4), . dispatcher (Figure 3). 6.1.1 Communication structure To enable communication between objects which are represented by entities or type objects inside different proc- esses, the communication pathways between the objects are implemented by VHDL signals. Beside the signal which carries the message additional signals for the protocol may be necessary. To avoid resolution functions, the signals are unidirectional. Consequently, this requires two opposite communication pathways if the message request produces return values. This physical connection is depicted in Figure 3. Figure 3: Communication structure So the target object of a communication can be addressed by the port which connects the sender with the re- ceiver. Further a communication pathway is restricted to connect only two objects. However, VHDL signals do not provide the abstraction expected from object-oriented message passing. To enable the connecting signals to provide the required abstraction, they will be implemented as polymorphic type classes. So such a signal gets the ability to hold different messages and to encapsulate a communication protocol. The messages and the communication protocol can be modelled by a structure which is shown in Figure 4. But in following main emphasis is given to the modelling of the messages. 6.1.2 The messages and the protocol The messages and the protocol are implemented and encapsulated by an abstract type class ('message'). The class is abstract because it is not intended to be instantiated and it should serve as base class for the classes representing real messages. Furthermore, the class provides the interface which allows to apply the message passing among the ob- jects. The interface is inherited to the derived classes and declares methods like 'send' (a message), `receive' (a mes- sage, or 'dispatch' etc. Figure 4: Modelling of messages To enable a communication pathway to hold all messages to a target object, the type class 'message' is refined by inheritance. Because (target) classes differ in the methods their instantiations can receive, for each kind of (target) receiver method M1 (.) method M2 (.) dispatching signal "message_to_receiver'CLASS" signal "message_to_receiver'CLASS" unidirectional communication pathway for message & protocol sender Entity Object Entity Object class receiver method M1 (X in .; Y inout .); method M2 (Z inout .); message exec_method send dispatch return_results receive message_to_receiver Y exec_method general functionality address special classes address methods parameters Z exec_method object an abstract subclass is derived (Figure 4, 2nd-stage). This class implements no additional functionality, its purpose is only to distinguish the different kinds of objects. (Different objects which are instantiations of the same class are represented only one time!) Finally, a message must correspond to one method which should be invoked and the message must contain the parameters for method invocation. To allow this, each class of the second stage of Figure 4 is refined with subclasses representing the methods of the (target) object (Figure 4, 3rd stage). The subclasses are named by the methods and the method parameters are represented as attributes. A signal which is an instantiation of such a class can hold a message corresponding to a special method and provides the communication protocol if one was defined in the superclasses. A communication pathway, however, should be able to hold all messages which can be sent to an object. By instantiation of the communication pathway as signal of a class-wide type belonging to the representation of a kind of object Figure 4, 2nd stage) this ability is given to the signal. Finally, each of the classes representing a method (Fig- ure 4, third stage) implements a method ('exec_method'), which invokes the corresponding method of the target ob- ject. As mentioned before, the methods of an entity object cannot be invoked directly because the entity encapsulates its methods completely. Nevertheless, to allow this method invocation a new construct 'for entity . end for' is intro- duced. The semantics of the construct is to make the declarations of an entity class visible inside a type class in order to allow its invocation. The implementation of such a method must be available wherever 'exec_method' is invoked with an instance of the message class. Results produced by the method execution can be sent back by the same mechanism. 6.1.3 Dispatching Each entity object contains one or more dispatching processes. These processes are sensitive to the ports which carry the incoming messages and must be specified by the user. To dispatch the incoming messages, functionality provided by the class 'message' can be used if implemented (method 'dispatch'). But basically, within the dispatcher process only the method 'exec_method' of the received message has to be invoked which calls the desired method of the entity object. By the type of the received message it can be decided during runtime which 'exec_method' has to be called. The number of the dispatching processes can be chosen by the user and determines the number of concurrently executable methods. If concurrent methods are allowed, the user has to take care about conflicting concurrent access to the attributes. To ensure atomic access to attributes, shared variables with the protect mechanism (cf. Chapter 3.3.4)[15] can be used. 6.1.4 Synchronization Synchronization can be very flexible. Synchronous, asynchronous, and data driven sychronization (c.f Chapter 3.3.3) can be modelled. The proposed mechanism does not require that an object is blocked after sending a message to a concurrent ob- ject. But it should wait at least until it is ensured by the protocol that the target object accepts the request. Of course, additional synchronization is necessary if the results of the request are needed. For asynchon communication message queues have to be integrated into the message passing mechanism. 6.2 Summary of message passing The provided message passing mechanism provides easy means to send messages. The sender identifies the target object through the port(s) by which they are connected. needs only to call the 'send' method of the class 'message' with the port names and the message itself as pa- rameters. The 'send' method will perform the protocol and assign the message to the interconnecting signal if the receiver is ready to accept it. After sending the sender can be sure the message is received and can continue its execution until potential results of the request are needed. In this case the sender is blocked until the results are available. The receiving object possesses a dispatching process which is sensitive to the connecting port signal. If a new message arrives, the dispatching process performs the re- ceiver's part of the communication protocol and invokes the desired method. Of course, this functionality can be encapsulated by a 'receive' or `dispatch' method of the class 'message'. The potential results can be send back implicit and immediately after their computation or explicit by a new communication Finally, it is important to note that the modelling of messages (cf. chapter 6.1.2) is really straight forward. So the effort to use this communication mechanism can be reduced significantly by a tool which produces the messages automatically. 6.3 Classification According to the proposed classification scheme the message passing can be classified as shown in Table 4. Different protocols can be defined and stored in a library. If required they can be refined. Concurrency of method execution depends only on the number of dispatching processes the user defines. In the proposed communication mechanism a method call is blocking and passing a message can be blocking as well as not blocking. Queues to buffer messages and to allow asynchronous communication can be modelled in principle but are not integrated in the communication up to now. So it is important to note that a special implementation of the message passing mechanism may result in a slightly other classification. 7 Future work A precompiler which translates Objective VHDL to VHDL will be implemented. With the availability of this tool the desired benefits of Objective VHDL and especially the message passing mechanism can be evaluated. The proposed mechanism for message passing has potential for improvements. For better protocol reuse a stronger separation of the protocol and the messages will be useful. The modelling effort which is caused by the restriction of unidirectional communication pathways can be reduced if resolution functions for the connecting signals can be defined. 8 Conclusion By analyzing and discussing the several aspects of abstract communication the design space for message passing has been shown and a classification scheme for message passing mechanisms has been developed. The scheme has been applied to the message passing mechanisms of two of the currently most discussed proposals for object-oriented extensions to VHDL. Finally, a new idea for message passing mechanism developed for Objective VHDL was introduced and classi- fied. The new approach targets especially flexibility of pro- tocols, reuse of protocols and consistency to VHDL (concurrency). 9 --R SUAVE: Painless Extensions for an Object-Oriented VH- DL Object Oriented Extensions to VHDL - The LaMI proposal An Object-Oriented Model for Extensible Concurrent Systems: The Composition-Filters Approach Concurrency And Reusability: From Sequential To Parallel Specification and Design of Embedded Systems Communicating Sequential Processes Language Architecture Document on Objective VHDL Inheritance Concept for signals in Object Oriented Extensions to VHDL Shared Variable Language --TR Communicating sequential processes Concurrency and reusability: from sequential to parallel Real-time object-oriented modeling Specification and design of embedded systems Inheritance concept for signals in object-oriented extensions to VHDL Object oriented extensions to VHDL, the LaMI proposal --CTR Cristina Barna , Wolfgang Rosenstiel, Object-oriented reuse methodology for VHDL, Proceedings of the conference on Design, automation and test in Europe, p.133-es, January 1999, Munich, Germany Annette Bunker , Ganesh Gopalakrishnan , Sally A. Mckee, Formal hardware specification languages for protocol compliance verification, ACM Transactions on Design Automation of Electronic Systems (TODAES), v.9 n.1, p.1-32, January 2004

message passing;communication;object-oriented hardware modelling

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Optimal temporal partitioning and synthesis for reconfigurable architectures.

We develop a 0-1 non-linear programming (NLP) model for combined temporal partitioning and high-level synthesis from behavioral specifications destined to be implemented on reconfigurable processors. We present tight linearizations of the NLP model. We present effective variable selection heuristics for a branch and bound solution of the derived linear programming model. We show how tight linearizations combined with good variable selection techniques during branch and bound yield optimal results in relatively short execution times.

Introduction Dynamically reconfigurable processors are becoming increasingly viable with the advent of modern field-programmable devices, especially the SRAM-based FPGAs. Execution of hardware computations using reconfigurable processors necessitates a temporal partitioning of the specification. Temporal partitioning divides the specification into a number of specification segments that are destined to be executed one after another on the target processor. While the processor is being reconfigured between executing the specification segments, results to be carried from one segment to a future segment must be stored in a memory. Reconfiguration time itself, along with the time it takes to save and restore the active data, is considered an overhead; it is desirable to minimize the number of reconfiguration steps, ie., the number of segments resulting from temporal partitioning, as well as the total amount of data to be stored and restored during the course of execution of the specification. In addition to the traditional synthesis process, a temporal partitioning step must be undertaken to implement hardware computations using reconfigurable processors. While techniques exist to partition and synthesize behavior level descriptions for spatial partitioning on multiple chips, no previous attempts have been published for temporal partitioning combined with behavioral synthesis. The paper presents a 0-1 non-linear programming (NLP) formulation of the combined problem of temporal partitioning and scheduling, function unit allocation and function unit binding steps in high-level synthe- sis. The objective of the formulation is to minimize the communication, that is, the total amount of data This work is supported in part by the US Air Force, Wright Laboratory, WPAFB, under contract number F33615-97-C-1043. transferred among the partition segments, so that the reconfiguration costs are minimized. We propose compact linearizations of the NLP formulation to transform it into a mixed integer 0-1 linear programming (LP) model. We show the effectiveness of our linearizations through experimentation. In ad- dition, we develop efficient heuristics to select the best candidate variables upon which to branch-and-bound while solving the LP model. Again, we show the effectiveness of our heuristics through experimental results. The paper is organized as follows. In Section 2 we discuss related work, and in Section 3 we provide an overview of our approach. The basic formulation is presented in Section 3 and its solution in Section 4. Experimental results are presented in Section 7 and Section 9. Previous Work Synthesis for reconfigurable architectures involves synthesis and partitioning. The partitioning can be both temporal and spatial. There has been significant research on spatial partitioning and synthesis, though the issue of temporal partitioning has been largely ig- nored. Early research [11, 12] in the synthesis domain solved the spatial partitioning problem independently from the scheduling and allocation subproblems. The problem of simultaneous spatial partitioning and synthesis was first formulated as an IP by Gebotys in [1]. It produced synthesized designs which were 10% faster than previous research. However as mentioned by Gebotys in [2], the size of the model was too large and could solve only small examples. In both [1, 2] the dimension of the problem is reduced by not considering the binding problem. Though the model can handle pipelined functional units and and functional units whose latency is greater than one cycle, it cannot handle design explorations where two different types of functional units can implement the same op- eration. For example, we cannot explore the possibility of using a non-pipelined and a pipelined multiplier in the same design. Also heuristics were proposed to assign entire critical paths to partitions. This might lead to solutions that are not globally optimal. Both [1, 2] focus on synthesis for ASICs and hence attempt to minimize area. For reconfigurable processors based on FPGA technology, area (resources on the FPGA devices) should be treated as a constraint which must be satisfied by every temporal segment in the temporal partition. c c c c operation graph of T 2 Figure 1: Behavioral Specification The work of Niemann [3] presents an IP-based methodology for hardware software partitioning of codesign systems. MULTIPAR [4] also has a non-linear 0-1 model for spatial partitioning and synthesis, without involving binding. The linearization technique is however not the tightest for such a formulation, (see section 4 for details). To achieve faster runtime they solved the formulation by heuristic techniques, leading to suboptimal results. Since a functional unit is not explicitly modeled in the above [1, 2, 4] formulations, they cannot determine the 'actual' area utilization of a partition. This factor though not critical for ASIC designs, is critical for reconfigurable processors based on FPGA technol- ogy. For example, in an optimal solution, a temporal segment may contain 1 multiplier and 5 adders and another temporal segment may contain 2 multipliers and 2 adders. Our formulation, which explicitly models binding of operations to functional units, can determine whether a functional unit is actually being used in any temporal segment. Moreover, we explicitly model the usage of each functional unit in each temporal segment and hence can explore the design space using 5 adders and 2 multipliers, although all these functional units may not simultaneously fit on the processor resources (function generators or CLBs in the FPGAs). Our model can automatically determine the above optimal solution, that simultaneously meets the resource constraints at the FPGA resource level as well as at the functional unit level. 3 System Specification The behavior specification is captured in the form of a Task Graph, as shown in the Figure 1. The vertices in the graph denote a set of tasks, T . The dependencies among the tasks are represented by directed edges. Each task can be visualized as being composed of a number of operations which should stay together in one temporal partition. The edge labels in the task graph represent the amount of communication required if the two tasks connected by an edge are placed in different partitions. Let I be the set of all operations in the specification. The cost metrics of the target FPGA, FPGA resource capacity (C) and temporary on-board memory size (M s ) are the inputs to our system. Typical resources for an FPGA are combinational logic blocks and function generators. We assume a component li- Heuristic temporal partition estimator preprocessing Model formulation, NLP linearized to an ILP solution by a LP solver Solution Found ? Behavior specification FPGA cost metrics Characterized component library number of partitions Figure 2: Flow of the Temporal Partitioning and Synthesis system brary consisting of various functional units which can execute the operations in the specification. The components in the library are characterized by cost metrics in terms of delay times and FPGA resource requirements An outline of the temporal partitioning system is shown in Figure 2. The system proceeds by first heuristically estimating the number of segments (N), which becomes an upper bound on the number of temporal segments in the NLP formulation. It uses a fast, heuristic list scheduling technique to estimate the number of segments. Before the problem can be formulated, we need to determine the ASAP and ALAP schedules for the op- erations. These will be used to set the mobility ranges for the operations in the formulation. This is done in a preprocessing step over the combined operation graph of the specification. From this schedule we can also determine the set of functional units F , which must be used for the design exploration. While partitioning the specification, we honor the task boundaries in the specification. That is, a task cannot be split across two temporal segments. How- ever, if two different tasks are together on the same segment, then they share control steps and functional units among them. If it is desired to permit splitting of tasks across segments, then each operation in the specification may be modeled as a task in our system. In this case, each task would have only one operation associated with it. The entire formulation developed in this paper will work correctly. Formally the system is defined as follows - directed edge between tasks, t exists in the task graph. t i is at the tail of the directed edge and t j is at the head, when the execution of task depends on some output of t i . directed edge between operations, exists. number of data units to be communicated between tasks t i and t j . ffl Op(t), the set of all the operations in the operation graph of a task t. F , is the set of functional units required for the most parallel schedule of the operation graph. ffl Fu(i), the set of functional units from F , on which operation i can execute. (k), the set of operations which can execute on functional unit k. ffl CS(i), the set of control steps over which operation i can be scheduled. Ranges from is the user-specified relaxation over the maximum ALAP for the schedule. ASAP (i) and ALAP (i) are the As Soon As Possible and As Late As Possible control steps for operation i. (j), the set of operations which can be scheduled on control step, j. ffl N upper bound on the number of partitions. The partitions are the index of the partition specifies the order of execution of the partitions. Note that the generated optimal solution may have fewer than N partitions. scratch memory available for storage between partitions. ffl FG(k), the number of function-generators used for functional unit k, obtained from the characterized component library. ffl C, is the resource capacity of the FPGA. sectionNon- Linear 0-1 model In this section we describe the variables, constraints and cost function used in the formulation of our model. 3.1 Variables We have three sets of decision variables, which model the three important properties - y tp models the partitioning at the task level, x ijk models the synthesis subproblem at the operation level, w pt1 t 2 models the communication cost incurred if two tasks connected to each other are not placed in the same partition. All are 0-1 variables. placed in partition p, I is placed in control uses functional placed in any partition placed in any of As will be seen y tp and x ijk are the fundamental system modeling variables. All other variables are secondary and are constrained in terms of the fundamental variables 3.2 Temporal Partitioning The variables y tp model the partitioning behavior of the system. Temporal partitioning has the following constraints. Uniqueness Constraint: Each task should be placed in exactly one partition among the N temporal partitions. Temporal order Constraint: Because we are partitioning over time, a task t 1 on which another task t 2 is dependent cannot be placed in a later partition than the partition in which task t 2 is placed. It has to be placed either in the same partition as t 2 , or in an earlier one. p2!p1-N Scratch Memory Constraint: The amount of intermediate data stored between partitions should be less than the scratch pad memory M s . The variable , if 1, signifies that t 1 and t 2 have a data dependency and are being placed across temporal partition p. Therefore the data being communicated between have to be stored in the scratch memory of partition p. The sum of all the data being communicated across a partition should be less than the scratch pad memory. (w pt1 t 2 Notice in the Figure 3, how the variable w , models not only the communication among tasks which are on adjacent temporal partitions, but also on the non-adjacent ones. 3 tasks are to be placed optimally on 3 partitions. On the left of the figure are the original equations used to model the constraints for the exam- ple. The equations on the right of the figure show the variables which will be 1 in the mapping of tasks to partitions shown and the constraint which have to be satisfied. The variable w , is shown to range from 2 to N, because the data at partition 1 is actually the external input to the system. We can reasonably assume that there is enough memory for the external input at any time, since this is known a priori and is not a function of the partitioning of the system. w pt1 t 2 are 0-1 non-linear terms constrained as - Temporal Partitions s Figure 3: The memory constraints to be satisfied if tasks are mapped to partitions as shown Equation (4) constrains w pt1 t 2 to take a value 1, if any of the non-linear product terms in the associated equations are 1. Equation (5) constrains w pt1 t 2 to a value zero, when all the associated product terms are 0. The equation alone (4) does not stop w pt1 t 2 , from taking a value 1 when all the product terms are 0 because 1 - 0 is a valid solution to the constraint (4). 3.3 Synthesis For the sake of clarity and ease of understanding we will not describe in the equations for synthesis, the formulations for pipelining, chaining, and latency of functional units. This formulation is easily extendible to incorporate those features, as described in [6] and We assume for the current model that the latency of each functional unit is one control step, and the result of an operation is available at the end of the control step. Unique Operation Assignment Constraint: Each operation should be scheduled at one control step and on only one functional unit. Therefore only one variable x ijk for an operation i, will be 1. Temporal Mapping Constraint: This constraint prevents more than one operation from being scheduled at the same control step on the same functional unit. Dependency Constraint: To maintain the dependency relationship between operations, an operation i 1 whose output is necessary for operation i 2 , should not be assigned a later control step than the control step to which i 2 is assigned. 3.4 Combining Partitioning and Synthesis It is essential for partitioning for an FPGA to meet the area constraints of the FPGA. It is not necessary that all the functional units F , used in the design exploration are finally used in a partition. To determine whether a functional unit has been used in a partition we define the following decision variables - used in partition to perform some operation uses functional unit k 2 F to perform some operation These variables are constrained by, (y tp The variable u pk , defines the functional unit usage in a partition and the variable defines the functional unit usage in a task. These variables are also secondary and are defined in terms of the fundamental modeling variables y tp and x ijk . Equation (9), constrains u pk to take a value 1, when any of its associated non-linear terms is 1. Equation (10) is needed to make sure, that at least one task on that particular partition uses the functional unit. The derivation for variable will be described in Section 4. Resource Constraints We introduce resource constraints in terms of variables u pk . Typical FPGA resources include function generators, combinational logic blocks (CLB) etc. Similar equations can be added if multiple resource types exist in the FPGA. ff is a user defined logic-optimization factor in the range 0-1. Typical values of ff using Synopsis FPGA components are in the range 0.6-0.8 [5]. Unique Control Step Constraint: We introduce constraints to make sure that each control step is mapped uniquely to a temporal segment. We use a new derived variable c tj to formulate this constraint. operation mapped to control step j x ijk (12) If the operations of two distinct tasks use same control steps, then these tasks should be on the same partition. In this paper, we have not considered flip-flop resource constraints. To consider flip-flop resources, the formulation must estimate the number of registers necessary to synthesize the design. It is straight-forward to add register optimization to our formulation on the lines proposed by Gebotys et al. [6]. Cost Function Minimize the cost of data transfer between temporal partitions. This cost function will get an optimal solution using the least number of partitions and the least amount of inter-partition data transfer. (w pt1 t 2 4 Solving the 0-1 Non-linear Model We can use various solution techniques in the mathematical programming field, for solving models which are non-linear in their objective function and con- straints. The main approaches are (1) Linearization methods, (2) Enumeration methods, (3) Cutting plane methods. Refer to [8] for an interesting survey of all the approaches. Due to the existence of a good set of linearization techniques and the easy availability of LP codes for solving linear models we have chosen the linearization technique, though enumerative methods and cutting plane methods are viable alternatives. Linearization of Equations (9-10) For each non-linear product term of the form a b, we generate a new 0-1 variable c as, c = a b; which can be written in Fortet's linearization method[8] 1 as Constraint both a and b are 1. Constraint (16) implies either a or b 1 For each distinct product of variables, replace it by a new 0-1 variable x n+k and add the constraints: become We need both the constraints to get the correct solution. However Glover and Wolsey [9] proposed an im- provement, by defining c as a continuous real-valued variable (with an upper bound of 1), instead of, an integer variable by replacing equation (16) by the following two constraints, while retaining equation (15) intact - a - c (17) and b - c (18) Glover's linearization has been shown to be tighter than Fortet's. This has also been borne out by our experimentations. Equations (9) and (10) can be now replaced by the following compact linearized equations. Here z ptk is a continuous real-valued variable bounded between 0 and 1 (0 - z ptk - 1). z Constraint (19) implies z are 1. Constraints (20) and (21) imply z when either y tp or We need all three constraints to get the correct solution. Linearization of Equations (4) - (5) This linearization can be done exactly as we have done for equations by defining a new continuous variable for each distinct non-linear term. As will be shown shortly however in our final formulation we use lesser number of variables in linearizing w pt1 t 2 Formulating constraints for decision variable For any task t and functional unit k, variable is 1 if any of the x ijk variables used to denote the synthesis variables of task t using functional unit k are 1. Consider a small example, where variables are the synthesis variables for task 1 and functional unit k. Then logical - and product being interchangeable for 0-1 terms. Using a and using the Glover's linearization we get - Characteristics ILP Model Graph No. N A+M+S L Var Const Run-Time Table 1: Some Preliminary results Based on the above discussion we get the following constraints for 5 Preliminary Results To verify our formulation by experimentation, we generated various random graphs. We ran some experiments with the current formulation to see how it performs. However as we see in Table 1, only one of the formulation solved in reasonable time, even though the graphs for which the experiments are done are not very large. Others were terminated when their run time became too large. Refer to Table 4, for the actual sizes of the graphs in this experiment. In the result tables, N denotes the number of partitions, A, M, S the number of adders, multipliers and subtracters respectively used in the design exploration. L, is the user specified latency margin bound. Var and Const denote the number of variables and constraints respectively that are generated for each graph by the ILP formulation. The run times are in seconds and all experiments have been run on an UltraSparc machine running at 175 Mhz. We use lp solve, a public domain LP solver [10] for solving our ILP formulation. 6 Additional constraints that tighten the model further When an ILP problem is solved by using LP tech- niques, we are solving the LP relaxation of the model. The LP relaxation is the same as the ILP, except the 0-1 constraints on the variables are dropped. The LP relaxation has a much larger feasible region, and the ILP's feasible region lies within it. An important method of reducing the solution time is to modify the formulation with constraints which cut away some of the LP feasible region, without changing the feasible region of the ILP. This is called tightening the LP model. After studying the model carefully we could identify some more constraints which cut off a large amount of non-optimal integer and non-integer solutions and helped in reducing the time required to solve the model. ffl If a task t 1 is placed on some partition p, and exists then it means that t 2 definitely can only be placed in partition p or greater and therefore Case (1) Figure 4: Equations for variable w for 2 tasks and 4 partitions they cannot contribute to the scratch memory of any partitions less than or equal to p. ffl If a task t 2 is placed on some partition p, and exists then it means that t 1 definitely can only be placed in partition p or lesser and therefore they cannot contribute to the scratch memory of any partitions greater than p. exists, and both tasks are in the same partition, then they cannot contribute to the scratch memory of any of the partitions. In our formulation we have not introduced explicitly a new term for each distinct product term as we did for We have linearized as follows - Fewer new variables are introduced in the linearization because a single new variable reflects the constraints of several non-linear product terms. The main limitation in the above linearization is that though w pt1 t 2 will take a value 1 if one or more of its constituent products is 1, it will also l Graph No. N A+M+S L Var Const Run-Time Table 2: Results of tightening the Constraints take a value 1 even if none of the constituent products is 1. This is because there is no constraint which will limit the value of w pt1 t 2 to 0, if all its constituent product terms are 0. If the new linear term is in a minimizing cost function, then such a solution will get cutoff after being generated as the solution, since the objective value of the cost function is greater in this case. This is true in our case as w pt1 t 2 is part of the cost function. However as we shall see the new tightened inequalities will actually limit the solution space. w pt1 t 2 will never be 1 if all of its constituent non-linear terms are Equations (28), (29) and (30) eliminate the problem mentioned above. We will show by an example how this is possible. In Figure 4 two tasks are to be place in four parti- tions. The variable w 312 , will have the equations shown. Consider the following three cases, where none of the 4 product terms shown in the figure are 1, how the variable w 312 will still never be 1. cut off by equation (29), (2) if t value w will be cut off by equation (28), will be cut off by equation (30). ffl Another constraint observed, which reduced dramatically the solution time is that, if a task t uses a functional unit k and is placed in partition p, then the associated u pk variable should also reflect this. Equations (1), (2), (3), (6), (7), (8), (11), (12), (13), (19), (20), (21), (22), (23), (26), (27), (28), (29), (30), (31) and (32) are the constraints and Equation (14) is the cost function in our final model. 7 Experimental Results for Tightened Constraints With the tightened constraints in place, we again ran the same series of experiments as in Table 1 and observed a significant improvement in the run times. Examples in rows 1, 2 and 3 of Table 2 solved, though the run times are very large for the sizes of graphs under consideration. 8 Solution by branch and bound While solving a 0-1 LP by the branch and bound technique, an active node of the branching tree (ini- tially the given mixed 0-1 problem), is chosen and its LP relaxation is solved, and a fractional variable, if any is chosen to branch on. If the variable is a 0-1 variable, one branch sets the variable to 0 and the other to 1. In this framework, there are two choices to be made at any time: the active node to be developed and the choice of the fractional variable to branch on. The variable choice can be very critical in keeping the size of the b-and-b tree small. We formulated the following heuristic to guide the selection of the branch and bound and has worked very well in practice. For the task graph, do a topological ordering of the tasks. For dependency have a higher priority than . The priorities range from 1.n, highest to lowest, and when the variable for tasks are generated in the ILP i.e., y tp , the index t reflects this priority. While solving the model by an LP-solver, we always take the branch which sets the variable value to 1 first. To pick a variable to branch and bound we use the following ffl If there are variables y tp still fractional, then pick the variable y tp with the lowest t value and p value to branch on first. ffl Once no fractional y tp variables remain, pick any fractional u pk variable to branch on. As a naive approach at this point once all the tasks are assigned to partitions we might have chosen to branch on any fractional x ijk variable to continue the synthesis process, but our approach cuts off all solutions which are using some functional unit which does not fit in the partition, very early in the branching process. As you will observe in Section 9, this variable selection process has greatly reduced the run time of the experiments. This result emphasizes that careful study into the variable selection method must be done, rather than leave the variable selection to the solver (which randomly chooses a variable to branch on). We choose to pick y tp as a variable to branch on because once the task get assigned to partitions, the remaining problem is now just a scheduling-allocation problem whose linearization is quite tight and so produces less number of non-integer solutions. Observe that we are not forcing the tasks to lie on a particular partition, as in [2], where the variables in the critical path are forced into one partition, thus making the solution only locally optimal. Our method just guides the solution process to a quick solution (which may or may not be optimal), and this solution then acts as a lower bound on all other solutions undertaken after it in the solution process. This helps in eliminating many non-optimal branches of the solution tree. Since we are never forcing the value of any variable to some value, our solutions are always global optimal. 9 Experimental Results We first explore the effect of different design parameters on an example behavioral specification. Table 3 shows the results of fixing the number of functional units and varying the latency and number of partitions. This graph called graph 1 has 5 tasks and 22 opera- tions. 2 adders, 2 multipliers and 1 subtracter were used to schedule the design. The first row in Table 3 Characteristics ILP Model Results Graph No. Tasks Opers N A+M+S L Var Const RunTime Feasible Table 4: Temporal Partitioning Results for various Graphs Const RunTime Feasible Table 3: Results of variation of Latency and number of Partitions for Graph 1 shows that with no relaxation in latency, the design could not be feasibly partitioned onto 3 partitions. So the latency bound was relaxed by 1, and the design was optimally partitioned and synthesized onto 3 par- titions. The latency bound was relaxed by 2 and the design fit onto 2 partitions. It was further relaxed to 3, and it fit optimally onto a single partition though partitions were used in the design space exploration. In all these examples, the execution time for getting the optimal results is very small. For any ILP formulation, being solved by branch and bound using LP relaxation of the model, it is very important to have tight linear relaxations and good variable selection methods, so that a large amount of non-integer solutions are cutoff from the solution space. We ran a series of experiments to substantiate the variable selection technique and the tightened constraints we have imposed on our model. In Table 4, the results of running the experiments on larger graphs are shown. The columns Tasks and Opers give the size of the specification in terms of the number of tasks and operations in them. Medium sized graphs of upto 72 operations, can be optimally partitioned in very small execution times. To make our model an effective tool for temporal partitioning and synthesis, we need to add constraints to model the registers and buses used in the design. Note however that the number of variables (which largely influence the solution time) will not increase, as the current variable set is enough to model the additional constraints. In this paper we have presented a novel technique to perform temporal partitioning and synthesis optimally. Good linearization techniques and careful variable selection procedures were very helpful in solving the models in a short time. The effectiveness was demonstrated by the results. --R "Optimal Synthesis of Multichip Architectures" "An Optimal methodology of Synthesis of DSP Multichip Architectures" "An Algorithm for Hardware/Software Partitioning Using Mixed Integer Linear Programming" "MULTIPAR: Behavioral Partitioning for Synthesizing Application-Specific Multiprocessor Architectures" "Resource Constrained RTL Partitioning for Synthesis of Multi-FPGA Designs" "Optimal VLSI architectural synthesis Area, Performance and Testability" "OS- CAR:Optimum Simultaneous Scheduling, Allocation and Resource Binding Based on Integer Pro- gramming" "Con- strained Nonlinear 0-1 programming" "Converting the 0-1 Polynomial Programming Problem to a 0-1 Linear Program" "CHOP: A Constraint-Driven-System-Level Partitioner" "Partitioning of Functional Models of Synchronous Digital Systems" --TR Optimal VLSI architectural synthesis OSCAR An optimal methodology for synthesis of DSP multichip architectures Optimal synthesis of multichip architectures Hardware/Software Partitioning using Integer Programming Resource Constrained RTL Partitioning for Synthesis of Multi-FPGA Designs --CTR Michael Eisenring , Marco Platzner, A Framework for Run-time Reconfigurable Systems, The Journal of Supercomputing, v.21 n.2, p.145-159, February 2002 R. Maestre , M. Fernandez , R. Hermida , N. Bagherzadeh, A Framework for Scheduling and Context Allocation in Reconfigurable Computing, Proceedings of the 12th international symposium on System synthesis, p.134, November 01-04, 1999 R. Maestre , F. J. Kurdahi , N. Bagherzadeh , H. Singh , R. Hermida , M. Fernandez, Kernel scheduling in reconfigurable computing, Proceedings of the conference on Design, automation and test in Europe, p.21-es, January 1999, Munich, Germany Meenakshi Kaul , Ranga Vemuri , Sriram Govindarajan , Iyad Ouaiss, An automated temporal partitioning and loop fission approach for FPGA based reconfigurable synthesis of DSP applications, Proceedings of the 36th ACM/IEEE conference on Design automation, p.616-622, June 21-25, 1999, New Orleans, Louisiana, United States Meenakshi Kaul , Ranga Vemuri, Temporal partitioning combined with design space exploration for latency minimization of run-time reconfigured designs, Proceedings of the conference on Design, automation and test in Europe, p.43-es, January 1999, Munich, Germany Meenakshi Kaul , Ranga Vemuri, Design-Space Exploration for Block-Processing Based TemporalPartitioning of Run-Time Reconfigurable Systems, Journal of VLSI Signal Processing Systems, v.24 n.2-3, p.181-209, Mar. 2000 Hartej Singh , Guangming Lu , Eliseu Filho , Rafael Maestre , Ming-Hau Lee , Fadi Kurdahi , Nader Bagherzadeh, MorphoSys: case study of a reconfigurable computing system targeting multimedia applications, Proceedings of the 37th conference on Design automation, p.573-578, June 05-09, 2000, Los Angeles, California, United States Rafael Maestre , Fadi J. Kurdahi , Milagros Fernndez , Roman Hermida , Nader Bagherzadeh , Hartej Singh, A framework for reconfigurable computing: task scheduling and context management, IEEE Transactions on Very Large Scale Integration (VLSI) Systems, v.9 n.6, p.858-873, 12/1/2001

temporal partitioning;non-linear programming;integer linear programming;synthesis

368299

Fast sequential circuit test generation using high-level and gate-level techniques.

A new approach for sequential circuit test generation is proposed that combines software testing based techniques at the high level with test enhancement techniques at the gate level. Several sequences are derived to ensure 100% coverage of all statements in a high-level VHDL description, or to maximize coverage of paths. The sequences are then enhanced at the gate level to maximize coverage of single stuck-at faults. High fault coverages have been achieved very quickly on several benchmark circuits using this approach.

Introduction Most recent work in the area of sequential circuit test generation has focused on the gate level and has been targeted at single stuck-at faults. Both deterministic fault-oriented and simulation-based approaches have been used e#ectively, although execution times are often long. The key factor limiting the e#ciency of these approaches has been the lack of knowledge about circuit behavior. Architectural-level test generation has been proposed as a means of exploiting high-level information while maintaining the capability to handle stuck-at faults [1]. However, the high-level information must be derived from a structural description at the register transfer level (RTL), and sequences generated are targeted at detecting specific stuck-at faults in modules for which gate-level descriptions are available. Circuits with modules for which gate-level descriptions are not available can be handled, but better fault coverages are obtained by a gate-level test generator in less time [2]. Several approaches have been proposed for automatic generation of functional test vectors for circuits described at a high level, including [3][4][5]. The functional test vectors can be used for design verification and power estimation, in addition to screening for manufacturing defects. Vemuri and Kalyanaraman enumerate paths in an annotated VHDL description and trans- # This research was supported in part by DARPA under Contract DABT63-95-C-0069, in part by the European Union through the FOST Project, and by Hewlett-Packard under an equipment grant. late them into a set of constraints [3]. A constraint solver is then used to obtain a test sequence to traverse the specified path. Fault coverages of test sets generated for statement coverage were low, but higher fault coverages were obtained by covering each statement multiple times. Cheng and Krishnakumar transform the high-level description in VHDL or C into an extended finite state machine (EFSM) model and then use the EFSM model to generate test sequences that exercise all specified functions [4]. Traversing all transitions in an EFSM model was shown to guarantee coverage of all functions. Execution time was very low for generation of test sequences, and good fault coverages were achieved for several circuits. The approach proposed by Corno et al., implemented in the test generator RAGE, aims to generate test sequences that cover each read or write operation on a variable in a high-level VHDL description [5]. The operations are each covered a specified number of times. Good fault coverages were achieved for several benchmark circuits by covering each operation at least three times, and fault coverages were sometimes higher than those obtained by a deterministic, gate-level test generator. Execution was fast for all but the larger circuits. These functional test generation approaches are based upon a technique commonly used for software testing: generating tests that cover all statements in the system description. Another software testing tech- nique, which has not been implemented in the previous work on functional test generation, is to generate tests that traverse all possible paths in the system description [6]. In this work, we address path coverage, as well as statement coverage. A VHDL circuit description may contain multiple processes that execute concurrently. Since a path is defined only within a single process, we apply path coverage to single-process designs or to the main process of multiple-process designs only. Here, limitations must be placed upon path length to bound the number of tests generated. Generation of tests for path coverage, in addition to statement coverage, may enable higher fault coverages to be achieved. Whether statement coverage or path coverage is used as the coverage metric, generation of test sequences using software testing based techniques is limited by an inability to specify values of variables that will maximize detection of faults at the gate level. We propose to combine a software testing based approach at the high level with test sequence enhancement techniques at the gate level to achieve high fault coverages in sequential circuits very quickly. The gate-level test sequence enhancement techniques that we use borrow from techniques already developed for dynamic compaction of tests generated at the gate level [7][8]. The objective is to maximize the number of faults that can be detected by each test sequence generated at the high level. We begin with an overview of the test generation process in Section 2. Generation of test sequences at the high level using software testing based techniques is then described in Section 3, followed by a discussion about test sequence enhancement at the gate level in Section 4. Results are presented in Section 5 for several benchmark circuits, and Section 6 concludes the paper. Overview We propose to combine software testing based techniques for test sequence generation at a high level with gate-level techniques for test sequence enhancement. The overall test generation process is illustrated in Figure 1. Several partially-specified test sequences are de- High-Level Description Test Automatic Synthesis Gate-Level Description Sequences Test Test Sequence Gate-Level High-Level Generation Test Sequence Enhancement Figure 1: Overview of test generation. rived from the high-level circuit description using various coverage goals, e.g., coverage of all statements. An automatic synthesis tool is used to obtain a gate-level implementation of the circuit, and then the gate-level test sequence enhancement tool is executed to generate a complete test set targeted at high coverage of single stuck-at faults, using test sequences generated at the high level as input. The high-level sequences generated are aimed at traversing through a number of control states in the system, and values of variables are left unspecified as much as possible. The gate-level tool then has more freedom to select values that will maximize fault coverage. The same sequence may be reused a number of times, but modifications made at the gate level, which essentially specify the values of variables in the datapath, are likely to result in di#erent fully-specified test sequences. Furthermore, any sequences or subsequences that do not contribute to improving the fault coverage are not added to the test set. 3 High-Level Test Generation The first step in our test generation procedure is to obtain a set of partially-specified test sequences using the high-level circuit description and various coverage goals. Ideally, we would like to automate this process, but automatic generation of tests for both statement and path coverage is itself a very di#cult problem, and no implementation is currently available. Therefore, in the current work, the sequences are derived manually. One of our goals in this work is to provide guidelines on the types of high-level sequences that are most useful for stuck-at fault testing. It may be possible to avoid using sequences that are di#cult to derive automatically and still achieve high fault coverages. In particular, our experiments indicate that statement coverage usually su#ces and is easier to achieve than path coverage. Various high-level benchmark circuits are used in our work, and most of these have been derived from VHDL descriptions found at various ftp sites. Circuits b01-b08 range from simple filters to more complex microprocessor fetch and execution units and are available from the authors. The simplest coverage metric is statement coverage. A test set with 100% statement coverage exercises all statements in the VHDL description. Every branch must be exercised at least once in the set of sequences derived, but all paths are not necessarily taken. Path coverage is a more comprehensive metric that does aim to ensure that all paths are taken. To obtain a set of sequences with 100% statement coverage, the datapath and control portions of the description are identified, and the state transition graph (STG) for the control machine is derived. Then test sequences are assembled to traverse all control states and all blocks of code for each state. Each sequence begins by resetting the cir- cuit. In the benchmark circuits that we are using, a reset signal is available. However, the only necessary assumption is that the circuit is initializable. This assumption is satisfied at the gate level by either using a reset signal or an initialization sequence. Several vectors are then added to traverse between states and exercise various statements. Finally, vectors are added to the end of the sequence to ensure that the circuit ends in a state in which the output is observed. For many cir- cuits, the outputs are observable in any state, so these vectors are unnecessary. Portions of the test sequences that determine the values of variables are left unspecified as much as possible so that the gate-level tool has more freedom in choosing values to maximize stuck-at fault coverage. Derivation of test sequences for 100% statement coverage is best illustrated by an example. The STG for the control machine of benchmark circuit b03 is shown in Figure 2. When the reset signal is asserted, the cir- Reset read request[1-4] update grant read request[1-4] ASSIGN INIT Figure 2: STG for benchmark circuit b03. cuit is placed in state INIT. In state INIT, the (bit) variables request1 through request4 are read from the primary inputs, and the next state is set to ANALISI REQ. In state ANALISI REQ, the 4-bit grant variable is written to the primary outputs, one of four blocks of code is executed, depending on the request variables read in the previous state, and the next state is set to ASSIGN. In state ASSIGN, the grant variable is updated, the variables request1 through request4 are read from the primary inputs, and the next state is set to ANALISI REQ. The set of test sequences derived for 100% statement coverage therefore contains four partially-specified sequences, each having five vec- tors. The first vector resets the circuit. The second vector sets the request variables to exercise one of the four code blocks in the following state. The last three vectors are used to traverse from the ANALISI REQ state to the ASSIGN state, where the grant variable is set, and back to the ANALISI REQ state, where the grant variable is written to the primary outputs. In obtaining a set of sequences for path coverage, we start with a sequence with 100% statement coverage. Several sequences are then added to maximize coverage of paths. Paths are considered within each state of the STG and also across several states. The procedure for deriving sequences to cover paths within a state is first explained for benchmark circuit b04. The STG for b04 is shown in Figure 3 along with a flow chart for state read D_IN sA Reset RLAST D_IN D_IN State sC F F F F F Figure 3: STG for benchmark circuit b04 and flow chart for state sC. sC. All assignments in the flow chart are carried out in the same clock cycle. When the circuit is reset, state sA is entered. States sB and sC are reached in the next two clock cycles, regardless of the inputs, as long as the reset line is not asserted. No particular patterns are needed to reach all statements and to cover all paths in states sA and sB. However, many paths are possible in state sC. The circuit must be in state sC for a minimum of four clock cycles to exercise all statements at least once. Either four separate sequences or one long sequence can be used. We have opted to use a larger number of shorter sequences in order to provide more sequences for optimization by the gate-level tool. Fifteen sequences are needed to cover all paths. (Note that the ENA variable is used at two separate decision points.) Only five sequences are needed if the last two decision points are not considered. We consider the 8- bit variable D IN used in the last two decision points to be part of the datapath, and therefore, in our ex- periments, we have left specification of values for this variable to the gate-level tool. Paths that occur across multiple states must also be considered. Consider the STG for the control unit of benchmark circuit b06 shown in Figure 4. This STG contains several cycles. In order to limit the number of sequences derived, we place restrictions on sequences that traverse a cycle. Self-loops are traversed at most once in any sequence, and for other cycles, the s_init s_intr s_intr_w s_enin_w s_wait s_enin Reset Figure 4: STG for benchmark circuit b06. sequences are terminated when a state is repeated. Four sequences are required to fully cover paths involving states s wait, s enin, and s enin w. Five sequences are needed to cover paths involving states s wait, s intr 1, s intr, and s intr w. Nine sequences are thus required for path coverage. In general, both the STG and the statement flow for each reached state must be considered when deriving sequences for path coverage. 4 Gate-Level Test Enhancement Functional tests generated at a high level are e#ective in traversing through much of the control space of a machine. However, they cannot exercise all values of variables, except in very small circuits, due to the large number of possible values. Selecting good values to use at the high level is an unsolved problem, and a gate-level approach may be more e#ective in finding values that exercise potential faults. 4.1 Architecture of Gate-Level Tool Our gate-level test enhancement tool repeatedly selects a partially-specified sequence provided by the high-level test generator and attempts to evolve a fully-specified sequence that maximizes fault coverage. The number of times that test sequence evolution is attempted is a parameter specified by the user. Sequences may be selected randomly or sequentially from the list of sequences provided by the high-level test generator. If sequences are selected randomly, a random number generator is used to decide which sequence to select. Random selection does not guarantee that every sequence will be used, but it does not restrict the order in which the sequences are selected. If sequences are selected sequentially, the first sequence is selected first, the second sequence is selected second, and so on. Every sequence will be selected at least once if the number of attempts at test sequence evolution is greater than or equal to the number of sequences. The main function of the gate-level test enhancement tool is to repeatedly solve an optimization problem: maximizing the number of faults detected by each se- quence. Genetic algorithms (GAs) have been used e#ec- tively for many di#erent optimization problems, including sequential circuit test generation [9]-[11],[2]. Thus, we use a GA for test sequence enhancement. We simply seed the GA with a sequence obtained at the high-level and then set the GA fitness function to maximize fault detection. The GA will explore several alternative sequences through a number of generations, and the best one is added to the test set if it improves the fault cov- erage. Any vectors at the end of the sequence that do not contribute to the fault coverage are removed. Then the next high-level sequence is selected, and the genetic enhancement procedure is repeated. This process continues until the number of attempts at test sequence enhancement reaches the user-specified limit. 4.2 A GA for Test Sequence Enhancement In this work, we use a simple GA, rather than a steady-state GA [12], since exploration of the search space is paramount. The simple GA contains a population of strings, or individuals [13]. In our application, each individual represents a test sequence, with successive vectors in the sequence placed in adjacent positions along the string. Each individual has an associated fit- ness, and in our application, the fitness measure indicates the number of faults detected by each sequence. The population is initialized with a set of sequences derived from a single sequence generated at the high level, and the evolutionary processes of selection, crossover, and mutation are used to generate an entirely new population from the existing population. This process is repeated for several generations. To generate a new population from the existing one, two individuals are selected, with selection biased toward more highly fit individuals. The two individuals are crossed to create two entirely new individuals, and each character in a new string is mutated with some small mutation prob- ability. The two new individuals are then placed in the new population, and this process continues until the new generation is entirely filled. Binary tournament selection without replacement and uniform crossover are used, as was done previously for gate-level test generation [10]. The goal of the evolutionary process is to improve the fitness of the best individual in each successive generation by combining the good portions of fit individuals from the preceding generation. However the best individual may appear in any generation, so we save the best individual found. The GA is seeded with copies of the partially- specified test sequence provided by the high-level test generator. The specified bits are the same for every individual. Bits that are not specified are filled ran- domly. Each fully-specified test sequence is then fault simulated to obtain its fitness value; the fitness value measures the quality of the corresponding solution, primarily in terms of fault coverage. The GA is evolved over several generations, and by the time the last generation is reached, several of the values specified by the high-level test generator may have changed in many of the individuals due to the mutation operator; i.e., the sequences may no longer be covered by the original partially-specified sequence. However, such a sequence will only be added to the test set if it covers some additional faults not already covered by previous vectors in the test set and if it has the highest fault coverage. 4.3 Fitness Function The PROOFS sequential circuit fault simulator [14] is used to evaluate the fitness of each candidate test sequence and again to update the state of the circuit after the best test sequence is selected. The number of faults detected is the primary metric in the fitness function, since the objective of the GA is to maximize the number of faults detected by a given test sequence. To di#er- entiate test sequences that detect the same number of faults, we include the number of fault e#ects propagated to flip-flops in the fitness function, since fault e#ects at the flip-flops may be propagated to the primary outputs in subsequent time frames. However, the number of fault e#ects propagated is o#set by the number of faults simulated and the number of flip-flops to ensure that the number of faults detected is the dominant factor in the fitness function: detected effects propagated to f lip f lops While an accurate fitness function is essential in achieving a good solution, the high computational cost of fault simulation may be prohibitive, especially for large circuits. To avoid excessive computations, we can approximate the fitness of a candidate test by using a small random sample of faults. In this work, we use a sample size of about 100 faults if the number of faults remaining in the fault list is greater than 100. Experiments were carried out to evaluate the proposed approach for combining high-level and gate-level techniques for sequential circuit test generation. Test sequences were derived manually at the high level by extracting the STG of the control machine and then ensuring that all VHDL statements or paths were covered within each control state. For di#eq, a short C program was written to assist in obtaining high-level sequences. This circuit contains a single loop, and the loop must be exited to observe the output. The C program was used to determine the number of loop iterations executed for a given input. Gate-level implementations of the circuits were synthesized using a commercial synthesis tool. Test sequence enhancement was then performed at the gate level using a new GA-based tool implemented using the existing PROOFS [14] source code and 2100 additional lines of C++ code. A small GA population size of 32 was used, and the number of generations was limited to 8 to minimize execution time. Nonoverlapping generations and crossover and mutation probabilities of 1 and 1/64 were used. Tests were generated for several high-level benchmark circuits on an HP 9000 J200 with 256 MB mem- ory. Characteristics of the benchmark circuits are summarized in Table 1, including number of VHDL lines in the high-level description, number of control states, number of logic gates in the gate-level circuit, number of flip-flops (FFs), number of primary inputs (PIs), number of primary outputs (POs), and number of collapsed faults. Circuits b01-b08 have been used previously for research on functional test generation [5]. Circuits barcode, gcd, dhrc, and di#eq were taken from the HLSynth92 and HLSynth95 high-level synthesis bench- marks. All circuits were translated into a synthesizable subset of VHDL before they were used. Test generation results are shown in Table 2 for sequences derived at the high level to maximize path coverage. Results for HITEC [15], a gate-level deterministic test generator, and GATEST [10], a gate-level GA-based test generator, are also shown for compari- son. Three passes through the fault list were made by HITEC for all circuits unless all faults were identified as detected or untestable earlier. Time limits for the three passes were 0.5, 5, and 50 seconds per fault. For each circuit, the number of faults detected (Det), the number of test vectors generated (Vec), and the execution time are shown for each test generator. The execution time for the proposed approach includes the time for gate-level test enhancement only, but the time for generating sequences from high-level circuit descriptions is expected to be of the same order of magnitude, based on previous work [5]. The number of attempts at generating a useful test sequence (Seq) and the sequence selection strategy (Strat), whether sequential or ran- dom, are also shown in the table, as well as the number of faults identified as untestable by HITEC. Results are shown for the sequence selection strategy and number of attempts that gave the highest fault coverage, while using a minimal number of test vectors. If more Table 1: High-Level Benchmark Circuits High Level Gate Level Circuit VHDL Lines Control States Gates Flip-Flops PIs POs Faults di#eq Table 2: Combining High-Level Test Generation with Gate-Level Test Enhancement High-Level Gate-Level HITEC GATEST Circuit Det Vec Time Seq Strat Det Vec Time Unt Det Vec Time b04 1204 113 1.17m 20 rand 1177 303 1.42h 136 1217 220 4.60m 28 261 84 46.3s barcode 580 77 1.68m 20 rand 689 1816 28.7h 12 552 161 4.52m gcd 1988 356 17.8m 90 rand 1638 206 13.7h 3 1377 227 10.6m di#eq 17,881 335 1.80h 100 rand 17,730 803 23.6h 46 18,009 662 7.71h attempts are made at test sequence enhancement, the execution time will increase, but higher fault coverages were not achieved in our experiments. For most circuits, the fault coverages for the proposed approach are competitive with the fault coverages achieved by HITEC. For barcode, the fault coverage is about the same as that achieved by HITEC after two passes through the fault list and 51.1 minutes of exe- cution, although more faults are detected by HITEC in the third pass. For b08, HITEC achieves higher fault coverage in the first pass. In some cases, such as b07, and gcd, higher fault coverages are obtained by combining the high-level and gate-level techniques. Furthermore, for a given level of fault coverage, the test sets generated using the proposed approach are much more compact. Execution times for gate-level test enhancement are often orders of magnitude smaller than those for HITEC. Nevertheless, untestable faults cannot be identified using the proposed approach. Thus, the designer may choose to run a gate-level test generator such as HITEC in a postprocessing step. Fault coverages for the proposed approach are significantly higher than those for GATEST for several circuits. For some circuits, GATEST achieves the same fault coverage as the proposed approach, but test set lengths and execution times are significantly higher. For di#eq, the GAT- EST fault coverage was higher, but execution time was also significantly higher. The gate-level test enhancement is very similar to the procedure used in GATEST, except that GATEST uses random sequences in the initial GA population. The seeds used by the gate-level test enhancement tool are critical in providing information to the GA about sequences that can activate faults and propagate fault e#ects. The two sequence selection strategies are compared in Table 3 for sequences derived at the high level to maximize path coverage or for 100% statement cover- age. Statement and path coverage are the same for di#eq, since this circuit contains only a single path. For path coverage, the sequential selection strategy gives better results in terms of fault coverage and test set size for some circuits, but in a few cases, the fault coverages are significantly higher for random selection. Random selection is therefore preferred in general. For statement Table 3: Sequential vs. Random Selection of Sequences Derived for Path Coverage or Statement Coverage Path Coverage Statement Coverage Sequential Random Sequential Random Circuit Seq Det Vec Time Det Vec Time Det Vec Time Det Vec Time gcd 90 1914 302 18.6m 1988 356 17.8m 1662 304 16.8m 1769 283 13.6m dhrc di#eq 100 17,881 335 1.79h 17,881 335 1.80h 17,881 335 1.79h 17,881 335 1.80h coverage, the random selection strategy tends to give fault coverages that are as good as or better than those for sequential selection. Fault coverages are sometimes higher than those for sequences derived for path cov- erage. However, fault coverages may be significantly lower, as is the case for circuit b05. These results are not unexpected, since certain paths may need to be traversed in order to excite some faults and propagate their e#ects to the primary outputs. Nevertheless, since good results are often obtained for sequences derived for 100% statement coverage alone, and these sequences are easier to derive, this approach may be preferred. 6 Conclusions High fault coverages have been obtained very quickly by combining high-level and gate-level techniques for test generation. Sequences derived to maximize coverage of statements or paths in the high-level VHDL description are enhanced at the gate level to maximize coverage of single stuck-at faults. This approach may be used as a preprocessing step to gate-level test generation to speed up the process, and it sometimes results in improved fault coverages as well. Higher fault coverages were obtained for sequences derived for path coverage, but good results were also obtained for 100% statement coverage. 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automatic test generation;test sequence compaction;sequential circuits;software testing